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Theorem cscval 38788
Description: Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
cscval  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( csc `  A
)  =  ( 1  /  ( sin `  A
) ) )

Proof of Theorem cscval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5849 . . . 4  |-  ( y  =  A  ->  ( sin `  y )  =  ( sin `  A
) )
21neeq1d 2680 . . 3  |-  ( y  =  A  ->  (
( sin `  y
)  =/=  0  <->  ( sin `  A )  =/=  0 ) )
32elrab 3207 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( sin `  A
)  =/=  0 ) )
4 fveq2 5849 . . . 4  |-  ( x  =  A  ->  ( sin `  x )  =  ( sin `  A
) )
54oveq2d 6294 . . 3  |-  ( x  =  A  ->  (
1  /  ( sin `  x ) )  =  ( 1  /  ( sin `  A ) ) )
6 df-csc 38785 . . 3  |-  csc  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1  /  ( sin `  x ) ) )
7 ovex 6306 . . 3  |-  ( 1  /  ( sin `  A
) )  e.  _V
85, 6, 7fvmpt 5932 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  ->  ( csc `  A
)  =  ( 1  /  ( sin `  A
) ) )
93, 8sylbir 213 1  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( csc `  A
)  =  ( 1  /  ( sin `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2758   ` cfv 5569  (class class class)co 6278   CCcc 9520   0cc0 9522   1c1 9523    / cdiv 10247   sincsin 14008   cscccsc 38782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-csc 38785
This theorem is referenced by:  csccl  38791  recsccl  38794  reccsc  38797  cotsqcscsq  38802
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