Step |
Hyp |
Ref |
Expression |
1 |
|
eigorthi.1 |
⊢ 𝐴 ∈ ℋ |
2 |
|
eigorthi.2 |
⊢ 𝐵 ∈ ℋ |
3 |
|
eigorthi.3 |
⊢ 𝐶 ∈ ℂ |
4 |
|
eigorthi.4 |
⊢ 𝐷 ∈ ℂ |
5 |
|
oveq2 |
⊢ ( ( 𝑇 ‘ 𝐵 ) = ( 𝐷 ·ℎ 𝐵 ) → ( 𝐴 ·ih ( 𝑇 ‘ 𝐵 ) ) = ( 𝐴 ·ih ( 𝐷 ·ℎ 𝐵 ) ) ) |
6 |
|
his5 |
⊢ ( ( 𝐷 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ih ( 𝐷 ·ℎ 𝐵 ) ) = ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) ) |
7 |
4 1 2 6
|
mp3an |
⊢ ( 𝐴 ·ih ( 𝐷 ·ℎ 𝐵 ) ) = ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) |
8 |
5 7
|
eqtrdi |
⊢ ( ( 𝑇 ‘ 𝐵 ) = ( 𝐷 ·ℎ 𝐵 ) → ( 𝐴 ·ih ( 𝑇 ‘ 𝐵 ) ) = ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) ) |
9 |
|
oveq1 |
⊢ ( ( 𝑇 ‘ 𝐴 ) = ( 𝐶 ·ℎ 𝐴 ) → ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐵 ) = ( ( 𝐶 ·ℎ 𝐴 ) ·ih 𝐵 ) ) |
10 |
|
ax-his3 |
⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐶 ·ℎ 𝐴 ) ·ih 𝐵 ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) ) |
11 |
3 1 2 10
|
mp3an |
⊢ ( ( 𝐶 ·ℎ 𝐴 ) ·ih 𝐵 ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) |
12 |
9 11
|
eqtrdi |
⊢ ( ( 𝑇 ‘ 𝐴 ) = ( 𝐶 ·ℎ 𝐴 ) → ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐵 ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) ) |
13 |
8 12
|
eqeqan12rd |
⊢ ( ( ( 𝑇 ‘ 𝐴 ) = ( 𝐶 ·ℎ 𝐴 ) ∧ ( 𝑇 ‘ 𝐵 ) = ( 𝐷 ·ℎ 𝐵 ) ) → ( ( 𝐴 ·ih ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐵 ) ↔ ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) ) ) |
14 |
1 2
|
hicli |
⊢ ( 𝐴 ·ih 𝐵 ) ∈ ℂ |
15 |
4
|
cjcli |
⊢ ( ∗ ‘ 𝐷 ) ∈ ℂ |
16 |
|
mulcan2 |
⊢ ( ( ( ∗ ‘ 𝐷 ) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ( ( 𝐴 ·ih 𝐵 ) ∈ ℂ ∧ ( 𝐴 ·ih 𝐵 ) ≠ 0 ) ) → ( ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) ↔ ( ∗ ‘ 𝐷 ) = 𝐶 ) ) |
17 |
15 3 16
|
mp3an12 |
⊢ ( ( ( 𝐴 ·ih 𝐵 ) ∈ ℂ ∧ ( 𝐴 ·ih 𝐵 ) ≠ 0 ) → ( ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) ↔ ( ∗ ‘ 𝐷 ) = 𝐶 ) ) |
18 |
14 17
|
mpan |
⊢ ( ( 𝐴 ·ih 𝐵 ) ≠ 0 → ( ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) ↔ ( ∗ ‘ 𝐷 ) = 𝐶 ) ) |
19 |
|
eqcom |
⊢ ( ( ∗ ‘ 𝐷 ) = 𝐶 ↔ 𝐶 = ( ∗ ‘ 𝐷 ) ) |
20 |
18 19
|
syl6bb |
⊢ ( ( 𝐴 ·ih 𝐵 ) ≠ 0 → ( ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) ↔ 𝐶 = ( ∗ ‘ 𝐷 ) ) ) |
21 |
20
|
biimpcd |
⊢ ( ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) → ( ( 𝐴 ·ih 𝐵 ) ≠ 0 → 𝐶 = ( ∗ ‘ 𝐷 ) ) ) |
22 |
21
|
necon1d |
⊢ ( ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) → ( 𝐶 ≠ ( ∗ ‘ 𝐷 ) → ( 𝐴 ·ih 𝐵 ) = 0 ) ) |
23 |
22
|
com12 |
⊢ ( 𝐶 ≠ ( ∗ ‘ 𝐷 ) → ( ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) → ( 𝐴 ·ih 𝐵 ) = 0 ) ) |
24 |
|
oveq2 |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( ( ∗ ‘ 𝐷 ) · 0 ) ) |
25 |
|
oveq2 |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · 0 ) ) |
26 |
3
|
mul01i |
⊢ ( 𝐶 · 0 ) = 0 |
27 |
15
|
mul01i |
⊢ ( ( ∗ ‘ 𝐷 ) · 0 ) = 0 |
28 |
26 27
|
eqtr4i |
⊢ ( 𝐶 · 0 ) = ( ( ∗ ‘ 𝐷 ) · 0 ) |
29 |
25 28
|
eqtrdi |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) = ( ( ∗ ‘ 𝐷 ) · 0 ) ) |
30 |
24 29
|
eqtr4d |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) ) |
31 |
23 30
|
impbid1 |
⊢ ( 𝐶 ≠ ( ∗ ‘ 𝐷 ) → ( ( ( ∗ ‘ 𝐷 ) · ( 𝐴 ·ih 𝐵 ) ) = ( 𝐶 · ( 𝐴 ·ih 𝐵 ) ) ↔ ( 𝐴 ·ih 𝐵 ) = 0 ) ) |
32 |
13 31
|
sylan9bb |
⊢ ( ( ( ( 𝑇 ‘ 𝐴 ) = ( 𝐶 ·ℎ 𝐴 ) ∧ ( 𝑇 ‘ 𝐵 ) = ( 𝐷 ·ℎ 𝐵 ) ) ∧ 𝐶 ≠ ( ∗ ‘ 𝐷 ) ) → ( ( 𝐴 ·ih ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) ·ih 𝐵 ) ↔ ( 𝐴 ·ih 𝐵 ) = 0 ) ) |