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Theorem lclkrlem2s 35170
Description: Lemma for lclkr 35178. Thus, the sum has a closed kernel when  B is zero. (Contributed by NM, 18-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 )
lclkrlem2n.n  |-  N  =  ( LSpan `  U )
lclkrlem2n.l  |-  L  =  (LKer `  U )
lclkrlem2o.h  |-  H  =  ( LHyp `  K
)
lclkrlem2o.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lclkrlem2o.u  |-  U  =  ( ( DVecH `  K
) `  W )
lclkrlem2o.a  |-  .(+)  =  (
LSSum `  U )
lclkrlem2o.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lclkrlem2q.le  |-  ( ph  ->  ( L `  E
)  =  (  ._|_  `  { X } ) )
lclkrlem2q.lg  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { Y } ) )
lclkrlem2q.b  |-  B  =  ( X  .-  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )
lclkrlem2q.n  |-  ( ph  ->  ( ( E  .+  G ) `  Y
)  =/=  .0.  )
lclkrlem2r.bn  |-  ( ph  ->  B  =  ( 0g
`  U ) )
Assertion
Ref Expression
lclkrlem2s  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )

Proof of Theorem lclkrlem2s
StepHypRef Expression
1 lclkrlem2o.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2 lclkrlem2m.y . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
32snssd 4018 . . . . . . 7  |-  ( ph  ->  { Y }  C_  V )
4 lclkrlem2o.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
5 eqid 2443 . . . . . . . 8  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
6 lclkrlem2o.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
7 lclkrlem2m.v . . . . . . . 8  |-  V  =  ( Base `  U
)
8 lclkrlem2o.o . . . . . . . 8  |-  ._|_  =  ( ( ocH `  K
) `  W )
94, 5, 6, 7, 8dochcl 34998 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { Y }  C_  V )  ->  (  ._|_  `  { Y }
)  e.  ran  (
( DIsoH `  K ) `  W ) )
101, 3, 9syl2anc 661 . . . . . 6  |-  ( ph  ->  (  ._|_  `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)
114, 5, 8dochoc 35012 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  (  ._|_  `  (  ._|_  `  (  ._|_  `  { Y } ) ) )  =  ( 
._|_  `  { Y }
) )
121, 10, 11syl2anc 661 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  (  ._|_  `  { Y } ) ) )  =  (  ._|_  `  { Y } ) )
1312ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
(  ._|_  `  (  ._|_  `  (  ._|_  `  { Y } ) ) )  =  (  ._|_  `  { Y } ) )
14 lclkrlem2m.t . . . . . . . . . 10  |-  .x.  =  ( .s `  U )
15 lclkrlem2m.s . . . . . . . . . 10  |-  S  =  (Scalar `  U )
16 lclkrlem2m.q . . . . . . . . . 10  |-  .X.  =  ( .r `  S )
17 lclkrlem2m.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  S )
18 lclkrlem2m.i . . . . . . . . . 10  |-  I  =  ( invr `  S
)
19 lclkrlem2m.m . . . . . . . . . 10  |-  .-  =  ( -g `  U )
20 lclkrlem2m.f . . . . . . . . . 10  |-  F  =  (LFnl `  U )
21 lclkrlem2m.d . . . . . . . . . 10  |-  D  =  (LDual `  U )
22 lclkrlem2m.p . . . . . . . . . 10  |-  .+  =  ( +g  `  D )
23 lclkrlem2m.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  V )
24 lclkrlem2m.e . . . . . . . . . 10  |-  ( ph  ->  E  e.  F )
25 lclkrlem2m.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  F )
26 lclkrlem2n.n . . . . . . . . . 10  |-  N  =  ( LSpan `  U )
27 lclkrlem2n.l . . . . . . . . . 10  |-  L  =  (LKer `  U )
28 lclkrlem2o.a . . . . . . . . . 10  |-  .(+)  =  (
LSSum `  U )
29 lclkrlem2q.le . . . . . . . . . 10  |-  ( ph  ->  ( L `  E
)  =  (  ._|_  `  { X } ) )
30 lclkrlem2q.lg . . . . . . . . . 10  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { Y } ) )
31 lclkrlem2q.b . . . . . . . . . 10  |-  B  =  ( X  .-  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )
32 lclkrlem2q.n . . . . . . . . . 10  |-  ( ph  ->  ( ( E  .+  G ) `  Y
)  =/=  .0.  )
33 lclkrlem2r.bn . . . . . . . . . 10  |-  ( ph  ->  B  =  ( 0g
`  U ) )
347, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 2, 24, 25, 26, 27, 4, 8, 6, 28, 1, 29, 30, 31, 32, 33lclkrlem2r 35169 . . . . . . . . 9  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( E  .+  G
) ) )
3534ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
( L `  G
)  C_  ( L `  ( E  .+  G
) ) )
36 eqid 2443 . . . . . . . . 9  |-  (LSHyp `  U )  =  (LSHyp `  U )
374, 6, 1dvhlvec 34754 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LVec )
3837ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  ->  U  e.  LVec )
39 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
( L `  G
)  e.  (LSHyp `  U ) )
40 simpr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
( L `  ( E  .+  G ) )  e.  (LSHyp `  U
) )
4136, 38, 39, 40lshpcmp 32633 . . . . . . . 8  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
( ( L `  G )  C_  ( L `  ( E  .+  G ) )  <->  ( L `  G )  =  ( L `  ( E 
.+  G ) ) ) )
4235, 41mpbid 210 . . . . . . 7  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
( L `  G
)  =  ( L `
 ( E  .+  G ) ) )
4330ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
( L `  G
)  =  (  ._|_  `  { Y } ) )
4442, 43eqtr3d 2477 . . . . . 6  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
( L `  ( E  .+  G ) )  =  (  ._|_  `  { Y } ) )
4544fveq2d 5695 . . . . 5  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
(  ._|_  `  ( L `  ( E  .+  G
) ) )  =  (  ._|_  `  (  ._|_  `  { Y } ) ) )
4645fveq2d 5695 . . . 4  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
(  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( 
._|_  `  (  ._|_  `  (  ._|_  `  { Y }
) ) ) )
4713, 46, 443eqtr4d 2485 . . 3  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  e.  (LSHyp `  U ) )  -> 
(  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )
484, 6, 8, 7, 1dochoc1 35006 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  V ) )  =  V )
4948ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  =  V )  ->  (  ._|_  `  (  ._|_  `  V ) )  =  V )
50 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  =  V )  ->  ( L `  ( E  .+  G
) )  =  V )
5150fveq2d 5695 . . . . 5  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  =  V )  ->  (  ._|_  `  ( L `  ( E  .+  G ) ) )  =  (  ._|_  `  V ) )
5251fveq2d 5695 . . . 4  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  =  V )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  (  ._|_  `  (  ._|_  `  V ) ) )
5349, 52, 503eqtr4d 2485 . . 3  |-  ( ( ( ph  /\  ( L `  G )  e.  (LSHyp `  U )
)  /\  ( L `  ( E  .+  G
) )  =  V )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  ( L `
 ( E  .+  G ) ) )
544, 6, 1dvhlmod 34755 . . . . . 6  |-  ( ph  ->  U  e.  LMod )
5520, 21, 22, 54, 24, 25ldualvaddcl 32775 . . . . 5  |-  ( ph  ->  ( E  .+  G
)  e.  F )
567, 36, 20, 27, 37, 55lkrshpor 32752 . . . 4  |-  ( ph  ->  ( ( L `  ( E  .+  G ) )  e.  (LSHyp `  U )  \/  ( L `  ( E  .+  G ) )  =  V ) )
5756adantr 465 . . 3  |-  ( (
ph  /\  ( L `  G )  e.  (LSHyp `  U ) )  -> 
( ( L `  ( E  .+  G ) )  e.  (LSHyp `  U )  \/  ( L `  ( E  .+  G ) )  =  V ) )
5847, 53, 57mpjaodan 784 . 2  |-  ( (
ph  /\  ( L `  G )  e.  (LSHyp `  U ) )  -> 
(  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )
5948adantr 465 . . 3  |-  ( (
ph  /\  ( L `  G )  =  V )  ->  (  ._|_  `  (  ._|_  `  V ) )  =  V )
607, 20, 27, 54, 55lkrssv 32741 . . . . . . 7  |-  ( ph  ->  ( L `  ( E  .+  G ) ) 
C_  V )
6160adantr 465 . . . . . 6  |-  ( (
ph  /\  ( L `  G )  =  V )  ->  ( L `  ( E  .+  G
) )  C_  V
)
62 simpr 461 . . . . . . 7  |-  ( (
ph  /\  ( L `  G )  =  V )  ->  ( L `  G )  =  V )
6334adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( L `  G )  =  V )  ->  ( L `  G )  C_  ( L `  ( E  .+  G ) ) )
6462, 63eqsstr3d 3391 . . . . . 6  |-  ( (
ph  /\  ( L `  G )  =  V )  ->  V  C_  ( L `  ( E  .+  G ) ) )
6561, 64eqssd 3373 . . . . 5  |-  ( (
ph  /\  ( L `  G )  =  V )  ->  ( L `  ( E  .+  G
) )  =  V )
6665fveq2d 5695 . . . 4  |-  ( (
ph  /\  ( L `  G )  =  V )  ->  (  ._|_  `  ( L `  ( E  .+  G ) ) )  =  (  ._|_  `  V ) )
6766fveq2d 5695 . . 3  |-  ( (
ph  /\  ( L `  G )  =  V )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  (  ._|_  `  (  ._|_  `  V ) ) )
6859, 67, 653eqtr4d 2485 . 2  |-  ( (
ph  /\  ( L `  G )  =  V )  ->  (  ._|_  `  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) )  =  ( L `
 ( E  .+  G ) ) )
697, 36, 20, 27, 37, 25lkrshpor 32752 . 2  |-  ( ph  ->  ( ( L `  G )  e.  (LSHyp `  U )  \/  ( L `  G )  =  V ) )
7058, 68, 69mpjaodan 784 1  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `  ( E 
.+  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606    C_ wss 3328   {csn 3877   ran crn 4841   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   .rcmulr 14239  Scalarcsca 14241   .scvsca 14242   0gc0g 14378   -gcsg 15413   LSSumclsm 16133   invrcinvr 16763   LSpanclspn 17052   LVecclvec 17183  LSHypclsh 32620  LFnlclfn 32702  LKerclk 32730  LDualcld 32768   HLchlt 32995   LHypclh 33628   DVecHcdvh 34723   DIsoHcdih 34873   ocHcoch 34992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-riotaBAD 32604
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-undef 6792  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-0g 14380  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-cntz 15835  df-lsm 16135  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-lmod 16950  df-lss 17014  df-lsp 17053  df-lvec 17184  df-lsatoms 32621  df-lshyp 32622  df-lfl 32703  df-lkr 32731  df-ldual 32769  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-lplanes 33143  df-lvols 33144  df-lines 33145  df-psubsp 33147  df-pmap 33148  df-padd 33440  df-lhyp 33632  df-laut 33633  df-ldil 33748  df-ltrn 33749  df-trl 33803  df-tendo 34399  df-edring 34401  df-disoa 34674  df-dvech 34724  df-dib 34784  df-dic 34818  df-dih 34874  df-doch 34993
This theorem is referenced by:  lclkrlem2t  35171
  Copyright terms: Public domain W3C validator