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Definition df-tan 13685
Description: Define the tangent function. We define it this way for cmpt 4495, which requires the form  ( x  e.  A  |->  B ). (Contributed by Mario Carneiro, 14-Mar-2014.)
Assertion
Ref Expression
df-tan  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x
) ) )

Detailed syntax breakdown of Definition df-tan
StepHypRef Expression
1 ctan 13679 . 2  class  tan
2 vx . . 3  setvar  x
3 ccos 13678 . . . . 5  class  cos
43ccnv 4988 . . . 4  class  `' cos
5 cc 9493 . . . . 5  class  CC
6 cc0 9495 . . . . . 6  class  0
76csn 4014 . . . . 5  class  { 0 }
85, 7cdif 3458 . . . 4  class  ( CC 
\  { 0 } )
94, 8cima 4992 . . 3  class  ( `' cos " ( CC 
\  { 0 } ) )
102cv 1382 . . . . 5  class  x
11 csin 13677 . . . . 5  class  sin
1210, 11cfv 5578 . . . 4  class  ( sin `  x )
1310, 3cfv 5578 . . . 4  class  ( cos `  x )
14 cdiv 10212 . . . 4  class  /
1512, 13, 14co 6281 . . 3  class  ( ( sin `  x )  /  ( cos `  x
) )
162, 9, 15cmpt 4495 . 2  class  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  /  ( cos `  x ) ) )
171, 16wceq 1383 1  wff  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x
) ) )
Colors of variables: wff setvar class
This definition is referenced by:  tanval  13740  dvtan  30040
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