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Theorem coshval-named 47268
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 47265. See coshval 16042 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 7366 . . 3 (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
21fveq2d 6847 . 2 (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴)))
3 df-cosh 47265 . 2 cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
4 fvex 6856 . 2 (cos‘(i · 𝐴)) ∈ V
52, 3, 4fvmpt 6949 1 (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6497  (class class class)co 7358  cc 11054  ici 11058   · cmul 11061  cosccos 15952  coshccosh 47262
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-cosh 47265
This theorem is referenced by:  sinhpcosh  47271
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