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 ) |
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mulcanad.5 | |- ( ph -> ( C x. A ) = ( C x. B ) ) |
||
Assertion | mulcanad | |- ( ph -> A = B ) |
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 ) |