Metamath Proof Explorer


Theorem sec0

Description: The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014)

Ref Expression
Assertion sec0
|- ( sec ` 0 ) = 1

Proof

Step Hyp Ref Expression
1 0cn
 |-  0 e. CC
2 cos0
 |-  ( cos ` 0 ) = 1
3 ax-1ne0
 |-  1 =/= 0
4 2 3 eqnetri
 |-  ( cos ` 0 ) =/= 0
5 secval
 |-  ( ( 0 e. CC /\ ( cos ` 0 ) =/= 0 ) -> ( sec ` 0 ) = ( 1 / ( cos ` 0 ) ) )
6 1 4 5 mp2an
 |-  ( sec ` 0 ) = ( 1 / ( cos ` 0 ) )
7 2 oveq2i
 |-  ( 1 / ( cos ` 0 ) ) = ( 1 / 1 )
8 1div1e1
 |-  ( 1 / 1 ) = 1
9 6 7 8 3eqtri
 |-  ( sec ` 0 ) = 1