Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cscval Structured version   Unicode version

Theorem cscval 31088
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 5696 . . . 4  |-  ( y  =  A  ->  ( sin `  y )  =  ( sin `  A
) )
21neeq1d 2626 . . 3  |-  ( y  =  A  ->  (
( sin `  y
)  =/=  0  <->  ( sin `  A )  =/=  0 ) )
32elrab 3122 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( sin `  A
)  =/=  0 ) )
4 fveq2 5696 . . . 4  |-  ( x  =  A  ->  ( sin `  x )  =  ( sin `  A
) )
54oveq2d 6112 . . 3  |-  ( x  =  A  ->  (
1  /  ( sin `  x ) )  =  ( 1  /  ( sin `  A ) ) )
6 df-csc 31085 . . 3  |-  csc  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1  /  ( sin `  x ) ) )
7 ovex 6121 . . 3  |-  ( 1  /  ( sin `  A
) )  e.  _V
85, 6, 7fvmpt 5779 . 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 1369    e. wcel 1756    =/= wne 2611   {crab 2724   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288    / cdiv 9998   sincsin 13354   cscccsc 31082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-csc 31085
This theorem is referenced by:  csccl  31091  recsccl  31094  reccsc  31097  cotsqcscsq  31102
  Copyright terms: Public domain W3C validator