Description: Cancellation law for multiplication. Theorem I.7 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
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Hypotheses | mulcand.1 | |- ( ph -> A e. CC ) |
|
mulcand.2 | |- ( ph -> B e. CC ) |
||
mulcand.3 | |- ( ph -> C e. CC ) |
||
mulcand.4 | |- ( ph -> C =/= 0 ) |
||
Assertion | mulcan2d | |- ( ph -> ( ( A x. C ) = ( B x. C ) <-> A = B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcand.1 | |- ( ph -> A e. CC ) |
|
2 | mulcand.2 | |- ( ph -> B e. CC ) |
|
3 | mulcand.3 | |- ( ph -> C e. CC ) |
|
4 | mulcand.4 | |- ( ph -> C =/= 0 ) |
|
5 | 1 3 | mulcomd | |- ( ph -> ( A x. C ) = ( C x. A ) ) |
6 | 2 3 | mulcomd | |- ( ph -> ( B x. C ) = ( C x. B ) ) |
7 | 5 6 | eqeq12d | |- ( ph -> ( ( A x. C ) = ( B x. C ) <-> ( C x. A ) = ( C x. B ) ) ) |
8 | 1 2 3 4 | mulcand | |- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) ) |
9 | 7 8 | bitrd | |- ( ph -> ( ( A x. C ) = ( B x. C ) <-> A = B ) ) |