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Theorem sinhval-named 47267
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 47264. See sinhval 16041 for a theorem to convert this further. See sinh-conventional 47270 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 ∈ β„‚ β†’ (sinhβ€˜π΄) = ((sinβ€˜(i Β· 𝐴)) / i))

Proof of Theorem sinhval-named
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 oveq2 7366 . . . 4 (π‘₯ = 𝐴 β†’ (i Β· π‘₯) = (i Β· 𝐴))
21fveq2d 6847 . . 3 (π‘₯ = 𝐴 β†’ (sinβ€˜(i Β· π‘₯)) = (sinβ€˜(i Β· 𝐴)))
32oveq1d 7373 . 2 (π‘₯ = 𝐴 β†’ ((sinβ€˜(i Β· π‘₯)) / i) = ((sinβ€˜(i Β· 𝐴)) / i))
4 df-sinh 47264 . 2 sinh = (π‘₯ ∈ β„‚ ↦ ((sinβ€˜(i Β· π‘₯)) / i))
5 ovex 7391 . 2 ((sinβ€˜(i Β· 𝐴)) / i) ∈ V
63, 4, 5fvmpt 6949 1 (𝐴 ∈ β„‚ β†’ (sinhβ€˜π΄) = ((sinβ€˜(i Β· 𝐴)) / i))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  ici 11058   Β· cmul 11061   / cdiv 11817  sincsin 15951  sinhcsinh 47261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-sinh 47264
This theorem is referenced by:  sinh-conventional  47270  sinhpcosh  47271
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