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
normsub0.2 B
Assertion normsub0i norm A - B = 0 A = B

Proof

Step Hyp Ref Expression
1 normsub0.1 A
2 normsub0.2 B
3 1 2 hvsubcli A - B
4 3 norm-i-i norm A - B = 0 A - B = 0
5 1 2 hvsubeq0i A - B = 0 A = B
6 4 5 bitri norm A - B = 0 A = B