Metamath Proof Explorer


Theorem mulcanad

Description: Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand . (Contributed by David Moews, 28-Feb-2017)

Ref Expression
Hypotheses mulcanad.1
|- ( ph -> A e. CC )
mulcanad.2
|- ( ph -> B e. CC )
mulcanad.3
|- ( ph -> C e. CC )
mulcanad.4
|- ( ph -> C =/= 0 )
mulcanad.5
|- ( ph -> ( C x. A ) = ( C x. B ) )
Assertion mulcanad
|- ( ph -> A = B )

Proof

Step Hyp Ref Expression
1 mulcanad.1
 |-  ( ph -> A e. CC )
2 mulcanad.2
 |-  ( ph -> B e. CC )
3 mulcanad.3
 |-  ( ph -> C e. CC )
4 mulcanad.4
 |-  ( ph -> C =/= 0 )
5 mulcanad.5
 |-  ( ph -> ( C x. A ) = ( C x. B ) )
6 1 2 3 4 mulcand
 |-  ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) )
7 5 6 mpbid
 |-  ( ph -> A = B )