Theorem sinhval-named 50829

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 50826. See sinhval 16322 for a theorem to convert this further. See sinh-conventional 50832 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 6889	. . 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 50826	. 2 ⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
5		ovex 7453	. 2 ⊢ ((sin‘(i · 𝐴)) / i) ∈ V
6	3, 4, 5	fvmpt 6993	1 ⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6538  (class class class)co 7420  ℂcc 11198  ici 11202   · cmul 11205   / cdiv 11973  sincsin 16229  sinhcsinh 50823

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 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-sinh 50826

This theorem is used by:  sinh-conventional  50832  sinhpcosh  50833