Theorem cotval 46451

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 6774	. . . 4 ⊢ (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴))
2	1	neeq1d 3003	. . 3 ⊢ (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0))
3	2	elrab 3624	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0))
4		fveq2 6774	. . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴))
5		fveq2 6774	. . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴))
6	4, 5	oveq12d 7293	. . 3 ⊢ (𝑥 = 𝐴 → ((cos‘𝑥) / (sin‘𝑥)) = ((cos‘𝐴) / (sin‘𝐴)))
7		df-cot 46448	. . 3 ⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
8		ovex 7308	. . 3 ⊢ ((cos‘𝐴) / (sin‘𝐴)) ∈ V
9	6, 7, 8	fvmpt 6875	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))
10	3, 9	sylbir 234	1 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  0cc0 10871   / cdiv 11632  sincsin 15773  cosccos 15774  cotccot 46445

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-cot 46448

This theorem is referenced by:  cotcl  46454  recotcl  46457  reccot  46460  rectan  46461  cotsqcscsq  46464