Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  secval Structured version   Unicode version

Theorem secval 28564
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.
StepHypRef Expression
1 fveq2 5731 . . . 4  |-  ( y  =  A  ->  ( cos `  y )  =  ( cos `  A
) )
21neeq1d 2616 . . 3  |-  ( y  =  A  ->  (
( cos `  y
)  =/=  0  <->  ( cos `  A )  =/=  0 ) )
32elrab 3094 . 2  |-  ( A  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( cos `  A
)  =/=  0 ) )
4 fveq2 5731 . . . 4  |-  ( x  =  A  ->  ( cos `  x )  =  ( cos `  A
) )
54oveq2d 6100 . . 3  |-  ( x  =  A  ->  (
1  /  ( cos `  x ) )  =  ( 1  /  ( cos `  A ) ) )
6 df-sec 28561 . . 3  |-  sec  =  ( x  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1  /  ( cos `  x ) ) )
7 ovex 6109 . . 3  |-  ( 1  /  ( cos `  A
) )  e.  _V
85, 6, 7fvmpt 5809 . 2  |-  ( A  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 }  ->  ( sec `  A
)  =  ( 1  /  ( cos `  A
) ) )
93, 8sylbir 206 1  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sec `  A
)  =  ( 1  /  ( cos `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996    / cdiv 9682   cosccos 12672   seccsec 28558
This theorem is referenced by:  seccl  28567  reseccl  28570  recsec  28573  sec0  28577  onetansqsecsq  28578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-sec 28561
  Copyright terms: Public domain W3C validator