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Theorem cscval 28553
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 5730 . . . 4  |-  ( y  =  A  ->  ( sin `  y )  =  ( sin `  A
) )
21neeq1d 2616 . . 3  |-  ( y  =  A  ->  (
( sin `  y
)  =/=  0  <->  ( sin `  A )  =/=  0 ) )
32elrab 3094 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( sin `  A
)  =/=  0 ) )
4 fveq2 5730 . . . 4  |-  ( x  =  A  ->  ( sin `  x )  =  ( sin `  A
) )
54oveq2d 6099 . . 3  |-  ( x  =  A  ->  (
1  /  ( sin `  x ) )  =  ( 1  /  ( sin `  A ) ) )
6 df-csc 28550 . . 3  |-  csc  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1  /  ( sin `  x ) ) )
7 ovex 6108 . . 3  |-  ( 1  /  ( sin `  A
) )  e.  _V
85, 6, 7fvmpt 5808 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  ->  ( csc `  A
)  =  ( 1  /  ( sin `  A
) ) )
93, 8sylbir 206 1  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( csc `  A
)  =  ( 1  /  ( sin `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   1c1 8993    / cdiv 9679   sincsin 12668   cscccsc 28547
This theorem is referenced by:  csccl  28556  recsccl  28559  reccsc  28562  cotsqcscsq  28567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pr 4405
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-csc 28550
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