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Theorem cotval 48363
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 6896 . . . 4 (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴))
21neeq1d 2989 . . 3 (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0))
32elrab 3679 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0))
4 fveq2 6896 . . . 4 (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴))
5 fveq2 6896 . . . 4 (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴))
64, 5oveq12d 7437 . . 3 (𝑥 = 𝐴 → ((cos‘𝑥) / (sin‘𝑥)) = ((cos‘𝐴) / (sin‘𝐴)))
7 df-cot 48360 . . 3 cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
8 ovex 7452 . . 3 ((cos‘𝐴) / (sin‘𝐴)) ∈ V
96, 7, 8fvmpt 7004 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))
103, 9sylbir 234 1 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2929  {crab 3418  cfv 6549  (class class class)co 7419  cc 11138  0cc0 11140   / cdiv 11903  sincsin 16043  cosccos 16044  cotccot 48357
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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-cot 48360
This theorem is referenced by:  cotcl  48366  recotcl  48369  reccot  48372  rectan  48373  cotsqcscsq  48376
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