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Theorem cotval 43602
Description: Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
Assertion
Ref Expression
cotval ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))

Proof of Theorem cotval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6446 . . . 4 (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴))
21neeq1d 3028 . . 3 (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0))
32elrab 3572 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0))
4 fveq2 6446 . . . 4 (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴))
5 fveq2 6446 . . . 4 (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴))
64, 5oveq12d 6940 . . 3 (𝑥 = 𝐴 → ((cos‘𝑥) / (sin‘𝑥)) = ((cos‘𝐴) / (sin‘𝐴)))
7 df-cot 43599 . . 3 cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
8 ovex 6954 . . 3 ((cos‘𝐴) / (sin‘𝐴)) ∈ V
96, 7, 8fvmpt 6542 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))
103, 9sylbir 227 1 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wne 2969  {crab 3094  cfv 6135  (class class class)co 6922  cc 10270  0cc0 10272   / cdiv 11032  sincsin 15196  cosccos 15197  cotccot 43596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-cot 43599
This theorem is referenced by:  cotcl  43605  recotcl  43608  reccot  43611  rectan  43612  cotsqcscsq  43615
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