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Theorem islfld 29797
Description: Properties that determine a linear functional. TODO: use this in place of islfl 29795 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
islfld.v  |-  ( ph  ->  V  =  ( Base `  W ) )
islfld.a  |-  ( ph  ->  .+  =  ( +g  `  W ) )
islfld.d  |-  ( ph  ->  D  =  (Scalar `  W ) )
islfld.s  |-  ( ph  ->  .x.  =  ( .s
`  W ) )
islfld.k  |-  ( ph  ->  K  =  ( Base `  D ) )
islfld.p  |-  ( ph  -> 
.+^  =  ( +g  `  D ) )
islfld.t  |-  ( ph  ->  .X.  =  ( .r
`  D ) )
islfld.f  |-  ( ph  ->  F  =  (LFnl `  W ) )
islfld.u  |-  ( ph  ->  G : V --> K )
islfld.l  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( G `  (
( r  .x.  x
)  .+  y )
)  =  ( ( r  .X.  ( G `  x ) )  .+^  ( G `  y ) ) )
islfld.w  |-  ( ph  ->  W  e.  X )
Assertion
Ref Expression
islfld  |-  ( ph  ->  G  e.  F )
Distinct variable groups:    x, r,
y, G    K, r, x, y    x, V, y    W, r, x, y    ph, r, x, y
Allowed substitution hints:    D( x, y, r)    .+ ( x, y, r)    .+^ (
x, y, r)    .x. ( x, y, r)    .X. ( x, y, r)    F( x, y, r)    V( r)    X( x, y, r)

Proof of Theorem islfld
StepHypRef Expression
1 islfld.w . . 3  |-  ( ph  ->  W  e.  X )
2 islfld.u . . . 4  |-  ( ph  ->  G : V --> K )
3 islfld.v . . . . 5  |-  ( ph  ->  V  =  ( Base `  W ) )
4 islfld.k . . . . . 6  |-  ( ph  ->  K  =  ( Base `  D ) )
5 islfld.d . . . . . . 7  |-  ( ph  ->  D  =  (Scalar `  W ) )
65fveq2d 5724 . . . . . 6  |-  ( ph  ->  ( Base `  D
)  =  ( Base `  (Scalar `  W )
) )
74, 6eqtrd 2467 . . . . 5  |-  ( ph  ->  K  =  ( Base `  (Scalar `  W )
) )
83, 7feq23d 5580 . . . 4  |-  ( ph  ->  ( G : V --> K 
<->  G : ( Base `  W ) --> ( Base `  (Scalar `  W )
) ) )
92, 8mpbid 202 . . 3  |-  ( ph  ->  G : ( Base `  W ) --> ( Base `  (Scalar `  W )
) )
10 islfld.l . . . . 5  |-  ( (
ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  -> 
( G `  (
( r  .x.  x
)  .+  y )
)  =  ( ( r  .X.  ( G `  x ) )  .+^  ( G `  y ) ) )
1110ralrimivvva 2791 . . . 4  |-  ( ph  ->  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( ( r  .x.  x ) 
.+  y ) )  =  ( ( r 
.X.  ( G `  x ) )  .+^  ( G `  y ) ) )
12 islfld.a . . . . . . . . . 10  |-  ( ph  ->  .+  =  ( +g  `  W ) )
13 islfld.s . . . . . . . . . . 11  |-  ( ph  ->  .x.  =  ( .s
`  W ) )
1413oveqd 6090 . . . . . . . . . 10  |-  ( ph  ->  ( r  .x.  x
)  =  ( r ( .s `  W
) x ) )
15 eqidd 2436 . . . . . . . . . 10  |-  ( ph  ->  y  =  y )
1612, 14, 15oveq123d 6094 . . . . . . . . 9  |-  ( ph  ->  ( ( r  .x.  x )  .+  y
)  =  ( ( r ( .s `  W ) x ) ( +g  `  W
) y ) )
1716fveq2d 5724 . . . . . . . 8  |-  ( ph  ->  ( G `  (
( r  .x.  x
)  .+  y )
)  =  ( G `
 ( ( r ( .s `  W
) x ) ( +g  `  W ) y ) ) )
18 islfld.p . . . . . . . . . 10  |-  ( ph  -> 
.+^  =  ( +g  `  D ) )
195fveq2d 5724 . . . . . . . . . 10  |-  ( ph  ->  ( +g  `  D
)  =  ( +g  `  (Scalar `  W )
) )
2018, 19eqtrd 2467 . . . . . . . . 9  |-  ( ph  -> 
.+^  =  ( +g  `  (Scalar `  W )
) )
21 islfld.t . . . . . . . . . . 11  |-  ( ph  ->  .X.  =  ( .r
`  D ) )
225fveq2d 5724 . . . . . . . . . . 11  |-  ( ph  ->  ( .r `  D
)  =  ( .r
`  (Scalar `  W )
) )
2321, 22eqtrd 2467 . . . . . . . . . 10  |-  ( ph  ->  .X.  =  ( .r
`  (Scalar `  W )
) )
2423oveqd 6090 . . . . . . . . 9  |-  ( ph  ->  ( r  .X.  ( G `  x )
)  =  ( r ( .r `  (Scalar `  W ) ) ( G `  x ) ) )
25 eqidd 2436 . . . . . . . . 9  |-  ( ph  ->  ( G `  y
)  =  ( G `
 y ) )
2620, 24, 25oveq123d 6094 . . . . . . . 8  |-  ( ph  ->  ( ( r  .X.  ( G `  x ) )  .+^  ( G `  y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x ) ) ( +g  `  (Scalar `  W ) ) ( G `  y ) ) )
2717, 26eqeq12d 2449 . . . . . . 7  |-  ( ph  ->  ( ( G `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  ( G `  x )
)  .+^  ( G `  y ) )  <->  ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x
) ) ( +g  `  (Scalar `  W )
) ( G `  y ) ) ) )
283, 27raleqbidv 2908 . . . . . 6  |-  ( ph  ->  ( A. y  e.  V  ( G `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  ( G `  x )
)  .+^  ( G `  y ) )  <->  A. y  e.  ( Base `  W
) ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x
) ) ( +g  `  (Scalar `  W )
) ( G `  y ) ) ) )
293, 28raleqbidv 2908 . . . . 5  |-  ( ph  ->  ( A. x  e.  V  A. y  e.  V  ( G `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  ( G `  x )
)  .+^  ( G `  y ) )  <->  A. x  e.  ( Base `  W
) A. y  e.  ( Base `  W
) ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x
) ) ( +g  `  (Scalar `  W )
) ( G `  y ) ) ) )
307, 29raleqbidv 2908 . . . 4  |-  ( ph  ->  ( A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( ( r  .x.  x )  .+  y
) )  =  ( ( r  .X.  ( G `  x )
)  .+^  ( G `  y ) )  <->  A. r  e.  ( Base `  (Scalar `  W ) ) A. x  e.  ( Base `  W ) A. y  e.  ( Base `  W
) ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x
) ) ( +g  `  (Scalar `  W )
) ( G `  y ) ) ) )
3111, 30mpbid 202 . . 3  |-  ( ph  ->  A. r  e.  (
Base `  (Scalar `  W
) ) A. x  e.  ( Base `  W
) A. y  e.  ( Base `  W
) ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x
) ) ( +g  `  (Scalar `  W )
) ( G `  y ) ) )
32 eqid 2435 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
33 eqid 2435 . . . . 5  |-  ( +g  `  W )  =  ( +g  `  W )
34 eqid 2435 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
35 eqid 2435 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
36 eqid 2435 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
37 eqid 2435 . . . . 5  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
38 eqid 2435 . . . . 5  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
39 eqid 2435 . . . . 5  |-  (LFnl `  W )  =  (LFnl `  W )
4032, 33, 34, 35, 36, 37, 38, 39islfl 29795 . . . 4  |-  ( W  e.  X  ->  ( G  e.  (LFnl `  W
)  <->  ( G :
( Base `  W ) --> ( Base `  (Scalar `  W
) )  /\  A. r  e.  ( Base `  (Scalar `  W )
) A. x  e.  ( Base `  W
) A. y  e.  ( Base `  W
) ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x
) ) ( +g  `  (Scalar `  W )
) ( G `  y ) ) ) ) )
4140biimpar 472 . . 3  |-  ( ( W  e.  X  /\  ( G : ( Base `  W ) --> ( Base `  (Scalar `  W )
)  /\  A. r  e.  ( Base `  (Scalar `  W ) ) A. x  e.  ( Base `  W ) A. y  e.  ( Base `  W
) ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x
) ) ( +g  `  (Scalar `  W )
) ( G `  y ) ) ) )  ->  G  e.  (LFnl `  W ) )
421, 9, 31, 41syl12anc 1182 . 2  |-  ( ph  ->  G  e.  (LFnl `  W ) )
43 islfld.f . 2  |-  ( ph  ->  F  =  (LFnl `  W ) )
4442, 43eleqtrrd 2512 1  |-  ( ph  ->  G  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   .rcmulr 13522  Scalarcsca 13524   .scvsca 13525  LFnlclfn 29792
This theorem is referenced by:  lflvscl  29812
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-lfl 29793
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