This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ⊢ norm ℎ ⁡ 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 ⊢ norm ℎ ⁡ A - ℎ B = 0 ↔ A - ℎ B = 0 ℎ
5 1 2 hvsubeq0i ⊢ A - ℎ B = 0 ℎ ↔ A = B
6 4 5 bitri ⊢ norm ℎ ⁡ A - ℎ B = 0 ↔ A = B