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Theorem coshval-named 48091
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 48088. See coshval 16123 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 7422 . . 3 (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
21fveq2d 6895 . 2 (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴)))
3 df-cosh 48088 . 2 cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
4 fvex 6904 . 2 (cos‘(i · 𝐴)) ∈ V
52, 3, 4fvmpt 6999 1 (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6542  (class class class)co 7414  cc 11128  ici 11132   · cmul 11135  cosccos 16032  coshccosh 48085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-cosh 48088
This theorem is referenced by:  sinhpcosh  48094
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