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Definition df-tan 15946
Description: Define the tangent function. We define it this way for cmpt 5186, 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 15940 . 2 class tan
2 vx . . 3 setvar π‘₯
3 ccos 15939 . . . . 5 class cos
43ccnv 5630 . . . 4 class β—‘cos
5 cc 11045 . . . . 5 class β„‚
6 cc0 11047 . . . . . 6 class 0
76csn 4584 . . . . 5 class {0}
85, 7cdif 3905 . . . 4 class (β„‚ βˆ– {0})
94, 8cima 5634 . . 3 class (β—‘cos β€œ (β„‚ βˆ– {0}))
102cv 1540 . . . . 5 class π‘₯
11 csin 15938 . . . . 5 class sin
1210, 11cfv 6493 . . . 4 class (sinβ€˜π‘₯)
1310, 3cfv 6493 . . . 4 class (cosβ€˜π‘₯)
14 cdiv 11808 . . . 4 class /
1512, 13, 14co 7353 . . 3 class ((sinβ€˜π‘₯) / (cosβ€˜π‘₯))
162, 9, 15cmpt 5186 . 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  16002  dvtan  36095
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