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Theorem lfli 29251
Description: Property of a linear functional. (lnfnli 22620 analog.) (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lflset.v  |-  V  =  ( Base `  W
)
lflset.a  |-  .+  =  ( +g  `  W )
lflset.d  |-  D  =  (Scalar `  W )
lflset.s  |-  .x.  =  ( .s `  W )
lflset.k  |-  K  =  ( Base `  D
)
lflset.p  |-  .+^  =  ( +g  `  D )
lflset.t  |-  .X.  =  ( .r `  D )
lflset.f  |-  F  =  (LFnl `  W )
Assertion
Ref Expression
lfli  |-  ( ( W  e.  Z  /\  G  e.  F  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V
) )  ->  ( G `  ( ( R  .x.  X )  .+  Y ) )  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  Y ) ) )

Proof of Theorem lfli
Dummy variables  r  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lflset.v . . . . 5  |-  V  =  ( Base `  W
)
2 lflset.a . . . . 5  |-  .+  =  ( +g  `  W )
3 lflset.d . . . . 5  |-  D  =  (Scalar `  W )
4 lflset.s . . . . 5  |-  .x.  =  ( .s `  W )
5 lflset.k . . . . 5  |-  K  =  ( Base `  D
)
6 lflset.p . . . . 5  |-  .+^  =  ( +g  `  D )
7 lflset.t . . . . 5  |-  .X.  =  ( .r `  D )
8 lflset.f . . . . 5  |-  F  =  (LFnl `  W )
91, 2, 3, 4, 5, 6, 7, 8islfl 29250 . . . 4  |-  ( W  e.  Z  ->  ( G  e.  F  <->  ( G : V --> K  /\  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( (
r  .x.  x )  .+  y ) )  =  ( ( r  .X.  ( G `  x ) )  .+^  ( G `  y ) ) ) ) )
109simplbda 607 . . 3  |-  ( ( W  e.  Z  /\  G  e.  F )  ->  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( ( r  .x.  x ) 
.+  y ) )  =  ( ( r 
.X.  ( G `  x ) )  .+^  ( G `  y ) ) )
11103adant3 975 . 2  |-  ( ( W  e.  Z  /\  G  e.  F  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V
) )  ->  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  ( G `  x )
)  .+^  ( G `  y ) ) )
12 oveq1 5865 . . . . . . 7  |-  ( r  =  R  ->  (
r  .x.  x )  =  ( R  .x.  x ) )
1312oveq1d 5873 . . . . . 6  |-  ( r  =  R  ->  (
( r  .x.  x
)  .+  y )  =  ( ( R 
.x.  x )  .+  y ) )
1413fveq2d 5529 . . . . 5  |-  ( r  =  R  ->  ( G `  ( (
r  .x.  x )  .+  y ) )  =  ( G `  (
( R  .x.  x
)  .+  y )
) )
15 oveq1 5865 . . . . . 6  |-  ( r  =  R  ->  (
r  .X.  ( G `  x ) )  =  ( R  .X.  ( G `  x )
) )
1615oveq1d 5873 . . . . 5  |-  ( r  =  R  ->  (
( r  .X.  ( G `  x )
)  .+^  ( G `  y ) )  =  ( ( R  .X.  ( G `  x ) )  .+^  ( G `  y ) ) )
1714, 16eqeq12d 2297 . . . 4  |-  ( r  =  R  ->  (
( G `  (
( r  .x.  x
)  .+  y )
)  =  ( ( r  .X.  ( G `  x ) )  .+^  ( G `  y ) )  <->  ( G `  ( ( R  .x.  x )  .+  y
) )  =  ( ( R  .X.  ( G `  x )
)  .+^  ( G `  y ) ) ) )
18 oveq2 5866 . . . . . . 7  |-  ( x  =  X  ->  ( R  .x.  x )  =  ( R  .x.  X
) )
1918oveq1d 5873 . . . . . 6  |-  ( x  =  X  ->  (
( R  .x.  x
)  .+  y )  =  ( ( R 
.x.  X )  .+  y ) )
2019fveq2d 5529 . . . . 5  |-  ( x  =  X  ->  ( G `  ( ( R  .x.  x )  .+  y ) )  =  ( G `  (
( R  .x.  X
)  .+  y )
) )
21 fveq2 5525 . . . . . . 7  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
2221oveq2d 5874 . . . . . 6  |-  ( x  =  X  ->  ( R  .X.  ( G `  x ) )  =  ( R  .X.  ( G `  X )
) )
2322oveq1d 5873 . . . . 5  |-  ( x  =  X  ->  (
( R  .X.  ( G `  x )
)  .+^  ( G `  y ) )  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  y ) ) )
2420, 23eqeq12d 2297 . . . 4  |-  ( x  =  X  ->  (
( G `  (
( R  .x.  x
)  .+  y )
)  =  ( ( R  .X.  ( G `  x ) )  .+^  ( G `  y ) )  <->  ( G `  ( ( R  .x.  X )  .+  y
) )  =  ( ( R  .X.  ( G `  X )
)  .+^  ( G `  y ) ) ) )
25 oveq2 5866 . . . . . 6  |-  ( y  =  Y  ->  (
( R  .x.  X
)  .+  y )  =  ( ( R 
.x.  X )  .+  Y ) )
2625fveq2d 5529 . . . . 5  |-  ( y  =  Y  ->  ( G `  ( ( R  .x.  X )  .+  y ) )  =  ( G `  (
( R  .x.  X
)  .+  Y )
) )
27 fveq2 5525 . . . . . 6  |-  ( y  =  Y  ->  ( G `  y )  =  ( G `  Y ) )
2827oveq2d 5874 . . . . 5  |-  ( y  =  Y  ->  (
( R  .X.  ( G `  X )
)  .+^  ( G `  y ) )  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  Y ) ) )
2926, 28eqeq12d 2297 . . . 4  |-  ( y  =  Y  ->  (
( G `  (
( R  .x.  X
)  .+  y )
)  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  y ) )  <->  ( G `  ( ( R  .x.  X )  .+  Y
) )  =  ( ( R  .X.  ( G `  X )
)  .+^  ( G `  Y ) ) ) )
3017, 24, 29rspc3v 2893 . . 3  |-  ( ( R  e.  K  /\  X  e.  V  /\  Y  e.  V )  ->  ( A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  ( G `  x )
)  .+^  ( G `  y ) )  -> 
( G `  (
( R  .x.  X
)  .+  Y )
)  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  Y ) ) ) )
31303ad2ant3 978 . 2  |-  ( ( W  e.  Z  /\  G  e.  F  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V
) )  ->  ( A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( ( r  .x.  x ) 
.+  y ) )  =  ( ( r 
.X.  ( G `  x ) )  .+^  ( G `  y ) )  ->  ( G `  ( ( R  .x.  X )  .+  Y
) )  =  ( ( R  .X.  ( G `  X )
)  .+^  ( G `  Y ) ) ) )
3211, 31mpd 14 1  |-  ( ( W  e.  Z  /\  G  e.  F  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V
) )  ->  ( G `  ( ( R  .x.  X )  .+  Y ) )  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  LFnlclfn 29247
This theorem is referenced by:  lfl0  29255  lfladd  29256  lflsub  29257  lflmul  29258  lflnegcl  29265  lflvscl  29267  lkrlss  29285  hdmapln1  32099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-lfl 29248
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