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Theorem cscval 47279
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 6843 . . . 4 (𝑦 = 𝐴 β†’ (sinβ€˜π‘¦) = (sinβ€˜π΄))
21neeq1d 3000 . . 3 (𝑦 = 𝐴 β†’ ((sinβ€˜π‘¦) β‰  0 ↔ (sinβ€˜π΄) β‰  0))
32elrab 3646 . 2 (𝐴 ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↔ (𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) β‰  0))
4 fveq2 6843 . . . 4 (π‘₯ = 𝐴 β†’ (sinβ€˜π‘₯) = (sinβ€˜π΄))
54oveq2d 7374 . . 3 (π‘₯ = 𝐴 β†’ (1 / (sinβ€˜π‘₯)) = (1 / (sinβ€˜π΄)))
6 df-csc 47276 . . 3 csc = (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↦ (1 / (sinβ€˜π‘₯)))
7 ovex 7391 . . 3 (1 / (sinβ€˜π΄)) ∈ V
85, 6, 7fvmpt 6949 . 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  {crab 3406  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  0cc0 11056  1c1 11057   / cdiv 11817  sincsin 15951  cscccsc 47273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-csc 47276
This theorem is referenced by:  csccl  47282  recsccl  47285  reccsc  47288  cotsqcscsq  47293
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