| Step | Hyp | Ref
| Expression |
| 1 | | dmadjrn 31914 |
. . 3
⊢ (𝑇 ∈ dom
adjℎ → (adjℎ‘𝑇) ∈ dom
adjℎ) |
| 2 | | dmadjop 31907 |
. . 3
⊢
((adjℎ‘𝑇) ∈ dom adjℎ →
(adjℎ‘𝑇): ℋ⟶ ℋ) |
| 3 | 1, 2 | syl 17 |
. 2
⊢ (𝑇 ∈ dom
adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ) |
| 4 | | simp2 1138 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑤 ∈ ℋ) |
| 5 | | adjcl 31951 |
. . . . . . . . . . . . . . 15
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑦 ∈ ℋ) →
((adjℎ‘𝑇)‘𝑦) ∈ ℋ) |
| 6 | | hvmulcl 31032 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℂ ∧
((adjℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) ∈ ℋ) |
| 7 | 5, 6 | sylan2 593 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℂ ∧ (𝑇 ∈ dom
adjℎ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) ∈ ℋ) |
| 8 | 7 | an12s 649 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ dom
adjℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) ∈ ℋ) |
| 9 | 8 | adantrr 717 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) ∈ ℋ) |
| 10 | 9 | 3adant2 1132 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) ∈ ℋ) |
| 11 | | adjcl 31951 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑧 ∈ ℋ) →
((adjℎ‘𝑇)‘𝑧) ∈ ℋ) |
| 12 | 11 | adantrl 716 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) →
((adjℎ‘𝑇)‘𝑧) ∈ ℋ) |
| 13 | 12 | 3adant2 1132 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) →
((adjℎ‘𝑇)‘𝑧) ∈ ℋ) |
| 14 | | his7 31109 |
. . . . . . . . . . 11
⊢ ((𝑤 ∈ ℋ ∧ (𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) ∈ ℋ ∧
((adjℎ‘𝑇)‘𝑧) ∈ ℋ) → (𝑤 ·ih ((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧))) = ((𝑤 ·ih (𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦))) + (𝑤 ·ih
((adjℎ‘𝑇)‘𝑧)))) |
| 15 | 4, 10, 13, 14 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑤 ·ih ((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧))) = ((𝑤 ·ih (𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦))) + (𝑤 ·ih
((adjℎ‘𝑇)‘𝑧)))) |
| 16 | | adj2 31953 |
. . . . . . . . . . . . . . 15
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑤) ·ih 𝑦) = (𝑤 ·ih
((adjℎ‘𝑇)‘𝑦))) |
| 17 | 16 | 3adant3l 1181 |
. . . . . . . . . . . . . 14
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·ih 𝑦) = (𝑤 ·ih
((adjℎ‘𝑇)‘𝑦))) |
| 18 | 17 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) →
((∗‘𝑥)
· ((𝑇‘𝑤)
·ih 𝑦)) = ((∗‘𝑥) · (𝑤 ·ih
((adjℎ‘𝑇)‘𝑦)))) |
| 19 | | simp3l 1202 |
. . . . . . . . . . . . . 14
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ) |
| 20 | | dmadjop 31907 |
. . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ dom
adjℎ → 𝑇: ℋ⟶ ℋ) |
| 21 | 20 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . 15
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑤) ∈ ℋ) |
| 22 | 21 | 3adant3 1133 |
. . . . . . . . . . . . . 14
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ) |
| 23 | | simp3r 1203 |
. . . . . . . . . . . . . 14
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ) |
| 24 | | his5 31105 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑤) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑤) ·ih (𝑥
·ℎ 𝑦)) = ((∗‘𝑥) · ((𝑇‘𝑤) ·ih 𝑦))) |
| 25 | 19, 22, 23, 24 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·ih (𝑥
·ℎ 𝑦)) = ((∗‘𝑥) · ((𝑇‘𝑤) ·ih 𝑦))) |
| 26 | | simp2 1138 |
. . . . . . . . . . . . . 14
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑤 ∈ ℋ) |
| 27 | 5 | adantrl 716 |
. . . . . . . . . . . . . . 15
⊢ ((𝑇 ∈ dom
adjℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) →
((adjℎ‘𝑇)‘𝑦) ∈ ℋ) |
| 28 | 27 | 3adant2 1132 |
. . . . . . . . . . . . . 14
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) →
((adjℎ‘𝑇)‘𝑦) ∈ ℋ) |
| 29 | | his5 31105 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ ∧
((adjℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑤 ·ih (𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦))) = ((∗‘𝑥) · (𝑤 ·ih
((adjℎ‘𝑇)‘𝑦)))) |
| 30 | 19, 26, 28, 29 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑤 ·ih (𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦))) = ((∗‘𝑥) · (𝑤 ·ih
((adjℎ‘𝑇)‘𝑦)))) |
| 31 | 18, 25, 30 | 3eqtr4d 2787 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·ih (𝑥
·ℎ 𝑦)) = (𝑤 ·ih (𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)))) |
| 32 | 31 | 3adant3r 1182 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑇‘𝑤) ·ih (𝑥
·ℎ 𝑦)) = (𝑤 ·ih (𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)))) |
| 33 | | adj2 31953 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑤) ·ih 𝑧) = (𝑤 ·ih
((adjℎ‘𝑇)‘𝑧))) |
| 34 | 33 | 3adant3l 1181 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑇‘𝑤) ·ih 𝑧) = (𝑤 ·ih
((adjℎ‘𝑇)‘𝑧))) |
| 35 | 32, 34 | oveq12d 7449 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (((𝑇‘𝑤) ·ih (𝑥
·ℎ 𝑦)) + ((𝑇‘𝑤) ·ih 𝑧)) = ((𝑤 ·ih (𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦))) + (𝑤 ·ih
((adjℎ‘𝑇)‘𝑧)))) |
| 36 | 21 | 3adant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ) |
| 37 | | hvmulcl 31032 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥
·ℎ 𝑦) ∈ ℋ) |
| 38 | 37 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑥
·ℎ 𝑦) ∈ ℋ) |
| 39 | 38 | 3ad2ant3 1136 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·ℎ 𝑦) ∈
ℋ) |
| 40 | | simp3r 1203 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑧 ∈ ℋ) |
| 41 | | his7 31109 |
. . . . . . . . . . . 12
⊢ (((𝑇‘𝑤) ∈ ℋ ∧ (𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑤) ·ih ((𝑥
·ℎ 𝑦) +ℎ 𝑧)) = (((𝑇‘𝑤) ·ih (𝑥
·ℎ 𝑦)) + ((𝑇‘𝑤) ·ih 𝑧))) |
| 42 | 36, 39, 40, 41 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑇‘𝑤) ·ih ((𝑥
·ℎ 𝑦) +ℎ 𝑧)) = (((𝑇‘𝑤) ·ih (𝑥
·ℎ 𝑦)) + ((𝑇‘𝑤) ·ih 𝑧))) |
| 43 | | hvaddcl 31031 |
. . . . . . . . . . . . 13
⊢ (((𝑥
·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈
ℋ) |
| 44 | 37, 43 | sylan 580 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥
·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) |
| 45 | | adj2 31953 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) → ((𝑇‘𝑤) ·ih ((𝑥
·ℎ 𝑦) +ℎ 𝑧)) = (𝑤 ·ih
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)))) |
| 46 | 44, 45 | syl3an3 1166 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑇‘𝑤) ·ih ((𝑥
·ℎ 𝑦) +ℎ 𝑧)) = (𝑤 ·ih
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)))) |
| 47 | 42, 46 | eqtr3d 2779 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (((𝑇‘𝑤) ·ih (𝑥
·ℎ 𝑦)) + ((𝑇‘𝑤) ·ih 𝑧)) = (𝑤 ·ih
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)))) |
| 48 | 15, 35, 47 | 3eqtr2rd 2784 |
. . . . . . . . 9
⊢ ((𝑇 ∈ dom
adjℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑤 ·ih
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (𝑤 ·ih ((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)))) |
| 49 | 48 | 3com23 1127 |
. . . . . . . 8
⊢ ((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑤 ·ih
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (𝑤 ·ih ((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)))) |
| 50 | 49 | 3expa 1119 |
. . . . . . 7
⊢ (((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) ∧ 𝑤 ∈ ℋ) → (𝑤 ·ih
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (𝑤 ·ih ((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)))) |
| 51 | 50 | ralrimiva 3146 |
. . . . . 6
⊢ ((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ∀𝑤 ∈ ℋ (𝑤
·ih ((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (𝑤 ·ih ((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)))) |
| 52 | | adjcl 31951 |
. . . . . . . 8
⊢ ((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) →
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) ∈
ℋ) |
| 53 | 44, 52 | sylan2 593 |
. . . . . . 7
⊢ ((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) →
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) ∈
ℋ) |
| 54 | | hvaddcl 31031 |
. . . . . . . . 9
⊢ (((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) ∈ ℋ ∧
((adjℎ‘𝑇)‘𝑧) ∈ ℋ) → ((𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)) ∈ ℋ) |
| 55 | 8, 11, 54 | syl2an 596 |
. . . . . . . 8
⊢ (((𝑇 ∈ dom
adjℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ (𝑇 ∈ dom adjℎ ∧
𝑧 ∈ ℋ)) →
((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)) ∈ ℋ) |
| 56 | 55 | anandis 678 |
. . . . . . 7
⊢ ((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)) ∈ ℋ) |
| 57 | | hial2eq2 31126 |
. . . . . . 7
⊢
((((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) ∈ ℋ ∧ ((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)) ∈ ℋ) → (∀𝑤 ∈ ℋ (𝑤
·ih ((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (𝑤 ·ih ((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧))) ↔
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)))) |
| 58 | 53, 56, 57 | syl2anc 584 |
. . . . . 6
⊢ ((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (∀𝑤 ∈ ℋ (𝑤
·ih ((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (𝑤 ·ih ((𝑥
·ℎ ((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧))) ↔
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)))) |
| 59 | 51, 58 | mpbid 232 |
. . . . 5
⊢ ((𝑇 ∈ dom
adjℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) →
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧))) |
| 60 | 59 | exp32 420 |
. . . 4
⊢ (𝑇 ∈ dom
adjℎ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑧 ∈ ℋ →
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧))))) |
| 61 | 60 | ralrimdv 3152 |
. . 3
⊢ (𝑇 ∈ dom
adjℎ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ∀𝑧 ∈ ℋ
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)))) |
| 62 | 61 | ralrimivv 3200 |
. 2
⊢ (𝑇 ∈ dom
adjℎ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧))) |
| 63 | | ellnop 31877 |
. 2
⊢
((adjℎ‘𝑇) ∈ LinOp ↔
((adjℎ‘𝑇): ℋ⟶ ℋ ∧
∀𝑥 ∈ ℂ
∀𝑦 ∈ ℋ
∀𝑧 ∈ ℋ
((adjℎ‘𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ
((adjℎ‘𝑇)‘𝑦)) +ℎ
((adjℎ‘𝑇)‘𝑧)))) |
| 64 | 3, 62, 63 | sylanbrc 583 |
1
⊢ (𝑇 ∈ dom
adjℎ → (adjℎ‘𝑇) ∈ LinOp) |