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Definition df-tan 15420
 Description: Define the tangent function. We define it this way for cmpt 5111, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (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 15414 . 2 class tan
2 vx . . 3 setvar 𝑥
3 ccos 15413 . . . . 5 class cos
43ccnv 5519 . . . 4 class cos
5 cc 10527 . . . . 5 class
6 cc0 10529 . . . . . 6 class 0
76csn 4525 . . . . 5 class {0}
85, 7cdif 3878 . . . 4 class (ℂ ∖ {0})
94, 8cima 5523 . . 3 class (cos “ (ℂ ∖ {0}))
102cv 1537 . . . . 5 class 𝑥
11 csin 15412 . . . . 5 class sin
1210, 11cfv 6325 . . . 4 class (sin‘𝑥)
1310, 3cfv 6325 . . . 4 class (cos‘𝑥)
14 cdiv 11289 . . . 4 class /
1512, 13, 14co 7136 . . 3 class ((sin‘𝑥) / (cos‘𝑥))
162, 9, 15cmpt 5111 . 2 class (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
171, 16wceq 1538 1 wff tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
 Colors of variables: wff setvar class This definition is referenced by:  tanval  15476  dvtan  35126
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