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Theorem coshval-named 28480
Description: Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 28477. See coshval 12756 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
Assertion
Ref Expression
coshval-named  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )

Proof of Theorem coshval-named
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 6089 . . 3  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21fveq2d 5732 . 2  |-  ( x  =  A  ->  ( cos `  ( _i  x.  x ) )  =  ( cos `  (
_i  x.  A )
) )
3 df-cosh 28477 . 2  |- cosh  =  ( x  e.  CC  |->  ( cos `  ( _i  x.  x ) ) )
4 fvex 5742 . 2  |-  ( cos `  ( _i  x.  A
) )  e.  _V
52, 3, 4fvmpt 5806 1  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   _ici 8992    x. cmul 8995   cosccos 12667  coshccosh 28474
This theorem is referenced by:  sinhpcosh  28483
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-cosh 28477
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