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Theorem lclkrlem2m 32218
Description: Lemma for lclkr 32232. 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 16168 . . . . . 6  |-  ( U  e.  LVec  ->  U  e. 
LMod )
42, 3syl 16 . . . . 5  |-  ( ph  ->  U  e.  LMod )
5 lmodgrp 15947 . . . . 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 15948 . . . . . . 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 29829 . . . . . . 7  |-  ( ph  ->  ( E  .+  G
)  e.  F )
17 eqid 2435 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
18 lclkrlem2m.v . . . . . . . 8  |-  V  =  ( Base `  U
)
198, 17, 18, 11lflcl 29763 . . . . . . 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 16167 . . . . . . . 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 29763 . . . . . . . 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 15842 . . . . . . 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 15667 . . . . . 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 15957 . . . . 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 14859 . . . 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 2519 . 2  |-  ( ph  ->  B  e.  V )
411fveq2i 5723 . . 3  |-  ( ( E  .+  G ) `
 B )  =  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )
42 eqid 2435 . . . . . 6  |-  ( -g `  S )  =  (
-g `  S )
438, 42, 18, 37, 11lflsub 29766 . . . . 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 29767 . . . . . . 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 15670 . . . . . . . 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 2435 . . . . . . . . . 10  |-  ( 1r
`  S )  =  ( 1r `  S
)
5017, 27, 31, 49, 28drnginvrl 15844 . . . . . . . . 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 6089 . . . . . . 7  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  (
( I `  (
( E  .+  G
) `  Y )
)  .X.  ( ( E  .+  G ) `  Y ) ) )  =  ( ( ( E  .+  G ) `
 X )  .X.  ( 1r `  S ) ) )
5348, 52eqtrd 2467 . . . . . 6  |-  ( ph  ->  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
)  =  ( ( ( E  .+  G
) `  X )  .X.  ( 1r `  S
) ) )
5417, 31, 49rngridm 15678 . . . . . . 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 2471 . . . . 5  |-  ( ph  ->  ( ( E  .+  G ) `  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  =  ( ( E 
.+  G ) `  X ) )
5756oveq2d 6089 . . . 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 15659 . . . . . 6  |-  ( S  e.  Ring  ->  S  e. 
Grp )
5910, 58syl 16 . . . . 5  |-  ( ph  ->  S  e.  Grp )
6017, 27, 42grpsubid 14863 . . . . 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 2471 . . 3  |-  ( ph  ->  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  .0.  )
6341, 62syl5eq 2479 . 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 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   Basecbs 13459   +g cplusg 13519   .rcmulr 13520  Scalarcsca 13522   .scvsca 13523   0gc0g 13713   Grpcgrp 14675   -gcsg 14678   Ringcrg 15650   1rcur 15652   invrcinvr 15766   DivRingcdr 15825   LModclmod 15940   LVecclvec 16164  LFnlclfn 29756  LDualcld 29822
This theorem is referenced by:  lclkrlem2o  32220  lclkrlem2q  32222
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-sca 13535  df-vsca 13536  df-0g 13717  df-mnd 14680  df-grp 14802  df-minusg 14803  df-sbg 14804  df-cmn 15404  df-abl 15405  df-mgp 15639  df-rng 15653  df-ur 15655  df-oppr 15718  df-dvdsr 15736  df-unit 15737  df-invr 15767  df-drng 15827  df-lmod 15942  df-lvec 16165  df-lfl 29757  df-ldual 29823
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