Theorem normpyth

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
Assertion	normpyth	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ↑ 2 ) = ( ( ( normℎ ‘ 𝐴 ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ·ih 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) )
2	1	eqeq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 ·ih 𝐵 ) = 0 ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) = 0 ) )
3		fvoveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) = ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) )
4	3	oveq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ↑ 2 ) = ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) ↑ 2 ) )
5		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( normℎ ‘ 𝐴 ) = ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
6	5	oveq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) )
7	6	oveq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( normℎ ‘ 𝐴 ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) = ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) )
8	4 7	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ↑ 2 ) = ( ( ( normℎ ‘ 𝐴 ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) ↔ ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) ↑ 2 ) = ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) ) )
9	2 8	imbi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ↑ 2 ) = ( ( ( normℎ ‘ 𝐴 ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) ) ↔ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) = 0 → ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) ↑ 2 ) = ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) ) ) )
10		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
11	10	eqeq1d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) = 0 ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = 0 ) )
12		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
13	12	fveq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) = ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
14	13	oveq1d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) ↑ 2 ) = ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ↑ 2 ) )
15		fveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( normℎ ‘ 𝐵 ) = ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
16	15	oveq1d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( normℎ ‘ 𝐵 ) ↑ 2 ) = ( ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ↑ 2 ) )
17	16	oveq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) = ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ↑ 2 ) ) )
18	14 17	eqeq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) ↑ 2 ) = ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) ↔ ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ↑ 2 ) ) ) )
19	11 18	imbi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) = 0 → ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) ↑ 2 ) = ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) ) ↔ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = 0 → ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ↑ 2 ) ) ) ) )
20		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
21		ifhvhv0	⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ∈ ℋ
22	20 21	normpythi	⊢ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = 0 → ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) + ( ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ↑ 2 ) ) )
23	9 19 22	dedth2h	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ↑ 2 ) = ( ( ( normℎ ‘ 𝐴 ) ↑ 2 ) + ( ( normℎ ‘ 𝐵 ) ↑ 2 ) ) ) )