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Metamath Proof Explorer

Theorem normlem9at

Description: Lemma used to derive properties of norm. Part of Remark 3.4(B) of Beran p. 98. (Contributed by NM, 10-May-2005) (New usage is discouraged.)

Ref Expression
Assertion normlem9at ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐵 ) ·ih ( 𝐴 −ℎ 𝐵 ) ) = ( ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) − ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) ) )

Proof

Step Hyp Ref Expression
1 oveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 −ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) )
2 1 1 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 −ℎ 𝐵 ) ·ih ( 𝐴 −ℎ 𝐵 ) ) = ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ·ih ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) )
3 id ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
4 3 3 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ·ih 𝐴 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
5 4 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) = ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( 𝐵 ·ih 𝐵 ) ) )
6 oveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ·ih 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) )
7 oveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐵 ·ih 𝐴 ) = ( 𝐵 ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
8 6 7 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) = ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) + ( 𝐵 ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
9 5 8 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) − ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) ) = ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( 𝐵 ·ih 𝐵 ) ) − ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) + ( 𝐵 ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) )
10 2 9 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( 𝐴 −ℎ 𝐵 ) ·ih ( 𝐴 −ℎ 𝐵 ) ) = ( ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) − ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) ) ↔ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ·ih ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( 𝐵 ·ih 𝐵 ) ) − ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) + ( 𝐵 ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) ) )
11 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 ) ) )
12 11 11 oveq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ·ih ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ·ih ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
13 id ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) )
14 13 13 oveq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( 𝐵 ·ih 𝐵 ) = ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
15 14 oveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( 𝐵 ·ih 𝐵 ) ) = ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
16 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 ) ) )
17 oveq1 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( 𝐵 ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
18 16 17 oveq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) + ( 𝐵 ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) = ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) + ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
19 15 18 oveq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( 𝐵 ·ih 𝐵 ) ) − ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) + ( 𝐵 ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) = ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) − ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) + ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) )
20 12 19 eqeq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ·ih ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( 𝐵 ·ih 𝐵 ) ) − ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih 𝐵 ) + ( 𝐵 ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) ↔ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ·ih ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) − ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) + ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) ) )
21 ifhvhv0 ⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
22 ifhvhv0 ⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ∈ ℋ
23 21 22 21 22 normlem9 ⊢ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ·ih ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) − ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) + ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
24 10 20 23 dedth2h ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐵 ) ·ih ( 𝐴 −ℎ 𝐵 ) ) = ( ( ( 𝐴 ·ih 𝐴 ) + ( 𝐵 ·ih 𝐵 ) ) − ( ( 𝐴 ·ih 𝐵 ) + ( 𝐵 ·ih 𝐴 ) ) ) )