Theorem secval 43606

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

Assertion

Ref	Expression
secval	⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) = (1 / (cos‘𝐴)))

Proof of Theorem secval

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

Step	Hyp	Ref	Expression
1		fveq2 6448	. . . 4 ⊢ (𝑦 = 𝐴 → (cos‘𝑦) = (cos‘𝐴))
2	1	neeq1d 3028	. . 3 ⊢ (𝑦 = 𝐴 → ((cos‘𝑦) ≠ 0 ↔ (cos‘𝐴) ≠ 0))
3	2	elrab 3572	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0))
4		fveq2 6448	. . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴))
5	4	oveq2d 6940	. . 3 ⊢ (𝑥 = 𝐴 → (1 / (cos‘𝑥)) = (1 / (cos‘𝐴)))
6		df-sec 43603	. . 3 ⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
7		ovex 6956	. . 3 ⊢ (1 / (cos‘𝐴)) ∈ V
8	5, 6, 7	fvmpt 6544	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} → (sec‘𝐴) = (1 / (cos‘𝐴)))
9	3, 8	sylbir 227	1 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) = (1 / (cos‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  {crab 3094  ‘cfv 6137  (class class class)co 6924  ℂcc 10272  0cc0 10274  1c1 10275   / cdiv 11034  cosccos 15201  seccsec 43600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-sec 43603

This theorem is referenced by:  seccl  43609  reseccl  43612  recsec  43615  sec0  43619  onetansqsecsq  43620