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Definition df-cot 28216
Description: Define the cotangent function. We define it this way for cmpt 4077, which requires the form  ( x  e.  A  |->  B ). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
df-cot  |-  cot  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x
)  /  ( sin `  x ) ) )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-cot
StepHypRef Expression
1 ccot 28213 . 2  class  cot
2 vx . . 3  set  x
3 vy . . . . . . 7  set  y
43cv 1622 . . . . . 6  class  y
5 csin 12345 . . . . . 6  class  sin
64, 5cfv 5255 . . . . 5  class  ( sin `  y )
7 cc0 8737 . . . . 5  class  0
86, 7wne 2446 . . . 4  wff  ( sin `  y )  =/=  0
9 cc 8735 . . . 4  class  CC
108, 3, 9crab 2547 . . 3  class  { y  e.  CC  |  ( sin `  y )  =/=  0 }
112cv 1622 . . . . 5  class  x
12 ccos 12346 . . . . 5  class  cos
1311, 12cfv 5255 . . . 4  class  ( cos `  x )
1411, 5cfv 5255 . . . 4  class  ( sin `  x )
15 cdiv 9423 . . . 4  class  /
1613, 14, 15co 5858 . . 3  class  ( ( cos `  x )  /  ( sin `  x
) )
172, 10, 16cmpt 4077 . 2  class  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x )  /  ( sin `  x ) ) )
181, 17wceq 1623 1  wff  cot  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x
)  /  ( sin `  x ) ) )
Colors of variables: wff set class
This definition is referenced by:  cotval  28219
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