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Metamath Proof Explorer

Theorem norm-iii

Description: Theorem 3.3(iii) of Beran p. 97. (Contributed by NM, 25-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion norm-iii ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( normℎ ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( normℎ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 fvoveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( normℎ ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( normℎ ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) ) )
2 fveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( abs ‘ 𝐴 ) = ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) )
3 2 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( abs ‘ 𝐴 ) · ( normℎ ‘ 𝐵 ) ) = ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) · ( normℎ ‘ 𝐵 ) ) )
4 1 3 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( normℎ ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( normℎ ‘ 𝐵 ) ) ↔ ( normℎ ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) ) = ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) · ( normℎ ‘ 𝐵 ) ) ) )
5 oveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
6 5 fveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( normℎ ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) ) = ( normℎ ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
7 fveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( normℎ ‘ 𝐵 ) = ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
8 7 oveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) · ( normℎ ‘ 𝐵 ) ) = ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) · ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
9 6 8 eqeq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( normℎ ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) ) = ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) · ( normℎ ‘ 𝐵 ) ) ↔ ( normℎ ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) · ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ) )
10 0cn ⊢ 0 ∈ ℂ
11 10 elimel ⊢ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ∈ ℂ
12 ifhvhv0 ⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ∈ ℋ
13 11 12 norm-iii-i ⊢ ( normℎ ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( ( abs ‘ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ) · ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
14 4 9 13 dedth2h ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( normℎ ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( normℎ ‘ 𝐵 ) ) )