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Theorem sinhval-named 48967
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 48964. See sinhval 16187 for a theorem to convert this further. See sinh-conventional 48970 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 7439 . . . 4 (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
21fveq2d 6911 . . 3 (𝑥 = 𝐴 → (sin‘(i · 𝑥)) = (sin‘(i · 𝐴)))
32oveq1d 7446 . 2 (𝑥 = 𝐴 → ((sin‘(i · 𝑥)) / i) = ((sin‘(i · 𝐴)) / i))
4 df-sinh 48964 . 2 sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
5 ovex 7464 . 2 ((sin‘(i · 𝐴)) / i) ∈ V
63, 4, 5fvmpt 7016 1 (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  cc 11151  ici 11155   · cmul 11158   / cdiv 11918  sincsin 16096  sinhcsinh 48961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-sinh 48964
This theorem is referenced by:  sinh-conventional  48970  sinhpcosh  48971
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