Theorem coshval-named 50228

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 50225. See coshval 16117 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 7370	. . 3 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6840	. 2 ⊢ (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴)))
3		df-cosh 50225	. 2 ⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
4		fvex 6849	. 2 ⊢ (cos‘(i · 𝐴)) ∈ V
5	2, 3, 4	fvmpt 6943	1 ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6494  (class class class)co 7362  ℂcc 11031  ici 11035   · cmul 11038  cosccos 16024  coshccosh 50222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-cosh 50225

This theorem is referenced by:  sinhpcosh  50231