Description: Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | div1d.1 | |- ( ph -> A e. CC ) |
|
divcld.2 | |- ( ph -> B e. CC ) |
||
divmuld.3 | |- ( ph -> C e. CC ) |
||
divassd.4 | |- ( ph -> C =/= 0 ) |
||
Assertion | divmul3d | |- ( ph -> ( ( A / C ) = B <-> A = ( B x. C ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | |- ( ph -> A e. CC ) |
|
2 | divcld.2 | |- ( ph -> B e. CC ) |
|
3 | divmuld.3 | |- ( ph -> C e. CC ) |
|
4 | divassd.4 | |- ( ph -> C =/= 0 ) |
|
5 | divmul3 | |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / C ) = B <-> A = ( B x. C ) ) ) |
|
6 | 1 2 3 4 5 | syl112anc | |- ( ph -> ( ( A / C ) = B <-> A = ( B x. C ) ) ) |