Theorem cotval 42490

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 6191	. . . 4 ⊢ (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴))
2	1	neeq1d 2853	. . 3 ⊢ (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0))
3	2	elrab 3363	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0))
4		fveq2 6191	. . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴))
5		fveq2 6191	. . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴))
6	4, 5	oveq12d 6668	. . 3 ⊢ (𝑥 = 𝐴 → ((cos‘𝑥) / (sin‘𝑥)) = ((cos‘𝐴) / (sin‘𝐴)))
7		df-cot 42487	. . 3 ⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
8		ovex 6678	. . 3 ⊢ ((cos‘𝐴) / (sin‘𝐴)) ∈ V
9	6, 7, 8	fvmpt 6282	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))
10	3, 9	sylbir 225	1 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {crab 2916  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  0cc0 9936   / cdiv 10684  sincsin 14794  cosccos 14795  cotccot 42484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-cot 42487

This theorem is referenced by:  cotcl  42493  recotcl  42496  reccot  42499  rectan  42500  cotsqcscsq  42503