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Theorem cscval 30924
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 5688 . . . 4  |-  ( y  =  A  ->  ( sin `  y )  =  ( sin `  A
) )
21neeq1d 2619 . . 3  |-  ( y  =  A  ->  (
( sin `  y
)  =/=  0  <->  ( sin `  A )  =/=  0 ) )
32elrab 3114 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( sin `  A
)  =/=  0 ) )
4 fveq2 5688 . . . 4  |-  ( x  =  A  ->  ( sin `  x )  =  ( sin `  A
) )
54oveq2d 6106 . . 3  |-  ( x  =  A  ->  (
1  /  ( sin `  x ) )  =  ( 1  /  ( sin `  A ) ) )
6 df-csc 30921 . . 3  |-  csc  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1  /  ( sin `  x ) ) )
7 ovex 6115 . . 3  |-  ( 1  /  ( sin `  A
) )  e.  _V
85, 6, 7fvmpt 5771 . 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 369    = wceq 1364    e. wcel 1761    =/= wne 2604   {crab 2717   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    / cdiv 9989   sincsin 13345   cscccsc 30918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-csc 30921
This theorem is referenced by:  csccl  30927  recsccl  30930  reccsc  30933  cotsqcscsq  30938
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