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Theorem cotval 28419
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 5720 . . . 4  |-  ( y  =  A  ->  ( sin `  y )  =  ( sin `  A
) )
21neeq1d 2611 . . 3  |-  ( y  =  A  ->  (
( sin `  y
)  =/=  0  <->  ( sin `  A )  =/=  0 ) )
32elrab 3084 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( sin `  A
)  =/=  0 ) )
4 fveq2 5720 . . . 4  |-  ( x  =  A  ->  ( cos `  x )  =  ( cos `  A
) )
5 fveq2 5720 . . . 4  |-  ( x  =  A  ->  ( sin `  x )  =  ( sin `  A
) )
64, 5oveq12d 6091 . . 3  |-  ( x  =  A  ->  (
( cos `  x
)  /  ( sin `  x ) )  =  ( ( cos `  A
)  /  ( sin `  A ) ) )
7 df-cot 28416 . . 3  |-  cot  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x
)  /  ( sin `  x ) ) )
8 ovex 6098 . . 3  |-  ( ( cos `  A )  /  ( sin `  A
) )  e.  _V
96, 7, 8fvmpt 5798 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  ->  ( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )
103, 9sylbir 205 1  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982    / cdiv 9669   sincsin 12658   cosccos 12659   cotccot 28413
This theorem is referenced by:  cotcl  28422  recotcl  28425  reccot  28428  rectan  28429  cotsqcscsq  28432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-cot 28416
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