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Definition df-csc 28215
Description: Define the cosecant function. We define it this way for cmpt 4077, which requires the form  ( x  e.  A  |->  B ). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 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-csc  |-  csc  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1  /  ( sin `  x ) ) )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-csc
StepHypRef Expression
1 ccsc 28212 . 2  class  csc
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 }
11 c1 8738 . . . 4  class  1
122cv 1622 . . . . 5  class  x
1312, 5cfv 5255 . . . 4  class  ( sin `  x )
14 cdiv 9423 . . . 4  class  /
1511, 13, 14co 5858 . . 3  class  ( 1  /  ( sin `  x
) )
162, 10, 15cmpt 4077 . 2  class  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1  / 
( sin `  x
) ) )
171, 16wceq 1623 1  wff  csc  =  ( x  e.  { y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1  /  ( sin `  x ) ) )
Colors of variables: wff set class
This definition is referenced by:  cscval  28218
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