| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fvco3 7008 | . . . . . 6
⊢ ((𝑈: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
(((𝐴
·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op 𝑇)‘(𝑈‘𝑥))) | 
| 2 | 1 | 3ad2antl3 1188 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
∧ 𝑥 ∈ ℋ)
→ (((𝐴
·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op 𝑇)‘(𝑈‘𝑥))) | 
| 3 |  | fvco3 7008 | . . . . . . . 8
⊢ ((𝑈: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥))) | 
| 4 | 3 | 3ad2antl3 1188 | . . . . . . 7
⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
∧ 𝑥 ∈ ℋ)
→ ((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥))) | 
| 5 | 4 | oveq2d 7447 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
∧ 𝑥 ∈ ℋ)
→ (𝐴
·ℎ ((𝑇 ∘ 𝑈)‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) | 
| 6 |  | ffvelcdm 7101 | . . . . . . . . . 10
⊢ ((𝑈: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
(𝑈‘𝑥) ∈ ℋ) | 
| 7 |  | homval 31760 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
(𝑈‘𝑥) ∈ ℋ) → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) | 
| 8 | 6, 7 | syl3an3 1166 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
(𝑈: ℋ⟶ ℋ
∧ 𝑥 ∈ ℋ))
→ ((𝐴
·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) | 
| 9 | 8 | 3expa 1119 | . . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧
(𝑈: ℋ⟶ ℋ
∧ 𝑥 ∈ ℋ))
→ ((𝐴
·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) | 
| 10 | 9 | exp43 436 | . . . . . . 7
⊢ (𝐴 ∈ ℂ → (𝑇: ℋ⟶ ℋ →
(𝑈: ℋ⟶ ℋ
→ (𝑥 ∈ ℋ
→ ((𝐴
·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))))) | 
| 11 | 10 | 3imp1 1348 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
∧ 𝑥 ∈ ℋ)
→ ((𝐴
·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) | 
| 12 | 5, 11 | eqtr4d 2780 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
∧ 𝑥 ∈ ℋ)
→ (𝐴
·ℎ ((𝑇 ∘ 𝑈)‘𝑥)) = ((𝐴 ·op 𝑇)‘(𝑈‘𝑥))) | 
| 13 | 2, 12 | eqtr4d 2780 | . . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
∧ 𝑥 ∈ ℋ)
→ (((𝐴
·op 𝑇) ∘ 𝑈)‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))) | 
| 14 |  | fco 6760 | . . . . . . . 8
⊢ ((𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ 𝑈): ℋ⟶
ℋ) | 
| 15 |  | homval 31760 | . . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op
(𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))) | 
| 16 | 14, 15 | syl3an2 1165 | . . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
∧ 𝑥 ∈ ℋ)
→ ((𝐴
·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))) | 
| 17 | 16 | 3expia 1122 | . . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶
ℋ)) → (𝑥 ∈
ℋ → ((𝐴
·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)))) | 
| 18 | 17 | 3impb 1115 | . . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ (𝑥 ∈ ℋ
→ ((𝐴
·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)))) | 
| 19 | 18 | imp 406 | . . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
∧ 𝑥 ∈ ℋ)
→ ((𝐴
·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))) | 
| 20 | 13, 19 | eqtr4d 2780 | . . 3
⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
∧ 𝑥 ∈ ℋ)
→ (((𝐴
·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥)) | 
| 21 | 20 | ralrimiva 3146 | . 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ ∀𝑥 ∈
ℋ (((𝐴
·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥)) | 
| 22 |  | homulcl 31778 | . . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)
→ (𝐴
·op 𝑇): ℋ⟶ ℋ) | 
| 23 |  | fco 6760 | . . . 4
⊢ (((𝐴 ·op
𝑇): ℋ⟶ ℋ
∧ 𝑈: ℋ⟶
ℋ) → ((𝐴
·op 𝑇) ∘ 𝑈): ℋ⟶ ℋ) | 
| 24 | 22, 23 | stoic3 1776 | . . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ ((𝐴
·op 𝑇) ∘ 𝑈): ℋ⟶ ℋ) | 
| 25 |  | homulcl 31778 | . . . . 5
⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ) → (𝐴 ·op
(𝑇 ∘ 𝑈)): ℋ⟶
ℋ) | 
| 26 | 14, 25 | sylan2 593 | . . . 4
⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶
ℋ)) → (𝐴
·op (𝑇 ∘ 𝑈)): ℋ⟶
ℋ) | 
| 27 | 26 | 3impb 1115 | . . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ (𝐴
·op (𝑇 ∘ 𝑈)): ℋ⟶
ℋ) | 
| 28 |  | hoeq 31779 | . . 3
⊢ ((((𝐴 ·op
𝑇) ∘ 𝑈): ℋ⟶ ℋ ∧
(𝐴
·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ) →
(∀𝑥 ∈ ℋ
(((𝐴
·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) ↔ ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈)))) | 
| 29 | 24, 27, 28 | syl2anc 584 | . 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ (∀𝑥 ∈
ℋ (((𝐴
·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) ↔ ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈)))) | 
| 30 | 21, 29 | mpbid 232 | 1
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ ((𝐴
·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈))) |