Theorem: LocallyConstant.toContinuousMapAlgHom_apply
File: Mathlib/Topology/ContinuousFunction/LocallyConstant.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
⊢ ↑1 = 1
state_after: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
a✝ : X
⊢ ↑1 a✝ = 1 a✝
tactic: ext
state_before: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
a✝ : X
⊢ ↑1 a✝ = 1 a✝
state_after: no goals
tactic: simp
state_before: X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
x y : LocallyConstant X Y
⊢ { toFun := toContinuousMap, map_one' := ⋯ }.toFun (x * y) =
    { toFun := toContinuousMap, map_one' := ⋯ }.toFun x * { toFun := toContinuousMap, map_one' := ⋯ }.toFun y
state_after: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
x y : LocallyConstant X Y
a✝ : X
⊢ ({ toFun := toContinuousMap, map_one' := ⋯ }.toFun (x * y)) a✝ =
    ({ toFun := toContinuousMap, map_one' := ⋯ }.toFun x * { toFun := toContinuousMap, map_one' := ⋯ }.toFun y) a✝
tactic: ext
state_before: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
x y : LocallyConstant X Y
a✝ : X
⊢ ({ toFun := toContinuousMap, map_one' := ⋯ }.toFun (x * y)) a✝ =
    ({ toFun := toContinuousMap, map_one' := ⋯ }.toFun x * { toFun := toContinuousMap, map_one' := ⋯ }.toFun y) a✝
state_after: no goals
tactic: simp
state_before: X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
⊢ (↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun 0 = 0
state_after: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
a✝ : X
⊢ ((↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun 0) a✝ = 0 a✝
tactic: ext
state_before: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
a✝ : X
⊢ ((↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun 0) a✝ = 0 a✝
state_after: no goals
tactic: simp
state_before: X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
x y : LocallyConstant X Y
⊢ (↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun (x + y) =
    (↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun x +
      (↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun y
state_after: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
x y : LocallyConstant X Y
a✝ : X
⊢ ((↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun (x + y)) a✝ =
    ((↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun x +
        (↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun y)
      a✝
tactic: ext
state_before: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
x y : LocallyConstant X Y
a✝ : X
⊢ ((↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun (x + y)) a✝ =
    ((↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun x +
        (↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯ }).toFun y)
      a✝
state_after: no goals
tactic: simp
state_before: X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
r : R
⊢ (↑↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }).toFun
      ((algebraMap R (LocallyConstant X Y)) r) =
    (algebraMap R C(X, Y)) r
state_after: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
r : R
x : X
⊢ ((↑↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }).toFun
        ((algebraMap R (LocallyConstant X Y)) r))
      x =
    ((algebraMap R C(X, Y)) r) x
tactic: ext x
state_before: case h
X : Type u_1
Y : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
f : LocallyConstant X Y
R : Type u_3
inst✝³ : CommSemiring R
inst✝² : Semiring Y
inst✝¹ : Algebra R Y
inst✝ : TopologicalSemiring Y
r : R
x : X
⊢ ((↑↑{ toFun := toContinuousMap, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }).toFun
        ((algebraMap R (LocallyConstant X Y)) r))
      x =
    ((algebraMap R C(X, Y)) r) x
state_after: no goals
tactic: simp [Algebra.smul_def]