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Theorem secval 28217
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 5525 . . . 4  |-  ( y  =  A  ->  ( cos `  y )  =  ( cos `  A
) )
21neeq1d 2459 . . 3  |-  ( y  =  A  ->  (
( cos `  y
)  =/=  0  <->  ( cos `  A )  =/=  0 ) )
32elrab 2923 . 2  |-  ( A  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( cos `  A
)  =/=  0 ) )
4 fveq2 5525 . . . 4  |-  ( x  =  A  ->  ( cos `  x )  =  ( cos `  A
) )
54oveq2d 5874 . . 3  |-  ( x  =  A  ->  (
1  /  ( cos `  x ) )  =  ( 1  /  ( cos `  A ) ) )
6 df-sec 28214 . . 3  |-  sec  =  ( x  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1  /  ( cos `  x ) ) )
7 ovex 5883 . . 3  |-  ( 1  /  ( cos `  A
) )  e.  _V
85, 6, 7fvmpt 5602 . 2  |-  ( A  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 }  ->  ( sec `  A
)  =  ( 1  /  ( cos `  A
) ) )
93, 8sylbir 204 1  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sec `  A
)  =  ( 1  /  ( cos `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    / cdiv 9423   cosccos 12346   seccsec 28211
This theorem is referenced by:  seccl  28220  reseccl  28223  recsec  28226  sec0  28230  onetansqsecsq  28231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-sec 28214
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