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Definition df-csc 47276
Description: Define the cosecant function. We define it this way for cmpt 5189, which requires the form (π‘₯ ∈ 𝐴 ↦ 𝐡). 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 5189. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
df-csc csc = (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↦ (1 / (sinβ€˜π‘₯)))
Distinct variable group:   π‘₯,𝑦

Detailed syntax breakdown of Definition df-csc
StepHypRef Expression
1 ccsc 47273 . 2 class csc
2 vx . . 3 setvar π‘₯
3 vy . . . . . . 7 setvar 𝑦
43cv 1541 . . . . . 6 class 𝑦
5 csin 15951 . . . . . 6 class sin
64, 5cfv 6497 . . . . 5 class (sinβ€˜π‘¦)
7 cc0 11056 . . . . 5 class 0
86, 7wne 2940 . . . 4 wff (sinβ€˜π‘¦) β‰  0
9 cc 11054 . . . 4 class β„‚
108, 3, 9crab 3406 . . 3 class {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0}
11 c1 11057 . . . 4 class 1
122cv 1541 . . . . 5 class π‘₯
1312, 5cfv 6497 . . . 4 class (sinβ€˜π‘₯)
14 cdiv 11817 . . . 4 class /
1511, 13, 14co 7358 . . 3 class (1 / (sinβ€˜π‘₯))
162, 10, 15cmpt 5189 . 2 class (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↦ (1 / (sinβ€˜π‘₯)))
171, 16wceq 1542 1 wff csc = (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↦ (1 / (sinβ€˜π‘₯)))
Colors of variables: wff setvar class
This definition is referenced by:  cscval  47279
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