Description: Swap denominators in a division. (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 ) |
||
divmuld.4 | |- ( ph -> B =/= 0 ) |
||
divdiv23d.5 | |- ( ph -> C =/= 0 ) |
||
Assertion | divdiv32d | |- ( ph -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) |
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 | divmuld.4 | |- ( ph -> B =/= 0 ) |
|
5 | divdiv23d.5 | |- ( ph -> C =/= 0 ) |
|
6 | divdiv32 | |- ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) |
|
7 | 1 2 4 3 5 6 | syl122anc | |- ( ph -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) |