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Theorem coshval-named 42277
 Description: Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 42274. See coshval 14724 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
Assertion
Ref Expression
coshval-named (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))

Proof of Theorem coshval-named
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . 3 (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
21fveq2d 6107 . 2 (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴)))
3 df-cosh 42274 . 2 cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
4 fvex 6113 . 2 (cos‘(i · 𝐴)) ∈ V
52, 3, 4fvmpt 6191 1 (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ici 9817   · cmul 9820  cosccos 14634  coshccosh 42271 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-cosh 42274 This theorem is referenced by:  sinhpcosh  42280
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