Step |
Hyp |
Ref |
Expression |
1 |
|
recn |
|- ( A e. RR -> A e. CC ) |
2 |
|
cotval |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) ) |
3 |
1 2
|
sylan |
|- ( ( A e. RR /\ ( sin ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) ) |
4 |
|
resincl |
|- ( A e. RR -> ( sin ` A ) e. RR ) |
5 |
|
recoscl |
|- ( A e. RR -> ( cos ` A ) e. RR ) |
6 |
|
redivcl |
|- ( ( ( cos ` A ) e. RR /\ ( sin ` A ) e. RR /\ ( sin ` A ) =/= 0 ) -> ( ( cos ` A ) / ( sin ` A ) ) e. RR ) |
7 |
5 6
|
syl3an1 |
|- ( ( A e. RR /\ ( sin ` A ) e. RR /\ ( sin ` A ) =/= 0 ) -> ( ( cos ` A ) / ( sin ` A ) ) e. RR ) |
8 |
4 7
|
syl3an2 |
|- ( ( A e. RR /\ A e. RR /\ ( sin ` A ) =/= 0 ) -> ( ( cos ` A ) / ( sin ` A ) ) e. RR ) |
9 |
8
|
3anidm12 |
|- ( ( A e. RR /\ ( sin ` A ) =/= 0 ) -> ( ( cos ` A ) / ( sin ` A ) ) e. RR ) |
10 |
3 9
|
eqeltrd |
|- ( ( A e. RR /\ ( sin ` A ) =/= 0 ) -> ( cot ` A ) e. RR ) |