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Theorem ellnfn 26494
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 5864 . . . . . 6  |-  ( t  =  T  ->  (
t `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( x  .h  y )  +h  z
) ) )
2 fveq1 5864 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
32oveq2d 6299 . . . . . . 7  |-  ( t  =  T  ->  (
x  x.  ( t `
 y ) )  =  ( x  x.  ( T `  y
) ) )
4 fveq1 5864 . . . . . . 7  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
53, 4oveq12d 6301 . . . . . 6  |-  ( t  =  T  ->  (
( x  x.  (
t `  y )
)  +  ( t `
 z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) )
61, 5eqeq12d 2489 . . . . 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 2903 . . . 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 2908 . . 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 26459 . . 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 3263 . 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 9572 . . . 4  |-  CC  e.  _V
12 ax-hilex 25608 . . . 4  |-  ~H  e.  _V
1311, 12elmap 7447 . . 3  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
1413anbi1i 695 . 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 249 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   -->wf 5583   ` cfv 5587  (class class class)co 6283    ^m cmap 7420   CCcc 9489    + caddc 9494    x. cmul 9496   ~Hchil 25528    +h cva 25529    .h csm 25530   LinFnclf 25563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-hilex 25608
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-map 7422  df-lnfn 26459
This theorem is referenced by:  lnfnf  26495  lnfnl  26542  bralnfn  26559  0lnfn  26596  cnlnadjlem2  26679
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