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Theorem cotval 30973
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 5686 . . . 4  |-  ( y  =  A  ->  ( sin `  y )  =  ( sin `  A
) )
21neeq1d 2616 . . 3  |-  ( y  =  A  ->  (
( sin `  y
)  =/=  0  <->  ( sin `  A )  =/=  0 ) )
32elrab 3112 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( sin `  A
)  =/=  0 ) )
4 fveq2 5686 . . . 4  |-  ( x  =  A  ->  ( cos `  x )  =  ( cos `  A
) )
5 fveq2 5686 . . . 4  |-  ( x  =  A  ->  ( sin `  x )  =  ( sin `  A
) )
64, 5oveq12d 6104 . . 3  |-  ( x  =  A  ->  (
( cos `  x
)  /  ( sin `  x ) )  =  ( ( cos `  A
)  /  ( sin `  A ) ) )
7 df-cot 30970 . . 3  |-  cot  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x
)  /  ( sin `  x ) ) )
8 ovex 6111 . . 3  |-  ( ( cos `  A )  /  ( sin `  A
) )  e.  _V
96, 7, 8fvmpt 5769 . 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 1369    e. wcel 1756    =/= wne 2601   {crab 2714   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274    / cdiv 9985   sincsin 13341   cosccos 13342   cotccot 30967
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 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-ov 6089  df-cot 30970
This theorem is referenced by:  cotcl  30976  recotcl  30979  reccot  30982  rectan  30983  cotsqcscsq  30986
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