Theorem normsub0i

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
Hypotheses	normsub0.1	⊢ 𝐴 ∈ ℋ
	normsub0.2	⊢ 𝐵 ∈ ℋ
Assertion	normsub0i	⊢ ( ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		normsub0.1	⊢ 𝐴 ∈ ℋ
2		normsub0.2	⊢ 𝐵 ∈ ℋ
3	1 2	hvsubcli	⊢ ( 𝐴 −ℎ 𝐵 ) ∈ ℋ
4	3	norm-i-i	⊢ ( ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ ( 𝐴 −ℎ 𝐵 ) = 0ℎ )
5	1 2	hvsubeq0i	⊢ ( ( 𝐴 −ℎ 𝐵 ) = 0ℎ ↔ 𝐴 = 𝐵 )
6	4 5	bitri	⊢ ( ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ 𝐴 = 𝐵 )