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Definition df-tan 11365
Description: Define the tangent function. We define it this way for cmpt 3989, which requires the form (𝑥𝐴𝐵). (Contributed by Mario Carneiro, 14-Mar-2014.)
Assertion
Ref Expression
df-tan tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))

Detailed syntax breakdown of Definition df-tan
StepHypRef Expression
1 ctan 11359 . 2 class tan
2 vx . . 3 setvar 𝑥
3 ccos 11358 . . . . 5 class cos
43ccnv 4538 . . . 4 class cos
5 cc 7625 . . . . 5 class
6 cc0 7627 . . . . . 6 class 0
76csn 3527 . . . . 5 class {0}
85, 7cdif 3068 . . . 4 class (ℂ ∖ {0})
94, 8cima 4542 . . 3 class (cos “ (ℂ ∖ {0}))
102cv 1330 . . . . 5 class 𝑥
11 csin 11357 . . . . 5 class sin
1210, 11cfv 5123 . . . 4 class (sin‘𝑥)
1310, 3cfv 5123 . . . 4 class (cos‘𝑥)
14 cdiv 8439 . . . 4 class /
1512, 13, 14co 5774 . . 3 class ((sin‘𝑥) / (cos‘𝑥))
162, 9, 15cmpt 3989 . 2 class (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
171, 16wceq 1331 1 wff tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
Colors of variables: wff set class
This definition is referenced by:  tanvalap  11421
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