Step | Hyp | Ref
| Expression |
1 | | simpl1 1057 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
𝐴 ∈
ℂ) |
2 | | simpl3 1059 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
𝑈: ℋ⟶
ℋ) |
3 | | simpr 476 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
𝑥 ∈
ℋ) |
4 | | homval 27984 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
((𝐴
·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥))) |
5 | 1, 2, 3, 4 | syl3anc 1318 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝐴
·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥))) |
6 | 5 | fveq2d 6107 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝑇‘((𝐴 ·op
𝑈)‘𝑥)) = (𝑇‘(𝐴 ·ℎ (𝑈‘𝑥)))) |
7 | | homulcl 28002 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ)
→ (𝐴
·op 𝑈): ℋ⟶ ℋ) |
8 | 7 | 3adant2 1073 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝐴
·op 𝑈): ℋ⟶ ℋ) |
9 | | fvco3 6185 |
. . . . 5
⊢ (((𝐴 ·op
𝑈): ℋ⟶ ℋ
∧ 𝑥 ∈ ℋ)
→ ((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = (𝑇‘((𝐴 ·op 𝑈)‘𝑥))) |
10 | 8, 9 | sylan 487 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = (𝑇‘((𝐴 ·op 𝑈)‘𝑥))) |
11 | | fvco3 6185 |
. . . . . . 7
⊢ ((𝑈: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥))) |
12 | 2, 3, 11 | syl2anc 691 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥))) |
13 | 12 | oveq2d 6565 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝐴
·ℎ ((𝑇 ∘ 𝑈)‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) |
14 | | lnopf 28102 |
. . . . . . . . 9
⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶
ℋ) |
15 | 14 | 3ad2ant2 1076 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ 𝑇: ℋ⟶
ℋ) |
16 | | simp3 1056 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ 𝑈: ℋ⟶
ℋ) |
17 | | fco 5971 |
. . . . . . . 8
⊢ ((𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ 𝑈): ℋ⟶
ℋ) |
18 | 15, 16, 17 | syl2anc 691 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ 𝑈): ℋ⟶
ℋ) |
19 | 18 | adantr 480 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝑇 ∘ 𝑈): ℋ⟶
ℋ) |
20 | | homval 27984 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op
(𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))) |
21 | 1, 19, 3, 20 | syl3anc 1318 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝐴
·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))) |
22 | | simpl2 1058 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
𝑇 ∈
LinOp) |
23 | 16 | ffvelrnda 6267 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝑈‘𝑥) ∈ ℋ) |
24 | | lnopmul 28210 |
. . . . . 6
⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ (𝑈‘𝑥) ∈ ℋ) → (𝑇‘(𝐴 ·ℎ (𝑈‘𝑥))) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) |
25 | 22, 1, 23, 24 | syl3anc 1318 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
(𝑇‘(𝐴 ·ℎ (𝑈‘𝑥))) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))) |
26 | 13, 21, 25 | 3eqtr4d 2654 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝐴
·op (𝑇 ∘ 𝑈))‘𝑥) = (𝑇‘(𝐴 ·ℎ (𝑈‘𝑥)))) |
27 | 6, 10, 26 | 3eqtr4d 2654 |
. . 3
⊢ (((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) ∧
𝑥 ∈ ℋ) →
((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥)) |
28 | 27 | ralrimiva 2949 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ ∀𝑥 ∈
ℋ ((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥)) |
29 | | fco 5971 |
. . . 4
⊢ ((𝑇: ℋ⟶ ℋ ∧
(𝐴
·op 𝑈): ℋ⟶ ℋ) → (𝑇 ∘ (𝐴 ·op 𝑈)): ℋ⟶
ℋ) |
30 | 15, 8, 29 | syl2anc 691 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ (𝐴 ·op
𝑈)): ℋ⟶
ℋ) |
31 | | simp1 1054 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ 𝐴 ∈
ℂ) |
32 | | homulcl 28002 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ) → (𝐴 ·op
(𝑇 ∘ 𝑈)): ℋ⟶
ℋ) |
33 | 31, 18, 32 | syl2anc 691 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝐴
·op (𝑇 ∘ 𝑈)): ℋ⟶
ℋ) |
34 | | hoeq 28003 |
. . 3
⊢ (((𝑇 ∘ (𝐴 ·op 𝑈)): ℋ⟶ ℋ
∧ (𝐴
·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ) →
(∀𝑥 ∈ ℋ
((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) ↔ (𝑇 ∘ (𝐴 ·op 𝑈)) = (𝐴 ·op (𝑇 ∘ 𝑈)))) |
35 | 30, 33, 34 | syl2anc 691 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (∀𝑥 ∈
ℋ ((𝑇 ∘ (𝐴 ·op
𝑈))‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) ↔ (𝑇 ∘ (𝐴 ·op 𝑈)) = (𝐴 ·op (𝑇 ∘ 𝑈)))) |
36 | 28, 35 | mpbid 221 |
1
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ (𝐴 ·op
𝑈)) = (𝐴 ·op (𝑇 ∘ 𝑈))) |