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Theorem sinhval-named 28416
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 28413. See sinhval 12747 for a theorem to convert this further. See sinh-conventional 28419 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 6081 . . . 4  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21fveq2d 5724 . . 3  |-  ( x  =  A  ->  ( sin `  ( _i  x.  x ) )  =  ( sin `  (
_i  x.  A )
) )
32oveq1d 6088 . 2  |-  ( x  =  A  ->  (
( sin `  (
_i  x.  x )
)  /  _i )  =  ( ( sin `  ( _i  x.  A
) )  /  _i ) )
4 df-sinh 28413 . 2  |- sinh  =  ( x  e.  CC  |->  ( ( sin `  (
_i  x.  x )
)  /  _i ) )
5 ovex 6098 . 2  |-  ( ( sin `  ( _i  x.  A ) )  /  _i )  e. 
_V
63, 4, 5fvmpt 5798 1  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  (
_i  x.  A )
)  /  _i ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   _ici 8984    x. cmul 8987    / cdiv 9669   sincsin 12658  sinhcsinh 28410
This theorem is referenced by:  sinh-conventional  28419  sinhpcosh  28420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-sinh 28413
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