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Theorem cscval 44154
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 6493 . . . 4 (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴))
21neeq1d 3020 . . 3 (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0))
32elrab 3589 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0))
4 fveq2 6493 . . . 4 (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴))
54oveq2d 6986 . . 3 (𝑥 = 𝐴 → (1 / (sin‘𝑥)) = (1 / (sin‘𝐴)))
6 df-csc 44151 . . 3 csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
7 ovex 7002 . . 3 (1 / (sin‘𝐴)) ∈ V
85, 6, 7fvmpt 6589 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (csc‘𝐴) = (1 / (sin‘𝐴)))
93, 8sylbir 227 1 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  wne 2961  {crab 3086  cfv 6182  (class class class)co 6970  cc 10325  0cc0 10327  1c1 10328   / cdiv 11090  sincsin 15267  cscccsc 44148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-iota 6146  df-fun 6184  df-fv 6190  df-ov 6973  df-csc 44151
This theorem is referenced by:  csccl  44157  recsccl  44160  reccsc  44163  cotsqcscsq  44168
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