Theorem cotval 42289

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.

Step	Hyp	Ref	Expression
1		fveq2 6103	. . . 4 ⊢ (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴))
2	1	neeq1d 2841	. . 3 ⊢ (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0))
3	2	elrab 3331	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0))
4		fveq2 6103	. . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴))
5		fveq2 6103	. . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴))
6	4, 5	oveq12d 6567	. . 3 ⊢ (𝑥 = 𝐴 → ((cos‘𝑥) / (sin‘𝑥)) = ((cos‘𝐴) / (sin‘𝐴)))
7		df-cot 42286	. . 3 ⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
8		ovex 6577	. . 3 ⊢ ((cos‘𝐴) / (sin‘𝐴)) ∈ V
9	6, 7, 8	fvmpt 6191	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))
10	3, 9	sylbir 224	1 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815   / cdiv 10563  sincsin 14633  cosccos 14634  cotccot 42283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-cot 42286

This theorem is referenced by:  cotcl  42292  recotcl  42295  reccot  42298  rectan  42299  cotsqcscsq  42302