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Theorem lfli 29859
Description: Property of a linear functional. (lnfnli 23543 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 29858 . . . 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 608 . . 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 977 . 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 6088 . . . . . . 7  |-  ( r  =  R  ->  (
r  .x.  x )  =  ( R  .x.  x ) )
1312oveq1d 6096 . . . . . 6  |-  ( r  =  R  ->  (
( r  .x.  x
)  .+  y )  =  ( ( R 
.x.  x )  .+  y ) )
1413fveq2d 5732 . . . . 5  |-  ( r  =  R  ->  ( G `  ( (
r  .x.  x )  .+  y ) )  =  ( G `  (
( R  .x.  x
)  .+  y )
) )
15 oveq1 6088 . . . . . 6  |-  ( r  =  R  ->  (
r  .X.  ( G `  x ) )  =  ( R  .X.  ( G `  x )
) )
1615oveq1d 6096 . . . . 5  |-  ( r  =  R  ->  (
( r  .X.  ( G `  x )
)  .+^  ( G `  y ) )  =  ( ( R  .X.  ( G `  x ) )  .+^  ( G `  y ) ) )
1714, 16eqeq12d 2450 . . . 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 6089 . . . . . . 7  |-  ( x  =  X  ->  ( R  .x.  x )  =  ( R  .x.  X
) )
1918oveq1d 6096 . . . . . 6  |-  ( x  =  X  ->  (
( R  .x.  x
)  .+  y )  =  ( ( R 
.x.  X )  .+  y ) )
2019fveq2d 5732 . . . . 5  |-  ( x  =  X  ->  ( G `  ( ( R  .x.  x )  .+  y ) )  =  ( G `  (
( R  .x.  X
)  .+  y )
) )
21 fveq2 5728 . . . . . . 7  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
2221oveq2d 6097 . . . . . 6  |-  ( x  =  X  ->  ( R  .X.  ( G `  x ) )  =  ( R  .X.  ( G `  X )
) )
2322oveq1d 6096 . . . . 5  |-  ( x  =  X  ->  (
( R  .X.  ( G `  x )
)  .+^  ( G `  y ) )  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  y ) ) )
2420, 23eqeq12d 2450 . . . 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 6089 . . . . . 6  |-  ( y  =  Y  ->  (
( R  .x.  X
)  .+  y )  =  ( ( R 
.x.  X )  .+  Y ) )
2625fveq2d 5732 . . . . 5  |-  ( y  =  Y  ->  ( G `  ( ( R  .x.  X )  .+  y ) )  =  ( G `  (
( R  .x.  X
)  .+  Y )
) )
27 fveq2 5728 . . . . . 6  |-  ( y  =  Y  ->  ( G `  y )  =  ( G `  Y ) )
2827oveq2d 6097 . . . . 5  |-  ( y  =  Y  ->  (
( R  .X.  ( G `  X )
)  .+^  ( G `  y ) )  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  Y ) ) )
2926, 28eqeq12d 2450 . . . 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 3061 . . 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 980 . 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 15 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 936    = wceq 1652    e. wcel 1725   A.wral 2705   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533  LFnlclfn 29855
This theorem is referenced by:  lfl0  29863  lfladd  29864  lflsub  29865  lflmul  29866  lflnegcl  29873  lflvscl  29875  lkrlss  29893  hdmapln1  32707
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-13 1727  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-pow 4377  ax-pr 4403  ax-un 4701
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-pw 3801  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-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-lfl 29856
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