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Theorem sinhval-named 27827
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 27824. See sinhval 12684 for a theorem to convert this further. See sinh-conventional 27830 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  e.  CC  ->  (sinh `  A )  =  ( ( sin `  (
_i  x.  A )
)  /  _i ) )

Proof of Theorem sinhval-named
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 6030 . . . 4  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21fveq2d 5674 . . 3  |-  ( x  =  A  ->  ( sin `  ( _i  x.  x ) )  =  ( sin `  (
_i  x.  A )
) )
32oveq1d 6037 . 2  |-  ( x  =  A  ->  (
( sin `  (
_i  x.  x )
)  /  _i )  =  ( ( sin `  ( _i  x.  A
) )  /  _i ) )
4 df-sinh 27824 . 2  |- sinh  =  ( x  e.  CC  |->  ( ( sin `  (
_i  x.  x )
)  /  _i ) )
5 ovex 6047 . 2  |-  ( ( sin `  ( _i  x.  A ) )  /  _i )  e. 
_V
63, 4, 5fvmpt 5747 1  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  (
_i  x.  A )
)  /  _i ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5396  (class class class)co 6022   CCcc 8923   _ici 8927    x. cmul 8930    / cdiv 9611   sincsin 12595  sinhcsinh 27821
This theorem is referenced by:  sinh-conventional  27830  sinhpcosh  27831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-sinh 27824
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