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Theorem sinhval-named 49310
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 49307. See sinhval 16191 for a theorem to convert this further. See sinh-conventional 49313 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 7440 . . . 4 (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
21fveq2d 6909 . . 3 (𝑥 = 𝐴 → (sin‘(i · 𝑥)) = (sin‘(i · 𝐴)))
32oveq1d 7447 . 2 (𝑥 = 𝐴 → ((sin‘(i · 𝑥)) / i) = ((sin‘(i · 𝐴)) / i))
4 df-sinh 49307 . 2 sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
5 ovex 7465 . 2 ((sin‘(i · 𝐴)) / i) ∈ V
63, 4, 5fvmpt 7015 1 (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cfv 6560  (class class class)co 7432  cc 11154  ici 11158   · cmul 11161   / cdiv 11921  sincsin 16100  sinhcsinh 49304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-sinh 49307
This theorem is referenced by:  sinh-conventional  49313  sinhpcosh  49314
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