Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tanhval-named Structured version   Unicode version

Theorem tanhval-named 28482
Description: Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 28479. (Contributed by David A. Wheeler, 10-May-2015.)
Assertion
Ref Expression
tanhval-named  |-  ( A  e.  ( `'cosh " ( CC  \  { 0 } ) )  ->  (tanh `  A )  =  ( ( tan `  (
_i  x.  A )
)  /  _i ) )

Proof of Theorem tanhval-named
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 6090 . . . 4  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21fveq2d 5733 . . 3  |-  ( x  =  A  ->  ( tan `  ( _i  x.  x ) )  =  ( tan `  (
_i  x.  A )
) )
32oveq1d 6097 . 2  |-  ( x  =  A  ->  (
( tan `  (
_i  x.  x )
)  /  _i )  =  ( ( tan `  ( _i  x.  A
) )  /  _i ) )
4 df-tanh 28479 . 2  |- tanh  =  ( x  e.  ( `'cosh " ( CC  \  { 0 } ) )  |->  ( ( tan `  ( _i  x.  x
) )  /  _i ) )
5 ovex 6107 . 2  |-  ( ( tan `  ( _i  x.  A ) )  /  _i )  e. 
_V
63, 4, 5fvmpt 5807 1  |-  ( A  e.  ( `'cosh " ( CC  \  { 0 } ) )  ->  (tanh `  A )  =  ( ( tan `  (
_i  x.  A )
)  /  _i ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    \ cdif 3318   {csn 3815   `'ccnv 4878   "cima 4882   ` cfv 5455  (class class class)co 6082   CCcc 8989   0cc0 8991   _ici 8993    x. cmul 8996    / cdiv 9678   tanctan 12669  coshccosh 28475  tanhctanh 28476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-tanh 28479
  Copyright terms: Public domain W3C validator