Theorem secval 42488

Description: Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)

Assertion

Ref	Expression
secval	|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( sec ` A ) = ( 1 / ( cos ` A ) ) )

Proof of Theorem secval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( y = A -> ( cos ` y ) = ( cos ` A ) )
2	1	neeq1d 2853	. . 3 |- ( y = A -> ( ( cos ` y ) =/= 0 <-> ( cos ` A ) =/= 0 ) )
3	2	elrab 3363	. 2 |- ( A e. { y e. CC | ( cos ` y ) =/= 0 } <-> ( A e. CC /\ ( cos ` A ) =/= 0 ) )
4		fveq2 6191	. . . 4 |- ( x = A -> ( cos ` x ) = ( cos ` A ) )
5	4	oveq2d 6666	. . 3 |- ( x = A -> ( 1 / ( cos ` x ) ) = ( 1 / ( cos ` A ) ) )
6		df-sec 42485	. . 3 |- sec = ( x e. { y e. CC | ( cos ` y ) =/= 0 } |-> ( 1 / ( cos ` x ) ) )
7		ovex 6678	. . 3 |- ( 1 / ( cos ` A ) ) e. _V
8	5, 6, 7	fvmpt 6282	. 2 |- ( A e. { y e. CC | ( cos ` y ) =/= 0 } -> ( sec ` A ) = ( 1 / ( cos ` A ) ) )
9	3, 8	sylbir 225	1 |- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( sec ` A ) = ( 1 / ( cos ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   / cdiv 10684   cosccos 14795   seccsec 42482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-sec 42485

This theorem is referenced by:  seccl  42491  reseccl  42494  recsec  42497  sec0  42501  onetansqsecsq  42502