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Theorem lclkrlem2m 32331
Description: Lemma for lclkr 32345. 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 15875 . . . . . 6  |-  ( U  e.  LVec  ->  U  e. 
LMod )
42, 3syl 15 . . . . 5  |-  ( ph  ->  U  e.  LMod )
5 lmodgrp 15650 . . . . 5  |-  ( U  e.  LMod  ->  U  e. 
Grp )
64, 5syl 15 . . . 4  |-  ( ph  ->  U  e.  Grp )
7 lclkrlem2m.x . . . 4  |-  ( ph  ->  X  e.  V )
8 lclkrlem2m.s . . . . . . . 8  |-  S  =  (Scalar `  U )
98lmodrng 15651 . . . . . . 7  |-  ( U  e.  LMod  ->  S  e. 
Ring )
104, 9syl 15 . . . . . 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 29942 . . . . . . 7  |-  ( ph  ->  ( E  .+  G
)  e.  F )
17 eqid 2296 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
18 lclkrlem2m.v . . . . . . . 8  |-  V  =  ( Base `  U
)
198, 17, 18, 11lflcl 29876 . . . . . . 7  |-  ( ( U  e.  LVec  /\  ( E  .+  G )  e.  F  /\  X  e.  V )  ->  (
( E  .+  G
) `  X )  e.  ( Base `  S
) )
202, 16, 7, 19syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( E  .+  G ) `  X
)  e.  ( Base `  S ) )
218lvecdrng 15874 . . . . . . . 8  |-  ( U  e.  LVec  ->  S  e.  DivRing )
222, 21syl 15 . . . . . . 7  |-  ( ph  ->  S  e.  DivRing )
23 lclkrlem2m.y . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
248, 17, 18, 11lflcl 29876 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  ( E  .+  G )  e.  F  /\  Y  e.  V )  ->  (
( E  .+  G
) `  Y )  e.  ( Base `  S
) )
252, 16, 23, 24syl3anc 1182 . . . . . . 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 15545 . . . . . . 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 1182 . . . . . 6  |-  ( ph  ->  ( I `  (
( E  .+  G
) `  Y )
)  e.  ( Base `  S ) )
31 lclkrlem2m.q . . . . . . 7  |-  .X.  =  ( .r `  S )
3217, 31rngcl 15370 . . . . . 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 1182 . . . . 5  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) )  e.  ( Base `  S
) )
34 lclkrlem2m.t . . . . . 6  |-  .x.  =  ( .s `  U )
3518, 8, 34, 17lmodvscl 15660 . . . . 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 1182 . . . 4  |-  ( ph  ->  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
)  e.  V )
37 lclkrlem2m.m . . . . 5  |-  .-  =  ( -g `  U )
3818, 37grpsubcl 14562 . . . 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 1182 . . 3  |-  ( ph  ->  ( X  .-  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  e.  V )
401, 39syl5eqel 2380 . 2  |-  ( ph  ->  B  e.  V )
411fveq2i 5544 . . 3  |-  ( ( E  .+  G ) `
 B )  =  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )
42 eqid 2296 . . . . . 6  |-  ( -g `  S )  =  (
-g `  S )
438, 42, 18, 37, 11lflsub 29879 . . . . 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 1186 . . . 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 29880 . . . . . . 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 1186 . . . . . 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 15373 . . . . . . . 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 1184 . . . . . . 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 2296 . . . . . . . . . 10  |-  ( 1r
`  S )  =  ( 1r `  S
)
5017, 27, 31, 49, 28drnginvrl 15547 . . . . . . . . 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 1182 . . . . . . . 8  |-  ( ph  ->  ( ( I `  ( ( E  .+  G ) `  Y
) )  .X.  (
( E  .+  G
) `  Y )
)  =  ( 1r
`  S ) )
5251oveq2d 5890 . . . . . . 7  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  (
( I `  (
( E  .+  G
) `  Y )
)  .X.  ( ( E  .+  G ) `  Y ) ) )  =  ( ( ( E  .+  G ) `
 X )  .X.  ( 1r `  S ) ) )
5348, 52eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
)  =  ( ( ( E  .+  G
) `  X )  .X.  ( 1r `  S
) ) )
5417, 31, 49rngridm 15381 . . . . . . 7  |-  ( ( S  e.  Ring  /\  (
( E  .+  G
) `  X )  e.  ( Base `  S
) )  ->  (
( ( E  .+  G ) `  X
)  .X.  ( 1r `  S ) )  =  ( ( E  .+  G ) `  X
) )
5510, 20, 54syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  ( 1r `  S ) )  =  ( ( E 
.+  G ) `  X ) )
5646, 53, 553eqtrd 2332 . . . . 5  |-  ( ph  ->  ( ( E  .+  G ) `  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  =  ( ( E 
.+  G ) `  X ) )
5756oveq2d 5890 . . . 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 15362 . . . . . 6  |-  ( S  e.  Ring  ->  S  e. 
Grp )
5910, 58syl 15 . . . . 5  |-  ( ph  ->  S  e.  Grp )
6017, 27, 42grpsubid 14566 . . . . 5  |-  ( ( S  e.  Grp  /\  ( ( E  .+  G ) `  X
)  e.  ( Base `  S ) )  -> 
( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 X ) )  =  .0.  )
6159, 20, 60syl2anc 642 . . . 4  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 X ) )  =  .0.  )
6244, 57, 613eqtrd 2332 . . 3  |-  ( ph  ->  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  .0.  )
6341, 62syl5eq 2340 . 2  |-  ( ph  ->  ( ( E  .+  G ) `  B
)  =  .0.  )
6440, 63jca 518 1  |-  ( ph  ->  ( B  e.  V  /\  ( ( E  .+  G ) `  B
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   Grpcgrp 14378   -gcsg 14381   Ringcrg 15353   1rcur 15355   invrcinvr 15469   DivRingcdr 15528   LModclmod 15643   LVecclvec 15871  LFnlclfn 29869  LDualcld 29935
This theorem is referenced by:  lclkrlem2o  32333  lclkrlem2q  32335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lvec 15872  df-lfl 29870  df-ldual 29936
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