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Theorem cscval 43002
Description: Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
cscval ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴)))

Proof of Theorem cscval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6352 . . . 4 (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴))
21neeq1d 2991 . . 3 (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0))
32elrab 3504 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0))
4 fveq2 6352 . . . 4 (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴))
54oveq2d 6829 . . 3 (𝑥 = 𝐴 → (1 / (sin‘𝑥)) = (1 / (sin‘𝐴)))
6 df-csc 42999 . . 3 csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
7 ovex 6841 . . 3 (1 / (sin‘𝐴)) ∈ V
85, 6, 7fvmpt 6444 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (csc‘𝐴) = (1 / (sin‘𝐴)))
93, 8sylbir 225 1 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wne 2932  {crab 3054  cfv 6049  (class class class)co 6813  cc 10126  0cc0 10128  1c1 10129   / cdiv 10876  sincsin 14993  cscccsc 42996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-csc 42999
This theorem is referenced by:  csccl  43005  recsccl  43008  reccsc  43011  cotsqcscsq  43016
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