Step |
Hyp |
Ref |
Expression |
1 |
|
1cnd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> 1 e. CC ) |
2 |
|
reccl |
|- ( ( B e. CC /\ B =/= 0 ) -> ( 1 / B ) e. CC ) |
3 |
2
|
adantl |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( 1 / B ) e. CC ) |
4 |
|
simpl |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( A e. CC /\ A =/= 0 ) ) |
5 |
|
divmul |
|- ( ( 1 e. CC /\ ( 1 / B ) e. CC /\ ( A e. CC /\ A =/= 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> ( A x. ( 1 / B ) ) = 1 ) ) |
6 |
1 3 4 5
|
syl3anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> ( A x. ( 1 / B ) ) = 1 ) ) |
7 |
|
simpll |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> A e. CC ) |
8 |
|
simprl |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> B e. CC ) |
9 |
|
simprr |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> B =/= 0 ) |
10 |
|
divrec |
|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) ) |
11 |
7 8 9 10
|
syl3anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( A / B ) = ( A x. ( 1 / B ) ) ) |
12 |
11
|
eqeq1d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A / B ) = 1 <-> ( A x. ( 1 / B ) ) = 1 ) ) |
13 |
|
diveq1 |
|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 1 <-> A = B ) ) |
14 |
7 8 9 13
|
syl3anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A / B ) = 1 <-> A = B ) ) |
15 |
6 12 14
|
3bitr2d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) ) |