Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
|- ( y = A -> ( cos ` y ) = ( cos ` A ) ) |
2 |
1
|
neeq1d |
|- ( y = A -> ( ( cos ` y ) =/= 0 <-> ( cos ` A ) =/= 0 ) ) |
3 |
2
|
elrab |
|- ( A e. { y e. CC | ( cos ` y ) =/= 0 } <-> ( A e. CC /\ ( cos ` A ) =/= 0 ) ) |
4 |
|
fveq2 |
|- ( x = A -> ( cos ` x ) = ( cos ` A ) ) |
5 |
4
|
oveq2d |
|- ( x = A -> ( 1 / ( cos ` x ) ) = ( 1 / ( cos ` A ) ) ) |
6 |
|
df-sec |
|- sec = ( x e. { y e. CC | ( cos ` y ) =/= 0 } |-> ( 1 / ( cos ` x ) ) ) |
7 |
|
ovex |
|- ( 1 / ( cos ` A ) ) e. _V |
8 |
5 6 7
|
fvmpt |
|- ( A e. { y e. CC | ( cos ` y ) =/= 0 } -> ( sec ` A ) = ( 1 / ( cos ` A ) ) ) |
9 |
3 8
|
sylbir |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( sec ` A ) = ( 1 / ( cos ` A ) ) ) |