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Theorem cotval 48042
Description: Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
cotval ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) β‰  0) β†’ (cotβ€˜π΄) = ((cosβ€˜π΄) / (sinβ€˜π΄)))

Proof of Theorem cotval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6882 . . . 4 (𝑦 = 𝐴 β†’ (sinβ€˜π‘¦) = (sinβ€˜π΄))
21neeq1d 2992 . . 3 (𝑦 = 𝐴 β†’ ((sinβ€˜π‘¦) β‰  0 ↔ (sinβ€˜π΄) β‰  0))
32elrab 3676 . 2 (𝐴 ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↔ (𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) β‰  0))
4 fveq2 6882 . . . 4 (π‘₯ = 𝐴 β†’ (cosβ€˜π‘₯) = (cosβ€˜π΄))
5 fveq2 6882 . . . 4 (π‘₯ = 𝐴 β†’ (sinβ€˜π‘₯) = (sinβ€˜π΄))
64, 5oveq12d 7420 . . 3 (π‘₯ = 𝐴 β†’ ((cosβ€˜π‘₯) / (sinβ€˜π‘₯)) = ((cosβ€˜π΄) / (sinβ€˜π΄)))
7 df-cot 48039 . . 3 cot = (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} ↦ ((cosβ€˜π‘₯) / (sinβ€˜π‘₯)))
8 ovex 7435 . . 3 ((cosβ€˜π΄) / (sinβ€˜π΄)) ∈ V
96, 7, 8fvmpt 6989 . 2 (𝐴 ∈ {𝑦 ∈ β„‚ ∣ (sinβ€˜π‘¦) β‰  0} β†’ (cotβ€˜π΄) = ((cosβ€˜π΄) / (sinβ€˜π΄)))
103, 9sylbir 234 1 ((𝐴 ∈ β„‚ ∧ (sinβ€˜π΄) β‰  0) β†’ (cotβ€˜π΄) = ((cosβ€˜π΄) / (sinβ€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  {crab 3424  β€˜cfv 6534  (class class class)co 7402  β„‚cc 11105  0cc0 11107   / cdiv 11870  sincsin 16009  cosccos 16010  cotccot 48036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-cot 48039
This theorem is referenced by:  cotcl  48045  recotcl  48048  reccot  48051  rectan  48052  cotsqcscsq  48055
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