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Metamath Proof Explorer

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 ) ) ) )