Metamath Proof Explorer


Theorem normpythi

Description: Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of Beran p. 98. (Contributed by NM, 17-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses normsub.1 𝐴 ∈ ℋ
normsub.2 𝐵 ∈ ℋ
Assertion normpythi ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( norm ‘ ( 𝐴 + 𝐵 ) ) ↑ 2 ) = ( ( ( norm𝐴 ) ↑ 2 ) + ( ( norm𝐵 ) ↑ 2 ) ) )

Proof

Step Hyp Ref Expression
1 normsub.1 𝐴 ∈ ℋ
2 normsub.2 𝐵 ∈ ℋ
3 1 2 1 2 normlem8 ( ( 𝐴 + 𝐵 ) ·ih ( 𝐴 + 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 + 𝐵 ) ·ih ( 𝐴 + 𝐵 ) ) = ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) )
18 1 2 hvaddcli ( 𝐴 + 𝐵 ) ∈ ℋ
19 18 normsqi ( ( norm ‘ ( 𝐴 + 𝐵 ) ) ↑ 2 ) = ( ( 𝐴 + 𝐵 ) ·ih ( 𝐴 + 𝐵 ) )
20 1 normsqi ( ( norm𝐴 ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 )
21 2 normsqi ( ( norm𝐵 ) ↑ 2 ) = ( 𝐵 ·ih 𝐵 )
22 20 21 oveq12i ( ( ( norm𝐴 ) ↑ 2 ) + ( ( norm𝐵 ) ↑ 2 ) ) = ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) )
23 17 19 22 3eqtr4g ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( norm ‘ ( 𝐴 + 𝐵 ) ) ↑ 2 ) = ( ( ( norm𝐴 ) ↑ 2 ) + ( ( norm𝐵 ) ↑ 2 ) ) )