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Theorem lclkrlem2m 31634
Description: Lemma for lclkr 31648. Construct a vector  B that makes the sum of functionals zero. Combine with  B  e.  V to shorten overall proof. (Contributed by NM, 17-Jan-2015.)
Hypotheses
Ref Expression
lclkrlem2m.v  |-  V  =  ( Base `  U
)
lclkrlem2m.t  |-  .x.  =  ( .s `  U )
lclkrlem2m.s  |-  S  =  (Scalar `  U )
lclkrlem2m.q  |-  .X.  =  ( .r `  S )
lclkrlem2m.z  |-  .0.  =  ( 0g `  S )
lclkrlem2m.i  |-  I  =  ( invr `  S
)
lclkrlem2m.m  |-  .-  =  ( -g `  U )
lclkrlem2m.f  |-  F  =  (LFnl `  U )
lclkrlem2m.d  |-  D  =  (LDual `  U )
lclkrlem2m.p  |-  .+  =  ( +g  `  D )
lclkrlem2m.x  |-  ( ph  ->  X  e.  V )
lclkrlem2m.y  |-  ( ph  ->  Y  e.  V )
lclkrlem2m.e  |-  ( ph  ->  E  e.  F )
lclkrlem2m.g  |-  ( ph  ->  G  e.  F )
lclkrlem2m.w  |-  ( ph  ->  U  e.  LVec )
lclkrlem2m.b  |-  B  =  ( X  .-  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )
lclkrlem2m.n  |-  ( ph  ->  ( ( E  .+  G ) `  Y
)  =/=  .0.  )
Assertion
Ref Expression
lclkrlem2m  |-  ( ph  ->  ( B  e.  V  /\  ( ( E  .+  G ) `  B
)  =  .0.  )
)

Proof of Theorem lclkrlem2m
StepHypRef Expression
1 lclkrlem2m.b . . 3  |-  B  =  ( X  .-  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )
2 lclkrlem2m.w . . . . . 6  |-  ( ph  ->  U  e.  LVec )
3 lveclmod 16105 . . . . . 6  |-  ( U  e.  LVec  ->  U  e. 
LMod )
42, 3syl 16 . . . . 5  |-  ( ph  ->  U  e.  LMod )
5 lmodgrp 15884 . . . . 5  |-  ( U  e.  LMod  ->  U  e. 
Grp )
64, 5syl 16 . . . 4  |-  ( ph  ->  U  e.  Grp )
7 lclkrlem2m.x . . . 4  |-  ( ph  ->  X  e.  V )
8 lclkrlem2m.s . . . . . . . 8  |-  S  =  (Scalar `  U )
98lmodrng 15885 . . . . . . 7  |-  ( U  e.  LMod  ->  S  e. 
Ring )
104, 9syl 16 . . . . . 6  |-  ( ph  ->  S  e.  Ring )
11 lclkrlem2m.f . . . . . . . 8  |-  F  =  (LFnl `  U )
12 lclkrlem2m.d . . . . . . . 8  |-  D  =  (LDual `  U )
13 lclkrlem2m.p . . . . . . . 8  |-  .+  =  ( +g  `  D )
14 lclkrlem2m.e . . . . . . . 8  |-  ( ph  ->  E  e.  F )
15 lclkrlem2m.g . . . . . . . 8  |-  ( ph  ->  G  e.  F )
1611, 12, 13, 4, 14, 15ldualvaddcl 29245 . . . . . . 7  |-  ( ph  ->  ( E  .+  G
)  e.  F )
17 eqid 2387 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
18 lclkrlem2m.v . . . . . . . 8  |-  V  =  ( Base `  U
)
198, 17, 18, 11lflcl 29179 . . . . . . 7  |-  ( ( U  e.  LVec  /\  ( E  .+  G )  e.  F  /\  X  e.  V )  ->  (
( E  .+  G
) `  X )  e.  ( Base `  S
) )
202, 16, 7, 19syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( E  .+  G ) `  X
)  e.  ( Base `  S ) )
218lvecdrng 16104 . . . . . . . 8  |-  ( U  e.  LVec  ->  S  e.  DivRing )
222, 21syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  DivRing )
23 lclkrlem2m.y . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
248, 17, 18, 11lflcl 29179 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  ( E  .+  G )  e.  F  /\  Y  e.  V )  ->  (
( E  .+  G
) `  Y )  e.  ( Base `  S
) )
252, 16, 23, 24syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( E  .+  G ) `  Y
)  e.  ( Base `  S ) )
26 lclkrlem2m.n . . . . . . 7  |-  ( ph  ->  ( ( E  .+  G ) `  Y
)  =/=  .0.  )
27 lclkrlem2m.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
28 lclkrlem2m.i . . . . . . . 8  |-  I  =  ( invr `  S
)
2917, 27, 28drnginvrcl 15779 . . . . . . 7  |-  ( ( S  e.  DivRing  /\  (
( E  .+  G
) `  Y )  e.  ( Base `  S
)  /\  ( ( E  .+  G ) `  Y )  =/=  .0.  )  ->  ( I `  ( ( E  .+  G ) `  Y
) )  e.  (
Base `  S )
)
3022, 25, 26, 29syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( I `  (
( E  .+  G
) `  Y )
)  e.  ( Base `  S ) )
31 lclkrlem2m.q . . . . . . 7  |-  .X.  =  ( .r `  S )
3217, 31rngcl 15604 . . . . . 6  |-  ( ( S  e.  Ring  /\  (
( E  .+  G
) `  X )  e.  ( Base `  S
)  /\  ( I `  ( ( E  .+  G ) `  Y
) )  e.  (
Base `  S )
)  ->  ( (
( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  e.  (
Base `  S )
)
3310, 20, 30, 32syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) )  e.  ( Base `  S
) )
34 lclkrlem2m.t . . . . . 6  |-  .x.  =  ( .s `  U )
3518, 8, 34, 17lmodvscl 15894 . . . . 5  |-  ( ( U  e.  LMod  /\  (
( ( E  .+  G ) `  X
)  .X.  ( I `  ( ( E  .+  G ) `  Y
) ) )  e.  ( Base `  S
)  /\  Y  e.  V )  ->  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y )  e.  V )
364, 33, 23, 35syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
)  e.  V )
37 lclkrlem2m.m . . . . 5  |-  .-  =  ( -g `  U )
3818, 37grpsubcl 14796 . . . 4  |-  ( ( U  e.  Grp  /\  X  e.  V  /\  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
)  e.  V )  ->  ( X  .-  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) )  e.  V
)
396, 7, 36, 38syl3anc 1184 . . 3  |-  ( ph  ->  ( X  .-  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  e.  V )
401, 39syl5eqel 2471 . 2  |-  ( ph  ->  B  e.  V )
411fveq2i 5671 . . 3  |-  ( ( E  .+  G ) `
 B )  =  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )
42 eqid 2387 . . . . . 6  |-  ( -g `  S )  =  (
-g `  S )
438, 42, 18, 37, 11lflsub 29182 . . . . 5  |-  ( ( U  e.  LMod  /\  ( E  .+  G )  e.  F  /\  ( X  e.  V  /\  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y )  e.  V ) )  -> 
( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) ) )
444, 16, 7, 36, 43syl112anc 1188 . . . 4  |-  ( ph  ->  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) ) )
458, 17, 31, 18, 34, 11lflmul 29183 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( E  .+  G )  e.  F  /\  ( ( ( ( E  .+  G ) `  X
)  .X.  ( I `  ( ( E  .+  G ) `  Y
) ) )  e.  ( Base `  S
)  /\  Y  e.  V ) )  -> 
( ( E  .+  G ) `  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  =  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
) )
464, 16, 33, 23, 45syl112anc 1188 . . . . . 6  |-  ( ph  ->  ( ( E  .+  G ) `  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  =  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
) )
4717, 31rngass 15607 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  (
( ( E  .+  G ) `  X
)  e.  ( Base `  S )  /\  (
I `  ( ( E  .+  G ) `  Y ) )  e.  ( Base `  S
)  /\  ( ( E  .+  G ) `  Y )  e.  (
Base `  S )
) )  ->  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.X.  ( ( E 
.+  G ) `  Y ) )  =  ( ( ( E 
.+  G ) `  X )  .X.  (
( I `  (
( E  .+  G
) `  Y )
)  .X.  ( ( E  .+  G ) `  Y ) ) ) )
4810, 20, 30, 25, 47syl13anc 1186 . . . . . . 7  |-  ( ph  ->  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
)  =  ( ( ( E  .+  G
) `  X )  .X.  ( ( I `  ( ( E  .+  G ) `  Y
) )  .X.  (
( E  .+  G
) `  Y )
) ) )
49 eqid 2387 . . . . . . . . . 10  |-  ( 1r
`  S )  =  ( 1r `  S
)
5017, 27, 31, 49, 28drnginvrl 15781 . . . . . . . . 9  |-  ( ( S  e.  DivRing  /\  (
( E  .+  G
) `  Y )  e.  ( Base `  S
)  /\  ( ( E  .+  G ) `  Y )  =/=  .0.  )  ->  ( ( I `
 ( ( E 
.+  G ) `  Y ) )  .X.  ( ( E  .+  G ) `  Y
) )  =  ( 1r `  S ) )
5122, 25, 26, 50syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( ( I `  ( ( E  .+  G ) `  Y
) )  .X.  (
( E  .+  G
) `  Y )
)  =  ( 1r
`  S ) )
5251oveq2d 6036 . . . . . . 7  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  (
( I `  (
( E  .+  G
) `  Y )
)  .X.  ( ( E  .+  G ) `  Y ) ) )  =  ( ( ( E  .+  G ) `
 X )  .X.  ( 1r `  S ) ) )
5348, 52eqtrd 2419 . . . . . 6  |-  ( ph  ->  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
)  =  ( ( ( E  .+  G
) `  X )  .X.  ( 1r `  S
) ) )
5417, 31, 49rngridm 15615 . . . . . . 7  |-  ( ( S  e.  Ring  /\  (
( E  .+  G
) `  X )  e.  ( Base `  S
) )  ->  (
( ( E  .+  G ) `  X
)  .X.  ( 1r `  S ) )  =  ( ( E  .+  G ) `  X
) )
5510, 20, 54syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  ( 1r `  S ) )  =  ( ( E 
.+  G ) `  X ) )
5646, 53, 553eqtrd 2423 . . . . 5  |-  ( ph  ->  ( ( E  .+  G ) `  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  =  ( ( E 
.+  G ) `  X ) )
5756oveq2d 6036 . . . 4  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 X ) ) )
58 rnggrp 15596 . . . . . 6  |-  ( S  e.  Ring  ->  S  e. 
Grp )
5910, 58syl 16 . . . . 5  |-  ( ph  ->  S  e.  Grp )
6017, 27, 42grpsubid 14800 . . . . 5  |-  ( ( S  e.  Grp  /\  ( ( E  .+  G ) `  X
)  e.  ( Base `  S ) )  -> 
( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 X ) )  =  .0.  )
6159, 20, 60syl2anc 643 . . . 4  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 X ) )  =  .0.  )
6244, 57, 613eqtrd 2423 . . 3  |-  ( ph  ->  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  .0.  )
6341, 62syl5eq 2431 . 2  |-  ( ph  ->  ( ( E  .+  G ) `  B
)  =  .0.  )
6440, 63jca 519 1  |-  ( ph  ->  ( B  e.  V  /\  ( ( E  .+  G ) `  B
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   .rcmulr 13457  Scalarcsca 13459   .scvsca 13460   0gc0g 13650   Grpcgrp 14612   -gcsg 14615   Ringcrg 15587   1rcur 15589   invrcinvr 15703   DivRingcdr 15762   LModclmod 15877   LVecclvec 16101  LFnlclfn 29172  LDualcld 29238
This theorem is referenced by:  lclkrlem2o  31636  lclkrlem2q  31638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-drng 15764  df-lmod 15879  df-lvec 16102  df-lfl 29173  df-ldual 29239
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