Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> A e. CC ) |
2 |
|
simp2 |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> B e. CC ) |
3 |
|
reccl |
|- ( ( C e. CC /\ C =/= 0 ) -> ( 1 / C ) e. CC ) |
4 |
3
|
3ad2ant3 |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( 1 / C ) e. CC ) |
5 |
1 2 4
|
adddird |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A + B ) x. ( 1 / C ) ) = ( ( A x. ( 1 / C ) ) + ( B x. ( 1 / C ) ) ) ) |
6 |
1 2
|
addcld |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( A + B ) e. CC ) |
7 |
|
simp3l |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> C e. CC ) |
8 |
|
simp3r |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> C =/= 0 ) |
9 |
|
divrec |
|- ( ( ( A + B ) e. CC /\ C e. CC /\ C =/= 0 ) -> ( ( A + B ) / C ) = ( ( A + B ) x. ( 1 / C ) ) ) |
10 |
6 7 8 9
|
syl3anc |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A + B ) / C ) = ( ( A + B ) x. ( 1 / C ) ) ) |
11 |
|
divrec |
|- ( ( A e. CC /\ C e. CC /\ C =/= 0 ) -> ( A / C ) = ( A x. ( 1 / C ) ) ) |
12 |
1 7 8 11
|
syl3anc |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( A / C ) = ( A x. ( 1 / C ) ) ) |
13 |
|
divrec |
|- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> ( B / C ) = ( B x. ( 1 / C ) ) ) |
14 |
2 7 8 13
|
syl3anc |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( B / C ) = ( B x. ( 1 / C ) ) ) |
15 |
12 14
|
oveq12d |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A / C ) + ( B / C ) ) = ( ( A x. ( 1 / C ) ) + ( B x. ( 1 / C ) ) ) ) |
16 |
5 10 15
|
3eqtr4d |
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) ) |