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Theorem sinhval-named 47993
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 47990. See sinhval 16096 for a theorem to convert this further. See sinh-conventional 47996 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.
StepHypRef Expression
1 oveq2 7410 . . . 4 (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
21fveq2d 6886 . . 3 (𝑥 = 𝐴 → (sin‘(i · 𝑥)) = (sin‘(i · 𝐴)))
32oveq1d 7417 . 2 (𝑥 = 𝐴 → ((sin‘(i · 𝑥)) / i) = ((sin‘(i · 𝐴)) / i))
4 df-sinh 47990 . 2 sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
5 ovex 7435 . 2 ((sin‘(i · 𝐴)) / i) ∈ V
63, 4, 5fvmpt 6989 1 (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6534  (class class class)co 7402  cc 11105  ici 11109   · cmul 11112   / cdiv 11869  sincsin 16005  sinhcsinh 47987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-sinh 47990
This theorem is referenced by:  sinh-conventional  47996  sinhpcosh  47997
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