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Theorem cscval 48179
Description: Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
cscval ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) β‰  0) β†’ (cscβ€˜π΄) = (1 / (sinβ€˜π΄)))

Proof of Theorem cscval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6897 . . . 4 (𝑦 = 𝐴 β†’ (sinβ€˜π‘¦) = (sinβ€˜π΄))
21neeq1d 2997 . . 3 (𝑦 = 𝐴 β†’ ((sinβ€˜π‘¦) β‰  0 ↔ (sinβ€˜π΄) β‰  0))
32elrab 3682 . 2 (𝐴 ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↔ (𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) β‰  0))
4 fveq2 6897 . . . 4 (π‘₯ = 𝐴 β†’ (sinβ€˜π‘₯) = (sinβ€˜π΄))
54oveq2d 7436 . . 3 (π‘₯ = 𝐴 β†’ (1 / (sinβ€˜π‘₯)) = (1 / (sinβ€˜π΄)))
6 df-csc 48176 . . 3 csc = (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↦ (1 / (sinβ€˜π‘₯)))
7 ovex 7453 . . 3 (1 / (sinβ€˜π΄)) ∈ V
85, 6, 7fvmpt 7005 . 2 (𝐴 ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} β†’ (cscβ€˜π΄) = (1 / (sinβ€˜π΄)))
93, 8sylbir 234 1 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) β‰  0) β†’ (cscβ€˜π΄) = (1 / (sinβ€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   β‰  wne 2937  {crab 3429  β€˜cfv 6548  (class class class)co 7420  β„‚cc 11137  0cc0 11139  1c1 11140   / cdiv 11902  sincsin 16040  cscccsc 48173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-csc 48176
This theorem is referenced by:  csccl  48182  recsccl  48185  reccsc  48188  cotsqcscsq  48193
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