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Theorem cotval 32630
Description: Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
cotval  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )

Proof of Theorem cotval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . 4  |-  ( y  =  A  ->  ( sin `  y )  =  ( sin `  A
) )
21neeq1d 2744 . . 3  |-  ( y  =  A  ->  (
( sin `  y
)  =/=  0  <->  ( sin `  A )  =/=  0 ) )
32elrab 3266 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( sin `  A
)  =/=  0 ) )
4 fveq2 5872 . . . 4  |-  ( x  =  A  ->  ( cos `  x )  =  ( cos `  A
) )
5 fveq2 5872 . . . 4  |-  ( x  =  A  ->  ( sin `  x )  =  ( sin `  A
) )
64, 5oveq12d 6313 . . 3  |-  ( x  =  A  ->  (
( cos `  x
)  /  ( sin `  x ) )  =  ( ( cos `  A
)  /  ( sin `  A ) ) )
7 df-cot 32627 . . 3  |-  cot  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x
)  /  ( sin `  x ) ) )
8 ovex 6320 . . 3  |-  ( ( cos `  A )  /  ( sin `  A
) )  e.  _V
96, 7, 8fvmpt 5957 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  ->  ( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )
103, 9sylbir 213 1  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504    / cdiv 10218   sincsin 13677   cosccos 13678   cotccot 32624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-cot 32627
This theorem is referenced by:  cotcl  32633  recotcl  32636  reccot  32639  rectan  32640  cotsqcscsq  32643
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