Theorem coshval-named 49898

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 49895. See coshval 16071 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.

Step	Hyp	Ref	Expression
1		oveq2 7363	. . 3 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6835	. 2 ⊢ (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴)))
3		df-cosh 49895	. 2 ⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
4		fvex 6844	. 2 ⊢ (cos‘(i · 𝐴)) ∈ V
5	2, 3, 4	fvmpt 6938	1 ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6489  (class class class)co 7355  ℂcc 11015  ici 11019   · cmul 11022  cosccos 15978  coshccosh 49892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-cosh 49895

This theorem is referenced by:  sinhpcosh  49901