Theorem: ContinuousMapZero.nonUnitalStarAlgHom_postcomp_apply
File: Mathlib/Topology/ContinuousFunction/ContinuousMapZero.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : TopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : TopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous ⇑φ
⊢ { toFun := ⇑φ, continuous_toFun := hφ } 0 = 0
state_after: no goals
tactic: simp
state_before: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : TopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : TopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous ⇑φ
⊢ ∀ (x : X), ({ toFun := { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp, map_smul' := ⋯ }.toFun 0) x = 0 x
state_after: no goals
tactic: simp
state_before: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : TopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : TopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous ⇑φ
x✝¹ x✝ : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp, map_smul' := ⋯ }.toFun (x✝¹ + x✝)) x =
      ({ toFun := { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp, map_smul' := ⋯ }.toFun x✝¹ +
          { toFun := { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp, map_smul' := ⋯ }.toFun x✝)
        x
state_after: no goals
tactic: simp
state_before: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : TopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : TopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous ⇑φ
x✝¹ x✝ : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp, map_smul' := ⋯, map_zero' := ⋯,
              map_add' := ⋯ }.toFun
          (x✝¹ * x✝))
        x =
      ({ toFun := { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp, map_smul' := ⋯, map_zero' := ⋯,
                map_add' := ⋯ }.toFun
            x✝¹ *
          { toFun := { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp, map_smul' := ⋯, map_zero' := ⋯,
                map_add' := ⋯ }.toFun
            x✝)
        x
state_after: no goals
tactic: simp
state_before: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : TopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : TopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous ⇑φ
x✝ : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp, map_smul' := ⋯, map_zero' := ⋯,
              map_add' := ⋯, map_mul' := ⋯ }.toFun
          (star x✝))
        x =
      (star
          ({ toFun := { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp, map_smul' := ⋯, map_zero' := ⋯,
                map_add' := ⋯, map_mul' := ⋯ }.toFun
            x✝))
        x
state_after: no goals
tactic: simp [map_star]
state_before: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : TopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : TopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous ⇑φ
r : M
f : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp (r • f)) x =
      ((MonoidHom.id M) r • { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp f) x
state_after: no goals
tactic: simp