Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> A e. RR ) |
2 |
1
|
recnd |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> A e. CC ) |
3 |
|
simp2 |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> B e. RR ) |
4 |
3
|
recnd |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> B e. CC ) |
5 |
|
simp3 |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> B =/= 0 ) |
6 |
|
divrec |
|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) ) |
7 |
2 4 5 6
|
syl3anc |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) ) |
8 |
|
rereccl |
|- ( ( B e. RR /\ B =/= 0 ) -> ( 1 / B ) e. RR ) |
9 |
8
|
3adant1 |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( 1 / B ) e. RR ) |
10 |
1 9
|
remulcld |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A x. ( 1 / B ) ) e. RR ) |
11 |
7 10
|
eqeltrd |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) e. RR ) |