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Definition df-cot 47791
Description: Define the cotangent function. We define it this way for cmpt 5232, which requires the form (π‘₯ ∈ 𝐴 ↦ 𝐡). 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 5232. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
df-cot cot = (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↦ ((cosβ€˜π‘₯) / (sinβ€˜π‘₯)))
Distinct variable group:   π‘₯,𝑦

Detailed syntax breakdown of Definition df-cot
StepHypRef Expression
1 ccot 47788 . 2 class cot
2 vx . . 3 setvar π‘₯
3 vy . . . . . . 7 setvar 𝑦
43cv 1541 . . . . . 6 class 𝑦
5 csin 16007 . . . . . 6 class sin
64, 5cfv 6544 . . . . 5 class (sinβ€˜π‘¦)
7 cc0 11110 . . . . 5 class 0
86, 7wne 2941 . . . 4 wff (sinβ€˜π‘¦) β‰  0
9 cc 11108 . . . 4 class β„‚
108, 3, 9crab 3433 . . 3 class {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0}
112cv 1541 . . . . 5 class π‘₯
12 ccos 16008 . . . . 5 class cos
1311, 12cfv 6544 . . . 4 class (cosβ€˜π‘₯)
1411, 5cfv 6544 . . . 4 class (sinβ€˜π‘₯)
15 cdiv 11871 . . . 4 class /
1613, 14, 15co 7409 . . 3 class ((cosβ€˜π‘₯) / (sinβ€˜π‘₯))
172, 10, 16cmpt 5232 . 2 class (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↦ ((cosβ€˜π‘₯) / (sinβ€˜π‘₯)))
181, 17wceq 1542 1 wff cot = (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↦ ((cosβ€˜π‘₯) / (sinβ€˜π‘₯)))
Colors of variables: wff setvar class
This definition is referenced by:  cotval  47794
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