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Theorem ellnfn 23391
Description: Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ellnfn  |-  ( T  e.  LinFn 
<->  ( T : ~H --> CC  /\  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) ) )
Distinct variable group:    x, y, z, T

Proof of Theorem ellnfn
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 fveq1 5730 . . . . . 6  |-  ( t  =  T  ->  (
t `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( x  .h  y )  +h  z
) ) )
2 fveq1 5730 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
32oveq2d 6100 . . . . . . 7  |-  ( t  =  T  ->  (
x  x.  ( t `
 y ) )  =  ( x  x.  ( T `  y
) ) )
4 fveq1 5730 . . . . . . 7  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
53, 4oveq12d 6102 . . . . . 6  |-  ( t  =  T  ->  (
( x  x.  (
t `  y )
)  +  ( t `
 z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) )
61, 5eqeq12d 2452 . . . . 5  |-  ( t  =  T  ->  (
( t `  (
( x  .h  y
)  +h  z ) )  =  ( ( x  x.  ( t `
 y ) )  +  ( t `  z ) )  <->  ( T `  ( ( x  .h  y )  +h  z
) )  =  ( ( x  x.  ( T `  y )
)  +  ( T `
 z ) ) ) )
76ralbidv 2727 . . . 4  |-  ( t  =  T  ->  ( A. z  e.  ~H  ( t `  (
( x  .h  y
)  +h  z ) )  =  ( ( x  x.  ( t `
 y ) )  +  ( t `  z ) )  <->  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z
) )  =  ( ( x  x.  ( T `  y )
)  +  ( T `
 z ) ) ) )
872ralbidv 2749 . . 3  |-  ( t  =  T  ->  ( A. x  e.  CC  A. y  e.  ~H  A. z  e.  ~H  (
t `  ( (
x  .h  y )  +h  z ) )  =  ( ( x  x.  ( t `  y ) )  +  ( t `  z
) )  <->  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) ) )
9 df-lnfn 23356 . . 3  |-  LinFn  =  {
t  e.  ( CC 
^m  ~H )  |  A. x  e.  CC  A. y  e.  ~H  A. z  e. 
~H  ( t `  ( ( x  .h  y )  +h  z
) )  =  ( ( x  x.  (
t `  y )
)  +  ( t `
 z ) ) }
108, 9elrab2 3096 . 2  |-  ( T  e.  LinFn 
<->  ( T  e.  ( CC  ^m  ~H )  /\  A. x  e.  CC  A. y  e.  ~H  A. z  e.  ~H  ( T `  ( (
x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) ) )
11 cnex 9076 . . . 4  |-  CC  e.  _V
12 ax-hilex 22507 . . . 4  |-  ~H  e.  _V
1311, 12elmap 7045 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
1413anbi1i 678 . 2  |-  ( ( T  e.  ( CC 
^m  ~H )  /\  A. x  e.  CC  A. y  e.  ~H  A. z  e. 
~H  ( T `  ( ( x  .h  y )  +h  z
) )  =  ( ( x  x.  ( T `  y )
)  +  ( T `
 z ) ) )  <->  ( T : ~H
--> CC  /\  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) ) )
1510, 14bitri 242 1  |-  ( T  e.  LinFn 
<->  ( T : ~H --> CC  /\  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^m cmap 7021   CCcc 8993    + caddc 8998    x. cmul 9000   ~Hchil 22427    +h cva 22428    .h csm 22429   LinFnclf 22462
This theorem is referenced by:  lnfnf  23392  lnfnl  23439  bralnfn  23456  0lnfn  23493  cnlnadjlem2  23576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-hilex 22507
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-lnfn 23356
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