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Theorem cotval 28473
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 5541 . . . 4  |-  ( y  =  A  ->  ( sin `  y )  =  ( sin `  A
) )
21neeq1d 2472 . . 3  |-  ( y  =  A  ->  (
( sin `  y
)  =/=  0  <->  ( sin `  A )  =/=  0 ) )
32elrab 2936 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 } 
<->  ( A  e.  CC  /\  ( sin `  A
)  =/=  0 ) )
4 fveq2 5541 . . . 4  |-  ( x  =  A  ->  ( cos `  x )  =  ( cos `  A
) )
5 fveq2 5541 . . . 4  |-  ( x  =  A  ->  ( sin `  x )  =  ( sin `  A
) )
64, 5oveq12d 5892 . . 3  |-  ( x  =  A  ->  (
( cos `  x
)  /  ( sin `  x ) )  =  ( ( cos `  A
)  /  ( sin `  A ) ) )
7 df-cot 28470 . . 3  |-  cot  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x
)  /  ( sin `  x ) ) )
8 ovex 5899 . . 3  |-  ( ( cos `  A )  /  ( sin `  A
) )  e.  _V
96, 7, 8fvmpt 5618 . 2  |-  ( A  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  ->  ( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )
103, 9sylbir 204 1  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    / cdiv 9439   sincsin 12361   cosccos 12362   cotccot 28467
This theorem is referenced by:  cotcl  28476  recotcl  28479  reccot  28482  rectan  28483  cotsqcscsq  28486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-cot 28470
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