Theorem norm-ii

Description: Triangle inequality for norms. Theorem 3.3(ii) of Beran p. 97. (Contributed by NM, 10-Mar-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	norm-ii	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ≤ ( ( normℎ ‘ 𝐴 ) + ( normℎ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fvoveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) = ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) )
2		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( normℎ ‘ 𝐴 ) = ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
3	2	oveq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( normℎ ‘ 𝐴 ) + ( normℎ ‘ 𝐵 ) ) = ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( normℎ ‘ 𝐵 ) ) )
4	1 3	breq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ≤ ( ( normℎ ‘ 𝐴 ) + ( normℎ ‘ 𝐵 ) ) ↔ ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) ≤ ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( normℎ ‘ 𝐵 ) ) ) )
5		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 ) ) )
6	5	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 ) ) ) )
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 ) → ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( normℎ ‘ 𝐵 ) ) = ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
9	6 8	breq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) ) ≤ ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( normℎ ‘ 𝐵 ) ) ↔ ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ≤ ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ) )
10		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
11		ifhvhv0	⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ∈ ℋ
12	10 11	norm-ii-i	⊢ ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ≤ ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) + ( normℎ ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
13	4 9 12	dedth2h	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( normℎ ‘ ( 𝐴 +ℎ 𝐵 ) ) ≤ ( ( normℎ ‘ 𝐴 ) + ( normℎ ‘ 𝐵 ) ) )