Theorem secval 50784

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 6877	. . . 4 ⊢ (𝑦 = 𝐴 → (cos‘𝑦) = (cos‘𝐴))
2	1	neeq1d 3015	. . 3 ⊢ (𝑦 = 𝐴 → ((cos‘𝑦) ≠ 0 ↔ (cos‘𝐴) ≠ 0))
3	2	elrab 3645	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0))
4		fveq2 6877	. . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴))
5	4	oveq2d 7428	. . 3 ⊢ (𝑥 = 𝐴 → (1 / (cos‘𝑥)) = (1 / (cos‘𝐴)))
6		df-sec 50781	. . 3 ⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
7		ovex 7445	. . 3 ⊢ (1 / (cos‘𝐴)) ∈ V
8	5, 6, 7	fvmpt 6985	. 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} → (sec‘𝐴) = (1 / (cos‘𝐴)))
9	3, 8	sylbir 238	1 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) = (1 / (cos‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  {crab 3413  ‘cfv 6531  (class class class)co 7412  ℂcc 11179  0cc0 11181  1c1 11182   / cdiv 11954  cosccos 16210  seccsec 50778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-sec 50781

This theorem is used by:  seccl  50787  reseccl  50790  recsec  50793  sec0  50797  onetansqsecsq  50798  dvsec  50800