Theorem sinhval-named 50727

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 50724. See sinhval 16267 for a theorem to convert this further. See sinh-conventional 50730 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 7424	. . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6885	. . 3 ⊢ (𝑥 = 𝐴 → (sin‘(i · 𝑥)) = (sin‘(i · 𝐴)))
3	2	oveq1d 7431	. 2 ⊢ (𝑥 = 𝐴 → ((sin‘(i · 𝑥)) / i) = ((sin‘(i · 𝐴)) / i))
4		df-sinh 50724	. 2 ⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
5		ovex 7449	. 2 ⊢ ((sin‘(i · 𝐴)) / i) ∈ V
6	3, 4, 5	fvmpt 6989	1 ⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6535  (class class class)co 7416  ℂcc 11147  ici 11151   · cmul 11154   / cdiv 11920  sincsin 16174  sinhcsinh 50721

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6491  df-fun 6537  df-fv 6543  df-ov 7419  df-sinh 50724

This theorem is used by:  sinh-conventional  50730  sinhpcosh  50731