Step |
Hyp |
Ref |
Expression |
1 |
|
normsub.1 |
⊢ 𝐴 ∈ ℋ |
2 |
|
normsub.2 |
⊢ 𝐵 ∈ ℋ |
3 |
1 2 1 2
|
normlem8 |
⊢ ( ( 𝐴 +ℎ 𝐵 ) ·ih ( 𝐴 +ℎ 𝐵 ) ) = ( ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) + ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) ) |
4 |
|
id |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( 𝐴 ·ih 𝐵 ) = 0 ) |
5 |
|
orthcom |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) = 0 ↔ ( 𝐵 ·ih 𝐴 ) = 0 ) ) |
6 |
1 2 5
|
mp2an |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 ↔ ( 𝐵 ·ih 𝐴 ) = 0 ) |
7 |
6
|
biimpi |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( 𝐵 ·ih 𝐴 ) = 0 ) |
8 |
4 7
|
oveq12d |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) = ( 0 + 0 ) ) |
9 |
|
00id |
⊢ ( 0 + 0 ) = 0 |
10 |
8 9
|
eqtrdi |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) = 0 ) |
11 |
10
|
oveq2d |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) + ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) ) = ( ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) + 0 ) ) |
12 |
1 1
|
hicli |
⊢ ( 𝐴 ·ih 𝐴 ) ∈ ℂ |
13 |
2 2
|
hicli |
⊢ ( 𝐵 ·ih 𝐵 ) ∈ ℂ |
14 |
12 13
|
addcli |
⊢ ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) ∈ ℂ |
15 |
14
|
addid1i |
⊢ ( ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) + 0 ) = ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) |
16 |
11 15
|
eqtrdi |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) + ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) ) = ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) ) |
17 |
3 16
|
syl5eq |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( 𝐴 +ℎ 𝐵 ) ·ih ( 𝐴 +ℎ 𝐵 ) ) = ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) ) |
18 |
1 2
|
hvaddcli |
⊢ ( 𝐴 +ℎ 𝐵 ) ∈ ℋ |
19 |
18
|
normsqi |
⊢ ( ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ↑ 2 ) = ( ( 𝐴 +ℎ 𝐵 ) ·ih ( 𝐴 +ℎ 𝐵 ) ) |
20 |
1
|
normsqi |
⊢ ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 ) |
21 |
2
|
normsqi |
⊢ ( ( normℎ ‘ 𝐵 ) ↑ 2 ) = ( 𝐵 ·ih 𝐵 ) |
22 |
20 21
|
oveq12i |
⊢ ( ( ( normℎ ‘ 𝐴 ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) = ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) |
23 |
17 19 22
|
3eqtr4g |
⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ↑ 2 ) = ( ( ( normℎ ‘ 𝐴 ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) ) |