Theorem cscval 49734

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 6826	. . . 4 ⊢ (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴))
2	1	neeq1d 2984	. . 3 ⊢ (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0))
3	2	elrab 3650	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0))
4		fveq2 6826	. . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴))
5	4	oveq2d 7369	. . 3 ⊢ (𝑥 = 𝐴 → (1 / (sin‘𝑥)) = (1 / (sin‘𝐴)))
6		df-csc 49731	. . 3 ⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
7		ovex 7386	. . 3 ⊢ (1 / (sin‘𝐴)) ∈ V
8	5, 6, 7	fvmpt 6934	. 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 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3396  ‘cfv 6486  (class class class)co 7353  ℂcc 11026  0cc0 11028  1c1 11029   / cdiv 11795  sincsin 15988  cscccsc 49728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-csc 49731

This theorem is referenced by:  csccl  49737  recsccl  49740  reccsc  49743  cotsqcscsq  49748