Algebraic structure on locally constant functions

This file puts algebraic structure (Group, AddGroup, etc) on the type of locally constant functions.

instance LocallyConstant.instOne {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [One Y] : One (LocallyConstant X Y)

instance LocallyConstant.instZero {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Zero Y] : Zero (LocallyConstant X Y)

@[simp]

theorem LocallyConstant.coe_one {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [One Y] : ⇑1 = 1

@[simp]

theorem LocallyConstant.coe_zero {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Zero Y] : ⇑0 = 0

theorem LocallyConstant.one_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [One Y] (x : X) : 1 x = 1

theorem LocallyConstant.zero_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Zero Y] (x : X) : 0 x = 0

instance LocallyConstant.instInv {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Inv Y] : Inv (LocallyConstant X Y)

instance LocallyConstant.instNeg {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Neg Y] : Neg (LocallyConstant X Y)

@[simp]

theorem LocallyConstant.coe_inv {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Inv Y] (f : LocallyConstant X Y) : ⇑f⁻¹ = (⇑f)⁻¹

@[simp]

theorem LocallyConstant.coe_neg {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Neg Y] (f : LocallyConstant X Y) : ⇑(-f) = -⇑f

theorem LocallyConstant.inv_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Inv Y] (f : LocallyConstant X Y) (x : X) : f⁻¹ x = (f x)⁻¹

theorem LocallyConstant.neg_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Neg Y] (f : LocallyConstant X Y) (x : X) : (-f) x = -f x

instance LocallyConstant.instMul {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Mul Y] : Mul (LocallyConstant X Y)

instance LocallyConstant.instAdd {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Add Y] : Add (LocallyConstant X Y)

@[simp]

theorem LocallyConstant.coe_mul {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Mul Y] (f g : LocallyConstant X Y) : ⇑(f * g) = ⇑f * ⇑g

@[simp]

theorem LocallyConstant.coe_add {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Add Y] (f g : LocallyConstant X Y) : ⇑(f + g) = ⇑f + ⇑g

theorem LocallyConstant.mul_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Mul Y] (f g : LocallyConstant X Y) (x : X) : (f * g) x = f x * g x

theorem LocallyConstant.add_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Add Y] (f g : LocallyConstant X Y) (x : X) : (f + g) x = f x + g x

instance LocallyConstant.instMulOneClass {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y] : MulOneClass (LocallyConstant X Y)

instance LocallyConstant.instAddZeroClass {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y] : AddZeroClass (LocallyConstant X Y)

def LocallyConstant.coeFnMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y] : LocallyConstant X Y →* X → Y

DFunLike.coe as a MonoidHom.

def LocallyConstant.coeFnAddMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y] : LocallyConstant X Y →+ X → Y

DFunLike.coe as an AddMonoidHom.

@[simp]

theorem LocallyConstant.coeFnAddMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y] (a✝ : LocallyConstant X Y) (a : X) : coeFnAddMonoidHom a✝ a = a✝ a

@[simp]

theorem LocallyConstant.coeFnMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y] (a✝ : LocallyConstant X Y) (a : X) : coeFnMonoidHom a✝ a = a✝ a

def LocallyConstant.constMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y] : Y →* LocallyConstant X Y

The constant-function embedding, as a multiplicative monoid hom.

def LocallyConstant.constAddMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y] : Y →+ LocallyConstant X Y

The constant-function embedding, as an additive monoid hom.

@[simp]

theorem LocallyConstant.constAddMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y] (y : Y) : constAddMonoidHom y = const X y

@[simp]

theorem LocallyConstant.constMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y] (y : Y) : constMonoidHom y = const X y

instance LocallyConstant.instMulZeroClass {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulZeroClass Y] : MulZeroClass (LocallyConstant X Y)

instance LocallyConstant.instMulZeroOneClass {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulZeroOneClass Y] : MulZeroOneClass (LocallyConstant X Y)

noncomputable def LocallyConstant.charFn {X : Type u_1} (Y : Type u_2) [TopologicalSpace X] [MulZeroOneClass Y] {U : Set X} (hU : IsClopen U) : LocallyConstant X Y

Characteristic functions are locally constant functions taking x : X to 1 if x ∈ U, where U is a clopen set, and 0 otherwise.

theorem LocallyConstant.coe_charFn {X : Type u_1} (Y : Type u_2) [TopologicalSpace X] [MulZeroOneClass Y] {U : Set X} (hU : IsClopen U) : ⇑(charFn Y hU) = U.indicator 1

theorem LocallyConstant.charFn_eq_one {X : Type u_1} (Y : Type u_2) [TopologicalSpace X] [MulZeroOneClass Y] {U : Set X} [Nontrivial Y] (x : X) (hU : IsClopen U) : (charFn Y hU) x = 1 ↔ x ∈ U

theorem LocallyConstant.charFn_eq_zero {X : Type u_1} (Y : Type u_2) [TopologicalSpace X] [MulZeroOneClass Y] {U : Set X} [Nontrivial Y] (x : X) (hU : IsClopen U) : (charFn Y hU) x = 0 ↔ x ∉ U

theorem LocallyConstant.charFn_inj {X : Type u_1} (Y : Type u_2) [TopologicalSpace X] [MulZeroOneClass Y] {U V : Set X} [Nontrivial Y] (hU : IsClopen U) (hV : IsClopen V) (h : charFn Y hU = charFn Y hV) : U = V

instance LocallyConstant.instDiv {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Div Y] : Div (LocallyConstant X Y)

instance LocallyConstant.instSub {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Sub Y] : Sub (LocallyConstant X Y)

theorem LocallyConstant.coe_div {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Div Y] (f g : LocallyConstant X Y) : ⇑(f / g) = ⇑f / ⇑g

theorem LocallyConstant.coe_sub {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Sub Y] (f g : LocallyConstant X Y) : ⇑(f - g) = ⇑f - ⇑g

theorem LocallyConstant.div_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Div Y] (f g : LocallyConstant X Y) (x : X) : (f / g) x = f x / g x

theorem LocallyConstant.sub_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Sub Y] (f g : LocallyConstant X Y) (x : X) : (f - g) x = f x - g x

instance LocallyConstant.instSemigroup {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Semigroup Y] : Semigroup (LocallyConstant X Y)

instance LocallyConstant.instAddSemigroup {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddSemigroup Y] : AddSemigroup (LocallyConstant X Y)

instance LocallyConstant.instSemigroupWithZero {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [SemigroupWithZero Y] : SemigroupWithZero (LocallyConstant X Y)

instance LocallyConstant.instCommSemigroup {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [CommSemigroup Y] : CommSemigroup (LocallyConstant X Y)

instance LocallyConstant.instAddCommSemigroup {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddCommSemigroup Y] : AddCommSemigroup (LocallyConstant X Y)

instance LocallyConstant.smul {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {α : Type u_3} [SMul α Y] : SMul α (LocallyConstant X Y)

instance LocallyConstant.vadd {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {α : Type u_3} [VAdd α Y] : VAdd α (LocallyConstant X Y)

@[simp]

theorem LocallyConstant.coe_smul {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_4} [SMul R Y] (r : R) (f : LocallyConstant X Y) : ⇑(r • f) = r • ⇑f

@[simp]

theorem LocallyConstant.coe_vadd {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_4} [VAdd R Y] (r : R) (f : LocallyConstant X Y) : ⇑(r +ᵥ f) = r +ᵥ ⇑f

theorem LocallyConstant.smul_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_4} [SMul R Y] (r : R) (f : LocallyConstant X Y) (x : X) : (r • f) x = r • f x

theorem LocallyConstant.vadd_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_4} [VAdd R Y] (r : R) (f : LocallyConstant X Y) (x : X) : (r +ᵥ f) x = r +ᵥ f x

instance LocallyConstant.instPow {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {α : Type u_3} [Pow Y α] : Pow (LocallyConstant X Y) α

instance LocallyConstant.instMonoid {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Monoid Y] : Monoid (LocallyConstant X Y)

instance LocallyConstant.instAddMonoid {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddMonoid Y] : AddMonoid (LocallyConstant X Y)

instance LocallyConstant.instNatCast {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NatCast Y] : NatCast (LocallyConstant X Y)

instance LocallyConstant.instIntCast {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [IntCast Y] : IntCast (LocallyConstant X Y)

instance LocallyConstant.instAddMonoidWithOne {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddMonoidWithOne Y] : AddMonoidWithOne (LocallyConstant X Y)

instance LocallyConstant.instCommMonoid {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [CommMonoid Y] : CommMonoid (LocallyConstant X Y)

instance LocallyConstant.instAddCommMonoid {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddCommMonoid Y] : AddCommMonoid (LocallyConstant X Y)

instance LocallyConstant.instGroup {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Group Y] : Group (LocallyConstant X Y)

instance LocallyConstant.instAddGroup {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddGroup Y] : AddGroup (LocallyConstant X Y)

instance LocallyConstant.instCommGroup {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [CommGroup Y] : CommGroup (LocallyConstant X Y)

instance LocallyConstant.instAddCommGroup {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddCommGroup Y] : AddCommGroup (LocallyConstant X Y)

instance LocallyConstant.instDistrib {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Distrib Y] : Distrib (LocallyConstant X Y)

instance LocallyConstant.instNonUnitalNonAssocSemiring {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonUnitalNonAssocSemiring Y] : NonUnitalNonAssocSemiring (LocallyConstant X Y)

instance LocallyConstant.instNonUnitalSemiring {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonUnitalSemiring Y] : NonUnitalSemiring (LocallyConstant X Y)

instance LocallyConstant.instNonAssocSemiring {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonAssocSemiring Y] : NonAssocSemiring (LocallyConstant X Y)

def LocallyConstant.constRingHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonAssocSemiring Y] : Y →+* LocallyConstant X Y

The constant-function embedding, as a ring hom.

@[simp]

theorem LocallyConstant.constRingHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonAssocSemiring Y] (y : Y) : constRingHom y = const X y

instance LocallyConstant.instSemiring {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Semiring Y] : Semiring (LocallyConstant X Y)

instance LocallyConstant.instNonUnitalCommSemiring {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonUnitalCommSemiring Y] : NonUnitalCommSemiring (LocallyConstant X Y)

instance LocallyConstant.instCommSemiring {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [CommSemiring Y] : CommSemiring (LocallyConstant X Y)

instance LocallyConstant.instNonUnitalNonAssocRing {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonUnitalNonAssocRing Y] : NonUnitalNonAssocRing (LocallyConstant X Y)

instance LocallyConstant.instNonUnitalRing {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonUnitalRing Y] : NonUnitalRing (LocallyConstant X Y)

instance LocallyConstant.instNonAssocRing {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonAssocRing Y] : NonAssocRing (LocallyConstant X Y)

instance LocallyConstant.instRing {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Ring Y] : Ring (LocallyConstant X Y)

instance LocallyConstant.instNonUnitalCommRing {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonUnitalCommRing Y] : NonUnitalCommRing (LocallyConstant X Y)

instance LocallyConstant.instCommRing {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [CommRing Y] : CommRing (LocallyConstant X Y)

instance LocallyConstant.instMulAction {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_5} [Monoid R] [MulAction R Y] : MulAction R (LocallyConstant X Y)

instance LocallyConstant.instDistribMulAction {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_5} [Monoid R] [AddMonoid Y] [DistribMulAction R Y] : DistribMulAction R (LocallyConstant X Y)

instance LocallyConstant.instModule {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_5} [Semiring R] [AddCommMonoid Y] [Module R Y] : Module R (LocallyConstant X Y)

instance LocallyConstant.instAlgebra {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_5} [CommSemiring R] [Semiring Y] [Algebra R Y] : Algebra R (LocallyConstant X Y)

@[simp]

theorem LocallyConstant.coe_algebraMap {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_5} [CommSemiring R] [Semiring Y] [Algebra R Y] (r : R) : ⇑((algebraMap R (LocallyConstant X Y)) r) = (algebraMap R (X → Y)) r

def LocallyConstant.coeFnRingHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Semiring Y] : LocallyConstant X Y →+* X → Y

DFunLike.coe as a RingHom.

@[simp]

theorem LocallyConstant.coeFnRingHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Semiring Y] (a✝ : LocallyConstant X Y) (a : X) : coeFnRingHom a✝ a = a✝ a

def LocallyConstant.coeFnₗ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [Semiring R] [AddCommMonoid Y] [Module R Y] : LocallyConstant X Y →ₗ[R] X → Y

DFunLike.coe as a linear map.

@[simp]

theorem LocallyConstant.coeFnₗ_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [Semiring R] [AddCommMonoid Y] [Module R Y] (a✝ : LocallyConstant X Y) (a✝¹ : X) : (coeFnₗ R) a✝ a✝¹ = a✝ a✝¹

def LocallyConstant.coeFnAlgHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [CommSemiring R] [Semiring Y] [Algebra R Y] : LocallyConstant X Y →ₐ[R] X → Y

DFunLike.coe as an AlgHom.

@[simp]

theorem LocallyConstant.coeFnAlgHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [CommSemiring R] [Semiring Y] [Algebra R Y] (a✝ : LocallyConstant X Y) (a : X) : (coeFnAlgHom R) a✝ a = a✝ a

def LocallyConstant.evalMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y] (x : X) : LocallyConstant X Y →* Y

Evaluation as a MonoidHom

def LocallyConstant.evalAddMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y] (x : X) : LocallyConstant X Y →+ Y

Evaluation as an AddMonoidHom

@[simp]

theorem LocallyConstant.evalAddMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y] (x : X) (a✝ : LocallyConstant X Y) : (evalAddMonoidHom x) a✝ = a✝ x

@[simp]

theorem LocallyConstant.evalMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y] (x : X) (a✝ : LocallyConstant X Y) : (evalMonoidHom x) a✝ = a✝ x

def LocallyConstant.evalₗ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [Semiring R] [AddCommMonoid Y] [Module R Y] (x : X) : LocallyConstant X Y →ₗ[R] Y

Evaluation as a linear map

@[simp]

theorem LocallyConstant.evalₗ_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [Semiring R] [AddCommMonoid Y] [Module R Y] (x : X) (a✝ : LocallyConstant X Y) : (evalₗ R x) a✝ = a✝ x

def LocallyConstant.evalRingHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Semiring Y] (x : X) : LocallyConstant X Y →+* Y

Evaluation as a RingHom

@[simp]

theorem LocallyConstant.evalRingHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Semiring Y] (x : X) (a✝ : LocallyConstant X Y) : (evalRingHom x) a✝ = a✝ x

def LocallyConstant.evalₐ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [CommSemiring R] [Semiring Y] [Algebra R Y] (x : X) : LocallyConstant X Y →ₐ[R] Y

Evaluation as an AlgHom

@[simp]

theorem LocallyConstant.evalₐ_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [CommSemiring R] [Semiring Y] [Algebra R Y] (x : X) (a✝ : LocallyConstant X Y) : (evalₐ R x) a✝ = a✝ x

def LocallyConstant.comapMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [MulOneClass Z] (f : C(X, Y)) : LocallyConstant Y Z →* LocallyConstant X Z

LocallyConstant.comap as a MonoidHom.

def LocallyConstant.comapAddMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [AddZeroClass Z] (f : C(X, Y)) : LocallyConstant Y Z →+ LocallyConstant X Z

LocallyConstant.comap as an AddMonoidHom.

@[simp]

theorem LocallyConstant.comapMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [MulOneClass Z] (f : C(X, Y)) (g : LocallyConstant Y Z) : (comapMonoidHom f) g = comap f g

@[simp]

theorem LocallyConstant.comapAddMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [AddZeroClass Z] (f : C(X, Y)) (g : LocallyConstant Y Z) : (comapAddMonoidHom f) g = comap f g

def LocallyConstant.comapₗ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y)) : LocallyConstant Y Z →ₗ[R] LocallyConstant X Z

LocallyConstant.comap as a linear map.

@[simp]

theorem LocallyConstant.comapₗ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y)) (g : LocallyConstant Y Z) (a✝ : X) : ((comapₗ R f) g) a✝ = g (f a✝)

def LocallyConstant.comapRingHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [Semiring Z] (f : C(X, Y)) : LocallyConstant Y Z →+* LocallyConstant X Z

LocallyConstant.comap as a RingHom.

@[simp]

theorem LocallyConstant.comapRingHom_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [Semiring Z] (f : C(X, Y)) (g : LocallyConstant Y Z) (a✝ : X) : ((comapRingHom f) g) a✝ = g (f a✝)

def LocallyConstant.comapₐ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Z] [Algebra R Z] (f : C(X, Y)) : LocallyConstant Y Z →ₐ[R] LocallyConstant X Z

LocallyConstant.comap as an AlgHom

@[simp]

theorem LocallyConstant.comapₐ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Z] [Algebra R Z] (f : C(X, Y)) (g : LocallyConstant Y Z) (a✝ : X) : ((comapₐ R f) g) a✝ = g (f a✝)

theorem LocallyConstant.ker_comapₗ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_5} [TopologicalSpace Y] {Z : Type u_6} [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y)) (hfs : Function.Surjective ⇑f) : (comapₗ R f).ker = ⊥

def LocallyConstant.congrLeftₗ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y) : LocallyConstant X Z ≃ₗ[R] LocallyConstant Y Z

LocallyConstant.congrLeft as a linear equivalence.

@[simp]

theorem LocallyConstant.congrLeftₗ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y) (g : LocallyConstant X Z) (a✝ : Y) : ((congrLeftₗ R e) g) a✝ = g (e.symm a✝)

@[simp]

theorem LocallyConstant.congrLeftₗ_symm_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y) (a✝ : LocallyConstant Y Z) (a✝¹ : X) : ((congrLeftₗ R e).symm a✝) a✝¹ = a✝ (e a✝¹)

def LocallyConstant.congrLeftRingEquiv {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [Semiring Z] (e : X ≃ₜ Y) : LocallyConstant X Z ≃+* LocallyConstant Y Z

LocallyConstant.congrLeft as a RingEquiv.

@[simp]

theorem LocallyConstant.congrLeftRingEquiv_symm_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [Semiring Z] (e : X ≃ₜ Y) (g : LocallyConstant Y Z) (a✝ : X) : ((congrLeftRingEquiv e).symm g) a✝ = g (e a✝)

@[simp]

theorem LocallyConstant.congrLeftRingEquiv_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [Semiring Z] (e : X ≃ₜ Y) (g : LocallyConstant X Z) (a✝ : Y) : ((congrLeftRingEquiv e) g) a✝ = g (e.symm a✝)

def LocallyConstant.congrLeftₐ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) : LocallyConstant X Z ≃ₐ[R] LocallyConstant Y Z

LocallyConstant.congrLeft as an AlgEquiv.

@[simp]

theorem LocallyConstant.congrLeftₐ_symm_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) (g : LocallyConstant Y Z) (a✝ : X) : ((congrLeftₐ R e).symm g) a✝ = g (e a✝)

@[simp]

theorem LocallyConstant.congrLeftₐ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) (g : LocallyConstant X Z) (a✝ : Y) : ((congrLeftₐ R e) g) a✝ = g (e.symm a✝)

def LocallyConstant.mapMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [MulOneClass Y] [MulOneClass Z] (f : Y →* Z) : LocallyConstant X Y →* LocallyConstant X Z

LocallyConstant.map as a MonoidHom.

def LocallyConstant.mapAddMonoidHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [AddZeroClass Y] [AddZeroClass Z] (f : Y →+ Z) : LocallyConstant X Y →+ LocallyConstant X Z

LocallyConstant.map as an AddMonoidHom.

@[simp]

theorem LocallyConstant.mapMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [MulOneClass Y] [MulOneClass Z] (f : Y →* Z) (g : LocallyConstant X Y) : (mapMonoidHom f) g = map (⇑f) g

@[simp]

theorem LocallyConstant.mapAddMonoidHom_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [AddZeroClass Y] [AddZeroClass Z] (f : Y →+ Z) (g : LocallyConstant X Y) : (mapAddMonoidHom f) g = map (⇑f) g

def LocallyConstant.mapₗ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (f : Y →ₗ[R] Z) : LocallyConstant X Y →ₗ[R] LocallyConstant X Z

LocallyConstant.map as a linear map.

@[simp]

theorem LocallyConstant.mapₗ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (f : Y →ₗ[R] Z) (g : LocallyConstant X Y) (a✝ : X) : ((mapₗ R f) g) a✝ = f (g a✝)

def LocallyConstant.mapRingHom {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [Semiring Y] [Semiring Z] (f : Y →+* Z) : LocallyConstant X Y →+* LocallyConstant X Z

LocallyConstant.map as a RingHom.

@[simp]

theorem LocallyConstant.mapRingHom_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [Semiring Y] [Semiring Z] (f : Y →+* Z) (g : LocallyConstant X Y) (a✝ : X) : ((mapRingHom f) g) a✝ = f (g a✝)

def LocallyConstant.mapₐ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (f : Y →ₐ[R] Z) : LocallyConstant X Y →ₐ[R] LocallyConstant X Z

LocallyConstant.map as an AlgHom

@[simp]

theorem LocallyConstant.mapₐ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (f : Y →ₐ[R] Z) (g : LocallyConstant X Y) (a✝ : X) : ((mapₐ R f) g) a✝ = f (g a✝)

def LocallyConstant.congrRightₗ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (e : Y ≃ₗ[R] Z) : LocallyConstant X Y ≃ₗ[R] LocallyConstant X Z

LocallyConstant.congrRight as a linear equivalence.

@[simp]

theorem LocallyConstant.congrRightₗ_symm_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (e : Y ≃ₗ[R] Z) (a✝ : LocallyConstant X Z) (a✝¹ : X) : ((congrRightₗ R e).symm a✝) a✝¹ = e.symm (a✝ a✝¹)

@[simp]

theorem LocallyConstant.congrRightₗ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (e : Y ≃ₗ[R] Z) (g : LocallyConstant X Y) (a✝ : X) : ((congrRightₗ R e) g) a✝ = e (g a✝)

def LocallyConstant.congrRightRingEquiv {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [Semiring Y] [Semiring Z] (e : Y ≃+* Z) : LocallyConstant X Y ≃+* LocallyConstant X Z

LocallyConstant.congrRight as a RingEquiv.

@[simp]

theorem LocallyConstant.congrRightRingEquiv_symm_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [Semiring Y] [Semiring Z] (e : Y ≃+* Z) (g : LocallyConstant X Z) (a✝ : X) : ((congrRightRingEquiv e).symm g) a✝ = (↑e).symm (g a✝)

@[simp]

theorem LocallyConstant.congrRightRingEquiv_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [Semiring Y] [Semiring Z] (e : Y ≃+* Z) (g : LocallyConstant X Y) (a✝ : X) : ((congrRightRingEquiv e) g) a✝ = e (g a✝)

def LocallyConstant.congrRightₐ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (e : Y ≃ₐ[R] Z) : LocallyConstant X Y ≃ₐ[R] LocallyConstant X Z

LocallyConstant.congrRight as an AlgEquiv.

@[simp]

theorem LocallyConstant.congrRightₐ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (e : Y ≃ₐ[R] Z) (g : LocallyConstant X Y) (a✝ : X) : ((congrRightₐ R e) g) a✝ = e (g a✝)

@[simp]

theorem LocallyConstant.congrRightₐ_symm_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (e : Y ≃ₐ[R] Z) (g : LocallyConstant X Z) (a✝ : X) : ((congrRightₐ R e).symm g) a✝ = e.symm (g a✝)

def LocallyConstant.constₗ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [Semiring R] [AddCommMonoid Y] [Module R Y] : Y →ₗ[R] LocallyConstant X Y

LocallyConstant.const as a linear map.

@[simp]

theorem LocallyConstant.constₗ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [Semiring R] [AddCommMonoid Y] [Module R Y] (y : Y) (a✝ : X) : ((constₗ R) y) a✝ = y

def LocallyConstant.constₐ {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [CommSemiring R] [Semiring Y] [Algebra R Y] : Y →ₐ[R] LocallyConstant X Y

LocallyConstant.const as an AlgHom

@[simp]

theorem LocallyConstant.constₐ_apply_apply {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [CommSemiring R] [Semiring Y] [Algebra R Y] (y : Y) (a✝ : X) : ((constₐ R) y) a✝ = y