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 ) ) )