Theorem coshval-named 50543

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 50540. See coshval 16215 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 7418	. . 3 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6885	. 2 ⊢ (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴)))
3		df-cosh 50540	. 2 ⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
4		fvex 6894	. 2 ⊢ (cos‘(i · 𝐴)) ∈ V
5	2, 3, 4	fvmpt 6989	1 ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  (class class class)co 7410  ℂcc 11102  ici 11106   · cmul 11109  cosccos 16122  coshccosh 50537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-cosh 50540

This theorem is used by:  sinhpcosh  50546