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Definition df-cot 42815
Description: Define the cotangent function. We define it this way for cmpt 4762, which requires the form (𝑥𝐴𝐵). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
df-cot cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-cot
StepHypRef Expression
1 ccot 42812 . 2 class cot
2 vx . . 3 setvar 𝑥
3 vy . . . . . . 7 setvar 𝑦
43cv 1522 . . . . . 6 class 𝑦
5 csin 14838 . . . . . 6 class sin
64, 5cfv 5926 . . . . 5 class (sin‘𝑦)
7 cc0 9974 . . . . 5 class 0
86, 7wne 2823 . . . 4 wff (sin‘𝑦) ≠ 0
9 cc 9972 . . . 4 class
108, 3, 9crab 2945 . . 3 class {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0}
112cv 1522 . . . . 5 class 𝑥
12 ccos 14839 . . . . 5 class cos
1311, 12cfv 5926 . . . 4 class (cos‘𝑥)
1411, 5cfv 5926 . . . 4 class (sin‘𝑥)
15 cdiv 10722 . . . 4 class /
1613, 14, 15co 6690 . . 3 class ((cos‘𝑥) / (sin‘𝑥))
172, 10, 16cmpt 4762 . 2 class (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
181, 17wceq 1523 1 wff cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  cotval  42818
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