Theorem cscval 49104

Description: Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)

Assertion

Ref	Expression
cscval	⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴)))

Proof of Theorem cscval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6914	. . . 4 ⊢ (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴))
2	1	neeq1d 3000	. . 3 ⊢ (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0))
3	2	elrab 3698	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0))
4		fveq2 6914	. . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴))
5	4	oveq2d 7454	. . 3 ⊢ (𝑥 = 𝐴 → (1 / (sin‘𝑥)) = (1 / (sin‘𝐴)))
6		df-csc 49101	. . 3 ⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
7		ovex 7471	. . 3 ⊢ (1 / (sin‘𝐴)) ∈ V
8	5, 6, 7	fvmpt 7023	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (csc‘𝐴) = (1 / (sin‘𝐴)))
9	3, 8	sylbir 235	1 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2940  {crab 3436  ‘cfv 6569  (class class class)co 7438  ℂcc 11160  0cc0 11162  1c1 11163   / cdiv 11927  sincsin 16105  cscccsc 49098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-iota 6522  df-fun 6571  df-fv 6577  df-ov 7441  df-csc 49101

This theorem is referenced by:  csccl  49107  recsccl  49110  reccsc  49113  cotsqcscsq  49118