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Definition df-csc 45264
Description: Define the cosecant function. We define it this way for cmpt 5113, which requires the form (𝑥𝐴𝐵). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5113. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
df-csc csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-csc
StepHypRef Expression
1 ccsc 45261 . 2 class csc
2 vx . . 3 setvar 𝑥
3 vy . . . . . . 7 setvar 𝑦
43cv 1537 . . . . . 6 class 𝑦
5 csin 15413 . . . . . 6 class sin
64, 5cfv 6328 . . . . 5 class (sin‘𝑦)
7 cc0 10530 . . . . 5 class 0
86, 7wne 2990 . . . 4 wff (sin‘𝑦) ≠ 0
9 cc 10528 . . . 4 class
108, 3, 9crab 3113 . . 3 class {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0}
11 c1 10531 . . . 4 class 1
122cv 1537 . . . . 5 class 𝑥
1312, 5cfv 6328 . . . 4 class (sin‘𝑥)
14 cdiv 11290 . . . 4 class /
1511, 13, 14co 7139 . . 3 class (1 / (sin‘𝑥))
162, 10, 15cmpt 5113 . 2 class (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
171, 16wceq 1538 1 wff csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  cscval  45267
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