Theorem normsub0

Description: Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	normsub0	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ 𝐴 = 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	1	eqeq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = 0 ) )
3		eqeq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 = 𝐵 ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) = 𝐵 ) )
4	2 3	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ 𝐴 = 𝐵 ) ↔ ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = 0 ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) = 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	fveqeq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = 0 ↔ ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = 0 ) )
7		eqeq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) = 𝐵 ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
8	6 7	bibi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = 0 ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) = 𝐵 ) ↔ ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = 0 ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
9		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
10		ifhvhv0	⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ∈ ℋ
11	9 10	normsub0i	⊢ ( ( normℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = 0 ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) )
12	4 8 11	dedth2h	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ 𝐴 = 𝐵 ) )