Theorem: Polynomial.toContinuousMapOnAlgHom_apply
File: Mathlib/Topology/ContinuousFunction/Polynomial.lean

Proof steps:
state_before: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
⊢ (↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun 0 = 0
state_after: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
a✝ : ↑X
⊢ ((↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun 0) a✝ = 0 a✝
tactic: ext
state_before: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
a✝ : ↑X
⊢ ((↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun 0) a✝ = 0 a✝
state_after: no goals
tactic: simp
state_before: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
x✝¹ x✝ : R[X]
⊢ (↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun (x✝¹ + x✝) =
    (↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun x✝¹ +
      (↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun x✝
state_after: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
x✝¹ x✝ : R[X]
a✝ : ↑X
⊢ ((↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun (x✝¹ + x✝)) a✝ =
    ((↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun x✝¹ +
        (↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun x✝)
      a✝
tactic: ext
state_before: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
x✝¹ x✝ : R[X]
a✝ : ↑X
⊢ ((↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun (x✝¹ + x✝)) a✝ =
    ((↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun x✝¹ +
        (↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯ }).toFun x✝)
      a✝
state_after: no goals
tactic: simp
state_before: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
⊢ (fun p => p.toContinuousMapOn X) 1 = 1
state_after: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
a✝ : ↑X
⊢ ((fun p => p.toContinuousMapOn X) 1) a✝ = 1 a✝
tactic: ext
state_before: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
a✝ : ↑X
⊢ ((fun p => p.toContinuousMapOn X) 1) a✝ = 1 a✝
state_after: no goals
tactic: simp
state_before: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
x✝¹ x✝ : R[X]
⊢ { toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯ }.toFun (x✝¹ * x✝) =
    { toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯ }.toFun x✝¹ *
      { toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯ }.toFun x✝
state_after: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
x✝¹ x✝ : R[X]
a✝ : ↑X
⊢ ({ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯ }.toFun (x✝¹ * x✝)) a✝ =
    ({ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯ }.toFun x✝¹ *
        { toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯ }.toFun x✝)
      a✝
tactic: ext
state_before: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
x✝¹ x✝ : R[X]
a✝ : ↑X
⊢ ({ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯ }.toFun (x✝¹ * x✝)) a✝ =
    ({ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯ }.toFun x✝¹ *
        { toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯ }.toFun x✝)
      a✝
state_after: no goals
tactic: simp
state_before: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
x✝ : R
⊢ (↑↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }).toFun
      ((algebraMap R R[X]) x✝) =
    (algebraMap R C(↑X, R)) x✝
state_after: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
x✝ : R
a✝ : ↑X
⊢ ((↑↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }).toFun
        ((algebraMap R R[X]) x✝))
      a✝ =
    ((algebraMap R C(↑X, R)) x✝) a✝
tactic: ext
state_before: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
X : Set R
x✝ : R
a✝ : ↑X
⊢ ((↑↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }).toFun
        ((algebraMap R R[X]) x✝))
      a✝ =
    ((algebraMap R C(↑X, R)) x✝) a✝
state_after: no goals
tactic: simp [Algebra.algebraMap_eq_smul_one]