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Metamath Proof Explorer

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 )