Description: Swap denominators in a division. (Contributed by NM, 15-Sep-1999)
Ref | Expression | ||
---|---|---|---|
Hypotheses | divclz.1 | |- A e. CC |
|
divclz.2 | |- B e. CC |
||
divmulz.3 | |- C e. CC |
||
Assertion | divdiv23zi | |- ( ( B =/= 0 /\ C =/= 0 ) -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divclz.1 | |- A e. CC |
|
2 | divclz.2 | |- B e. CC |
|
3 | divmulz.3 | |- C e. CC |
|
4 | divdiv32 | |- ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) |
|
5 | 1 4 | mp3an1 | |- ( ( ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) |
6 | 3 5 | mpanr1 | |- ( ( ( B e. CC /\ B =/= 0 ) /\ C =/= 0 ) -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) |
7 | 2 6 | mpanl1 | |- ( ( B =/= 0 /\ C =/= 0 ) -> ( ( A / B ) / C ) = ( ( A / C ) / B ) ) |