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Theorem secval 28204
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 5687 . . . 4  |-  ( y  =  A  ->  ( cos `  y )  =  ( cos `  A
) )
21neeq1d 2580 . . 3  |-  ( y  =  A  ->  (
( cos `  y
)  =/=  0  <->  ( cos `  A )  =/=  0 ) )
32elrab 3052 . 2  |-  ( A  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( cos `  A
)  =/=  0 ) )
4 fveq2 5687 . . . 4  |-  ( x  =  A  ->  ( cos `  x )  =  ( cos `  A
) )
54oveq2d 6056 . . 3  |-  ( x  =  A  ->  (
1  /  ( cos `  x ) )  =  ( 1  /  ( cos `  A ) ) )
6 df-sec 28201 . . 3  |-  sec  =  ( x  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1  /  ( cos `  x ) ) )
7 ovex 6065 . . 3  |-  ( 1  /  ( cos `  A
) )  e.  _V
85, 6, 7fvmpt 5765 . 2  |-  ( A  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 }  ->  ( sec `  A
)  =  ( 1  /  ( cos `  A
) ) )
93, 8sylbir 205 1  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sec `  A
)  =  ( 1  /  ( cos `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    / cdiv 9633   cosccos 12622   seccsec 28198
This theorem is referenced by:  seccl  28207  reseccl  28210  recsec  28213  sec0  28217  onetansqsecsq  28218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-sec 28201
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