| Step | Hyp | Ref
| Expression |
| 1 | | hmopf 31893 |
. . . 4
⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶
ℋ) |
| 2 | | hmopf 31893 |
. . . 4
⊢ (𝑈 ∈ HrmOp → 𝑈: ℋ⟶
ℋ) |
| 3 | | fco 6760 |
. . . 4
⊢ ((𝑇: ℋ⟶ ℋ ∧
𝑈: ℋ⟶ ℋ)
→ (𝑇 ∘ 𝑈): ℋ⟶
ℋ) |
| 4 | 1, 2, 3 | syl2an 596 |
. . 3
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ) |
| 5 | 4 | 3adant3 1133 |
. 2
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ) |
| 6 | | fvco3 7008 |
. . . . . . . . . 10
⊢ ((𝑈: ℋ⟶ ℋ ∧
𝑦 ∈ ℋ) →
((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦))) |
| 7 | 2, 6 | sylan 580 |
. . . . . . . . 9
⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦))) |
| 8 | 7 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥
·ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈‘𝑦)))) |
| 9 | 8 | ad2ant2l 746 |
. . . . . . 7
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥
·ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈‘𝑦)))) |
| 10 | | simpll 767 |
. . . . . . . 8
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ HrmOp) |
| 11 | | simprl 771 |
. . . . . . . 8
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈
ℋ) |
| 12 | 2 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈‘𝑦) ∈ ℋ) |
| 13 | 12 | ad2ant2l 746 |
. . . . . . . 8
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈‘𝑦) ∈ ℋ) |
| 14 | | hmop 31941 |
. . . . . . . 8
⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈‘𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·ih (𝑈‘𝑦))) |
| 15 | 10, 11, 13, 14 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥
·ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·ih (𝑈‘𝑦))) |
| 16 | | simplr 769 |
. . . . . . . 8
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp) |
| 17 | 1 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ) |
| 18 | 17 | ad2ant2r 747 |
. . . . . . . 8
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑥) ∈ ℋ) |
| 19 | | simprr 773 |
. . . . . . . 8
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈
ℋ) |
| 20 | | hmop 31941 |
. . . . . . . 8
⊢ ((𝑈 ∈ HrmOp ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦)) |
| 21 | 16, 18, 19, 20 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦)) |
| 22 | 9, 15, 21 | 3eqtrd 2781 |
. . . . . 6
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥
·ih ((𝑇 ∘ 𝑈)‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦)) |
| 23 | | fvco3 7008 |
. . . . . . . . 9
⊢ ((𝑇: ℋ⟶ ℋ ∧
𝑥 ∈ ℋ) →
((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥))) |
| 24 | 1, 23 | sylan 580 |
. . . . . . . 8
⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥))) |
| 25 | 24 | oveq1d 7446 |
. . . . . . 7
⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦)) |
| 26 | 25 | ad2ant2r 747 |
. . . . . 6
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) →
(((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦)) |
| 27 | 22, 26 | eqtr4d 2780 |
. . . . 5
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥
·ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦)) |
| 28 | 27 | 3adantl3 1169 |
. . . 4
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦)) |
| 29 | | fveq1 6905 |
. . . . . . 7
⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → ((𝑇 ∘ 𝑈)‘𝑥) = ((𝑈 ∘ 𝑇)‘𝑥)) |
| 30 | 29 | oveq1d 7446 |
. . . . . 6
⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → (((𝑇 ∘ 𝑈)‘𝑥) ·ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦)) |
| 31 | 30 | 3ad2ant3 1136 |
. . . . 5
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (((𝑇 ∘ 𝑈)‘𝑥) ·ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦)) |
| 32 | 31 | adantr 480 |
. . . 4
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇 ∘ 𝑈)‘𝑥) ·ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦)) |
| 33 | 28, 32 | eqtr4d 2780 |
. . 3
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑇 ∘ 𝑈)‘𝑥) ·ih 𝑦)) |
| 34 | 33 | ralrimivva 3202 |
. 2
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑇 ∘ 𝑈)‘𝑥) ·ih 𝑦)) |
| 35 | | elhmop 31892 |
. 2
⊢ ((𝑇 ∘ 𝑈) ∈ HrmOp ↔ ((𝑇 ∘ 𝑈): ℋ⟶ ℋ ∧
∀𝑥 ∈ ℋ
∀𝑦 ∈ ℋ
(𝑥
·ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑇 ∘ 𝑈)‘𝑥) ·ih 𝑦))) |
| 36 | 5, 34, 35 | sylanbrc 583 |
1
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈) ∈ HrmOp) |