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Theorem secval 47278
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.
StepHypRef Expression
1 fveq2 6843 . . . 4 (𝑦 = 𝐴 → (cos‘𝑦) = (cos‘𝐴))
21neeq1d 3000 . . 3 (𝑦 = 𝐴 → ((cos‘𝑦) ≠ 0 ↔ (cos‘𝐴) ≠ 0))
32elrab 3646 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0))
4 fveq2 6843 . . . 4 (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴))
54oveq2d 7374 . . 3 (𝑥 = 𝐴 → (1 / (cos‘𝑥)) = (1 / (cos‘𝐴)))
6 df-sec 47275 . . 3 sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
7 ovex 7391 . . 3 (1 / (cos‘𝐴)) ∈ V
85, 6, 7fvmpt 6949 . 2 (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} → (sec‘𝐴) = (1 / (cos‘𝐴)))
93, 8sylbir 234 1 ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) = (1 / (cos‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  {crab 3406  cfv 6497  (class class class)co 7358  cc 11054  0cc0 11056  1c1 11057   / cdiv 11817  cosccos 15952  seccsec 47272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-sec 47275
This theorem is referenced by:  seccl  47281  reseccl  47284  recsec  47287  sec0  47291  onetansqsecsq  47292
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