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Theorem secval 27907
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 5632 . . . 4  |-  ( y  =  A  ->  ( cos `  y )  =  ( cos `  A
) )
21neeq1d 2542 . . 3  |-  ( y  =  A  ->  (
( cos `  y
)  =/=  0  <->  ( cos `  A )  =/=  0 ) )
32elrab 3009 . 2  |-  ( A  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( cos `  A
)  =/=  0 ) )
4 fveq2 5632 . . . 4  |-  ( x  =  A  ->  ( cos `  x )  =  ( cos `  A
) )
54oveq2d 5997 . . 3  |-  ( x  =  A  ->  (
1  /  ( cos `  x ) )  =  ( 1  /  ( cos `  A ) ) )
6 df-sec 27904 . . 3  |-  sec  =  ( x  e.  { y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1  /  ( cos `  x ) ) )
7 ovex 6006 . . 3  |-  ( 1  /  ( cos `  A
) )  e.  _V
85, 6, 7fvmpt 5709 . 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 1647    e. wcel 1715    =/= wne 2529   {crab 2632   ` cfv 5358  (class class class)co 5981   CCcc 8882   0cc0 8884   1c1 8885    / cdiv 9570   cosccos 12554   seccsec 27901
This theorem is referenced by:  seccl  27910  reseccl  27913  recsec  27916  sec0  27920  onetansqsecsq  27921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-sec 27904
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