Step |
Hyp |
Ref |
Expression |
1 |
|
dmadjrn |
⊢ ( 𝑇 ∈ dom adjℎ → ( adjℎ ‘ 𝑇 ) ∈ dom adjℎ ) |
2 |
|
dmadjop |
⊢ ( ( adjℎ ‘ 𝑇 ) ∈ dom adjℎ → ( adjℎ ‘ 𝑇 ) : ℋ ⟶ ℋ ) |
3 |
1 2
|
syl |
⊢ ( 𝑇 ∈ dom adjℎ → ( adjℎ ‘ 𝑇 ) : ℋ ⟶ ℋ ) |
4 |
|
simp2 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → 𝑤 ∈ ℋ ) |
5 |
|
adjcl |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ ) → ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ∈ ℋ ) |
6 |
|
hvmulcl |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ∈ ℋ ) → ( 𝑥 ·ℎ ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ) ∈ ℋ ) |
7 |
5 6
|
sylan2 |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( 𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ ) ) → ( 𝑥 ·ℎ ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ) ∈ ℋ ) |
8 |
7
|
an12s |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → ( 𝑥 ·ℎ ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ) ∈ ℋ ) |
9 |
8
|
adantrr |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → ( 𝑥 ·ℎ ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ) ∈ ℋ ) |
10 |
9
|
3adant2 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → ( 𝑥 ·ℎ ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ) ∈ ℋ ) |
11 |
|
adjcl |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑧 ∈ ℋ ) → ( ( adjℎ ‘ 𝑇 ) ‘ 𝑧 ) ∈ ℋ ) |
12 |
11
|
adantrl |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → ( ( adjℎ ‘ 𝑇 ) ‘ 𝑧 ) ∈ ℋ ) |
13 |
12
|
3adant2 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → ( ( adjℎ ‘ 𝑇 ) ‘ 𝑧 ) ∈ ℋ ) |
14 |
|
his7 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑤 ) ·ih 𝑦 ) = ( 𝑤 ·ih ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ) ) |
17 |
16
|
3adant3l |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → ( ( 𝑇 ‘ 𝑤 ) ·ih 𝑦 ) = ( 𝑤 ·ih ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ) ) |
18 |
17
|
oveq2d |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → ( ( ∗ ‘ 𝑥 ) · ( ( 𝑇 ‘ 𝑤 ) ·ih 𝑦 ) ) = ( ( ∗ ‘ 𝑥 ) · ( 𝑤 ·ih ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) |
19 |
|
simp3l |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → 𝑥 ∈ ℂ ) |
20 |
|
dmadjop |
⊢ ( 𝑇 ∈ dom adjℎ → 𝑇 : ℋ ⟶ ℋ ) |
21 |
20
|
ffvelrnda |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ) → ( 𝑇 ‘ 𝑤 ) ∈ ℋ ) |
22 |
21
|
3adant3 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → ( 𝑇 ‘ 𝑤 ) ∈ ℋ ) |
23 |
|
simp3r |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → 𝑦 ∈ ℋ ) |
24 |
|
his5 |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( 𝑇 ‘ 𝑤 ) ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑤 ) ·ih ( 𝑥 ·ℎ 𝑦 ) ) = ( ( ∗ ‘ 𝑥 ) · ( ( 𝑇 ‘ 𝑤 ) ·ih 𝑦 ) ) ) |
25 |
19 22 23 24
|
syl3anc |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → ( ( 𝑇 ‘ 𝑤 ) ·ih ( 𝑥 ·ℎ 𝑦 ) ) = ( ( ∗ ‘ 𝑥 ) · ( ( 𝑇 ‘ 𝑤 ) ·ih 𝑦 ) ) ) |
26 |
|
simp2 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → 𝑤 ∈ ℋ ) |
27 |
5
|
adantrl |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ∈ ℋ ) |
28 |
27
|
3adant2 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ) → ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ∈ ℋ ) |
29 |
|
his5 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑤 ) ·ih 𝑧 ) = ( 𝑤 ·ih ( ( adjℎ ‘ 𝑇 ) ‘ 𝑧 ) ) ) |
34 |
33
|
3adant3l |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → ( ( 𝑇 ‘ 𝑤 ) ·ih 𝑧 ) = ( 𝑤 ·ih ( ( adjℎ ‘ 𝑇 ) ‘ 𝑧 ) ) ) |
35 |
32 34
|
oveq12d |
⊢ ( ( 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 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → ( 𝑇 ‘ 𝑤 ) ∈ ℋ ) |
37 |
|
hvmulcl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) → ( 𝑥 ·ℎ 𝑦 ) ∈ ℋ ) |
38 |
37
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) → ( 𝑥 ·ℎ 𝑦 ) ∈ ℋ ) |
39 |
38
|
3ad2ant3 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → ( 𝑥 ·ℎ 𝑦 ) ∈ ℋ ) |
40 |
|
simp3r |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ 𝑤 ∈ ℋ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → 𝑧 ∈ ℋ ) |
41 |
|
his7 |
⊢ ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( 𝑥 ·ℎ 𝑦 ) ∈ ℋ ∧ 𝑧 ∈ ℋ ) → ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ∈ ℋ ) |
44 |
37 43
|
sylan |
⊢ ( ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) → ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ∈ ℋ ) |
45 |
|
adj2 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ∈ ℋ ) → ( ( adjℎ ‘ 𝑇 ) ‘ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) ∈ ℋ ) |
53 |
44 52
|
sylan2 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → ( ( adjℎ ‘ 𝑇 ) ‘ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) ∈ ℋ ) |
54 |
|
hvaddcl |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 𝑇 ∈ dom adjℎ ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ) ∧ 𝑧 ∈ ℋ ) ) → ( ( 𝑥 ·ℎ ( ( adjℎ ‘ 𝑇 ) ‘ 𝑦 ) ) +ℎ ( ( adjℎ ‘ 𝑇 ) ‘ 𝑧 ) ) ∈ ℋ ) |
57 |
|
hial2eq2 |
⊢ ( ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( ( 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 |
⊢ ( 𝑇 ∈ dom adjℎ → ( adjℎ ‘ 𝑇 ) ∈ LinOp ) |