Theorem coshval-named 47944

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 47941. See coshval 16105 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 7420	. . 3 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6895	. 2 ⊢ (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴)))
3		df-cosh 47941	. 2 ⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
4		fvex 6904	. 2 ⊢ (cos‘(i · 𝐴)) ∈ V
5	2, 3, 4	fvmpt 6998	1 ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ‘cfv 6543  (class class class)co 7412  ℂcc 11114  ici 11118   · cmul 11121  cosccos 16015  coshccosh 47938

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-cosh 47941

This theorem is referenced by:  sinhpcosh  47947