Theorem sinhval-named 50591

Description: Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 50588. See sinhval 16237 for a theorem to convert this further. See sinh-conventional 50594 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)

Assertion

Ref	Expression
sinhval-named	⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i))

Proof of Theorem sinhval-named

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7428	. . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6890	. . 3 ⊢ (𝑥 = 𝐴 → (sin‘(i · 𝑥)) = (sin‘(i · 𝐴)))
3	2	oveq1d 7435	. 2 ⊢ (𝑥 = 𝐴 → ((sin‘(i · 𝑥)) / i) = ((sin‘(i · 𝐴)) / i))
4		df-sinh 50588	. 2 ⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
5		ovex 7453	. 2 ⊢ ((sin‘(i · 𝐴)) / i) ∈ V
6	3, 4, 5	fvmpt 6994	1 ⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ‘cfv 6541  (class class class)co 7420  ℂcc 11118  ici 11122   · cmul 11125   / cdiv 11891  sincsin 16144  sinhcsinh 50585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6497  df-fun 6543  df-fv 6549  df-ov 7423  df-sinh 50588

This theorem is used by:  sinh-conventional  50594  sinhpcosh  50595