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Definition df-tan 15999
Description: Define the tangent function. We define it this way for cmpt 5225, which requires the form (π‘₯ ∈ 𝐴 ↦ 𝐡). (Contributed by Mario Carneiro, 14-Mar-2014.)
Assertion
Ref Expression
df-tan tan = (π‘₯ ∈ (β—‘cos β€œ (β„‚ βˆ– {0})) ↦ ((sinβ€˜π‘₯) / (cosβ€˜π‘₯)))

Detailed syntax breakdown of Definition df-tan
StepHypRef Expression
1 ctan 15993 . 2 class tan
2 vx . . 3 setvar π‘₯
3 ccos 15992 . . . . 5 class cos
43ccnv 5669 . . . 4 class β—‘cos
5 cc 11092 . . . . 5 class β„‚
6 cc0 11094 . . . . . 6 class 0
76csn 4623 . . . . 5 class {0}
85, 7cdif 3942 . . . 4 class (β„‚ βˆ– {0})
94, 8cima 5673 . . 3 class (β—‘cos β€œ (β„‚ βˆ– {0}))
102cv 1540 . . . . 5 class π‘₯
11 csin 15991 . . . . 5 class sin
1210, 11cfv 6533 . . . 4 class (sinβ€˜π‘₯)
1310, 3cfv 6533 . . . 4 class (cosβ€˜π‘₯)
14 cdiv 11855 . . . 4 class /
1512, 13, 14co 7394 . . 3 class ((sinβ€˜π‘₯) / (cosβ€˜π‘₯))
162, 9, 15cmpt 5225 . 2 class (π‘₯ ∈ (β—‘cos β€œ (β„‚ βˆ– {0})) ↦ ((sinβ€˜π‘₯) / (cosβ€˜π‘₯)))
171, 16wceq 1541 1 wff tan = (π‘₯ ∈ (β—‘cos β€œ (β„‚ βˆ– {0})) ↦ ((sinβ€˜π‘₯) / (cosβ€˜π‘₯)))
Colors of variables: wff setvar class
This definition is referenced by:  tanval  16055  dvtan  36406
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