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Theorem lcfrlem25 36581
Description: Lemma for lcfr 36599. Special case of lcfrlem35 36591 when  ( ( J `
 Y ) `  I ) is zero. (Contributed by NM, 11-Mar-2015.)
Hypotheses
Ref Expression
lcfrlem17.h  |-  H  =  ( LHyp `  K
)
lcfrlem17.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lcfrlem17.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcfrlem17.v  |-  V  =  ( Base `  U
)
lcfrlem17.p  |-  .+  =  ( +g  `  U )
lcfrlem17.z  |-  .0.  =  ( 0g `  U )
lcfrlem17.n  |-  N  =  ( LSpan `  U )
lcfrlem17.a  |-  A  =  (LSAtoms `  U )
lcfrlem17.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lcfrlem17.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
lcfrlem17.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
lcfrlem17.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
lcfrlem22.b  |-  B  =  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  { ( X  .+  Y ) } ) )
lcfrlem24.t  |-  .x.  =  ( .s `  U )
lcfrlem24.s  |-  S  =  (Scalar `  U )
lcfrlem24.q  |-  Q  =  ( 0g `  S
)
lcfrlem24.r  |-  R  =  ( Base `  S
)
lcfrlem24.j  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
lcfrlem24.ib  |-  ( ph  ->  I  e.  B )
lcfrlem24.l  |-  L  =  (LKer `  U )
lcfrlem25.d  |-  D  =  (LDual `  U )
lcfrlem25.jz  |-  ( ph  ->  ( ( J `  Y ) `  I
)  =  Q )
lcfrlem25.in  |-  ( ph  ->  I  =/=  .0.  )
Assertion
Ref Expression
lcfrlem25  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  =  ( L `  ( J `
 Y ) ) )
Distinct variable groups:    v, k, w, x,  ._|_    .+ , k, v, w, x    R, k, v, x    S, k    .x. , k, v, w, x   
v, V, x    k, X, v, w, x    k, Y, v, w, x    x,  .0.
Allowed substitution hints:    ph( x, w, v, k)    A( x, w, v, k)    B( x, w, v, k)    D( x, w, v, k)    Q( x, w, v, k)    R( w)    S( x, w, v)    U( x, w, v, k)    H( x, w, v, k)    I( x, w, v, k)    J( x, w, v, k)    K( x, w, v, k)    L( x, w, v, k)    N( x, w, v, k)    V( w, k)    W( x, w, v, k)    .0. ( w, v, k)

Proof of Theorem lcfrlem25
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 lcfrlem17.h . . . 4  |-  H  =  ( LHyp `  K
)
2 lcfrlem17.o . . . 4  |-  ._|_  =  ( ( ocH `  K
) `  W )
3 lcfrlem17.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
4 lcfrlem17.v . . . 4  |-  V  =  ( Base `  U
)
5 lcfrlem17.p . . . 4  |-  .+  =  ( +g  `  U )
6 lcfrlem17.z . . . 4  |-  .0.  =  ( 0g `  U )
7 lcfrlem17.n . . . 4  |-  N  =  ( LSpan `  U )
8 lcfrlem17.a . . . 4  |-  A  =  (LSAtoms `  U )
9 lcfrlem17.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
10 lcfrlem17.x . . . 4  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
11 lcfrlem17.y . . . 4  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
12 lcfrlem17.ne . . . 4  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
13 lcfrlem22.b . . . 4  |-  B  =  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  { ( X  .+  Y ) } ) )
14 eqid 2467 . . . 4  |-  ( LSSum `  U )  =  (
LSSum `  U )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14lcfrlem23 36579 . . 3  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
) ( LSSum `  U
) B )  =  (  ._|_  `  { ( X  .+  Y ) } ) )
16 lcfrlem24.t . . . . . 6  |-  .x.  =  ( .s `  U )
17 lcfrlem24.s . . . . . 6  |-  S  =  (Scalar `  U )
18 lcfrlem24.q . . . . . 6  |-  Q  =  ( 0g `  S
)
19 lcfrlem24.r . . . . . 6  |-  R  =  ( Base `  S
)
20 lcfrlem24.j . . . . . 6  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
21 lcfrlem24.ib . . . . . 6  |-  ( ph  ->  I  e.  B )
22 lcfrlem24.l . . . . . 6  |-  L  =  (LKer `  U )
231, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22lcfrlem24 36580 . . . . 5  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  =  ( ( L `  ( J `  X ) )  i^i  ( L `
 ( J `  Y ) ) ) )
24 inss2 3719 . . . . 5  |-  ( ( L `  ( J `
 X ) )  i^i  ( L `  ( J `  Y ) ) )  C_  ( L `  ( J `  Y ) )
2523, 24syl6eqss 3554 . . . 4  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  C_  ( L `  ( J `
 Y ) ) )
261, 3, 9dvhlvec 36123 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
271, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13lcfrlem22 36578 . . . . . 6  |-  ( ph  ->  B  e.  A )
28 lcfrlem25.in . . . . . 6  |-  ( ph  ->  I  =/=  .0.  )
296, 7, 8, 26, 27, 21, 28lsatel 34019 . . . . 5  |-  ( ph  ->  B  =  ( N `
 { I }
) )
30 eqid 2467 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
311, 3, 9dvhlmod 36124 . . . . . 6  |-  ( ph  ->  U  e.  LMod )
32 eqid 2467 . . . . . . . 8  |-  (LFnl `  U )  =  (LFnl `  U )
33 lcfrlem25.d . . . . . . . 8  |-  D  =  (LDual `  U )
34 eqid 2467 . . . . . . . 8  |-  ( 0g
`  D )  =  ( 0g `  D
)
35 eqid 2467 . . . . . . . 8  |-  { f  e.  (LFnl `  U
)  |  (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f ) }  =  { f  e.  (LFnl `  U )  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
361, 2, 3, 4, 5, 16, 17, 19, 6, 32, 22, 33, 34, 35, 20, 9, 11lcfrlem10 36566 . . . . . . 7  |-  ( ph  ->  ( J `  Y
)  e.  (LFnl `  U ) )
3732, 22, 30lkrlss 34109 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( J `  Y )  e.  (LFnl `  U )
)  ->  ( L `  ( J `  Y
) )  e.  (
LSubSp `  U ) )
3831, 36, 37syl2anc 661 . . . . . 6  |-  ( ph  ->  ( L `  ( J `  Y )
)  e.  ( LSubSp `  U ) )
39 lcfrlem25.jz . . . . . . 7  |-  ( ph  ->  ( ( J `  Y ) `  I
)  =  Q )
404, 8, 31, 27lsatssv 34012 . . . . . . . . 9  |-  ( ph  ->  B  C_  V )
4140, 21sseldd 3505 . . . . . . . 8  |-  ( ph  ->  I  e.  V )
424, 17, 18, 32, 22, 31, 36, 41ellkr2 34105 . . . . . . 7  |-  ( ph  ->  ( I  e.  ( L `  ( J `
 Y ) )  <-> 
( ( J `  Y ) `  I
)  =  Q ) )
4339, 42mpbird 232 . . . . . 6  |-  ( ph  ->  I  e.  ( L `
 ( J `  Y ) ) )
4430, 7, 31, 38, 43lspsnel5a 17454 . . . . 5  |-  ( ph  ->  ( N `  {
I } )  C_  ( L `  ( J `
 Y ) ) )
4529, 44eqsstrd 3538 . . . 4  |-  ( ph  ->  B  C_  ( L `  ( J `  Y
) ) )
4630lsssssubg 17416 . . . . . . 7  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
4731, 46syl 16 . . . . . 6  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
4810eldifad 3488 . . . . . . . 8  |-  ( ph  ->  X  e.  V )
4911eldifad 3488 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
50 prssi 4183 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  { X ,  Y }  C_  V )
5148, 49, 50syl2anc 661 . . . . . . 7  |-  ( ph  ->  { X ,  Y }  C_  V )
521, 3, 4, 30, 2dochlss 36368 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { X ,  Y }  C_  V )  ->  (  ._|_  `  { X ,  Y }
)  e.  ( LSubSp `  U ) )
539, 51, 52syl2anc 661 . . . . . 6  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  e.  ( LSubSp `  U )
)
5447, 53sseldd 3505 . . . . 5  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  e.  (SubGrp `  U )
)
554, 30, 7, 31, 48, 49lspprcl 17436 . . . . . . . 8  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  U ) )
561, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12lcfrlem17 36573 . . . . . . . . . . 11  |-  ( ph  ->  ( X  .+  Y
)  e.  ( V 
\  {  .0.  }
) )
5756eldifad 3488 . . . . . . . . . 10  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
5857snssd 4172 . . . . . . . . 9  |-  ( ph  ->  { ( X  .+  Y ) }  C_  V )
591, 3, 4, 30, 2dochlss 36368 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { ( X 
.+  Y ) } 
C_  V )  -> 
(  ._|_  `  { ( X  .+  Y ) } )  e.  ( LSubSp `  U ) )
609, 58, 59syl2anc 661 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  e.  (
LSubSp `  U ) )
6130lssincl 17423 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  ( N `  { X ,  Y } )  e.  ( LSubSp `  U )  /\  (  ._|_  `  {
( X  .+  Y
) } )  e.  ( LSubSp `  U )
)  ->  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) } ) )  e.  ( LSubSp `  U )
)
6231, 55, 60, 61syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  { ( X  .+  Y ) } ) )  e.  ( LSubSp `  U ) )
6313, 62syl5eqel 2559 . . . . . 6  |-  ( ph  ->  B  e.  ( LSubSp `  U ) )
6447, 63sseldd 3505 . . . . 5  |-  ( ph  ->  B  e.  (SubGrp `  U ) )
6547, 38sseldd 3505 . . . . 5  |-  ( ph  ->  ( L `  ( J `  Y )
)  e.  (SubGrp `  U ) )
6614lsmlub 16498 . . . . 5  |-  ( ( (  ._|_  `  { X ,  Y } )  e.  (SubGrp `  U )  /\  B  e.  (SubGrp `  U )  /\  ( L `  ( J `  Y ) )  e.  (SubGrp `  U )
)  ->  ( (
(  ._|_  `  { X ,  Y } )  C_  ( L `  ( J `
 Y ) )  /\  B  C_  ( L `  ( J `  Y ) ) )  <-> 
( (  ._|_  `  { X ,  Y }
) ( LSSum `  U
) B )  C_  ( L `  ( J `
 Y ) ) ) )
6754, 64, 65, 66syl3anc 1228 . . . 4  |-  ( ph  ->  ( ( (  ._|_  `  { X ,  Y } )  C_  ( L `  ( J `  Y ) )  /\  B  C_  ( L `  ( J `  Y ) ) )  <->  ( (  ._|_  `  { X ,  Y } ) ( LSSum `  U ) B ) 
C_  ( L `  ( J `  Y ) ) ) )
6825, 45, 67mpbi2and 919 . . 3  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
) ( LSSum `  U
) B )  C_  ( L `  ( J `
 Y ) ) )
6915, 68eqsstr3d 3539 . 2  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  C_  ( L `  ( J `  Y ) ) )
70 eqid 2467 . . 3  |-  (LSHyp `  U )  =  (LSHyp `  U )
711, 2, 3, 4, 6, 70, 9, 56dochsnshp 36467 . . 3  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  e.  (LSHyp `  U ) )
721, 2, 3, 4, 5, 16, 17, 19, 6, 32, 22, 33, 34, 35, 20, 9, 11lcfrlem13 36569 . . . . 5  |-  ( ph  ->  ( J `  Y
)  e.  ( { f  e.  (LFnl `  U )  |  ( 
._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( L `  f ) }  \  { ( 0g `  D ) } ) )
73 eldifsni 4153 . . . . 5  |-  ( ( J `  Y )  e.  ( { f  e.  (LFnl `  U
)  |  (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f ) } 
\  { ( 0g
`  D ) } )  ->  ( J `  Y )  =/=  ( 0g `  D ) )
7472, 73syl 16 . . . 4  |-  ( ph  ->  ( J `  Y
)  =/=  ( 0g
`  D ) )
7570, 32, 22, 33, 34, 26, 36lduallkr3 34176 . . . 4  |-  ( ph  ->  ( ( L `  ( J `  Y ) )  e.  (LSHyp `  U )  <->  ( J `  Y )  =/=  ( 0g `  D ) ) )
7674, 75mpbird 232 . . 3  |-  ( ph  ->  ( L `  ( J `  Y )
)  e.  (LSHyp `  U ) )
7770, 26, 71, 76lshpcmp 34002 . 2  |-  ( ph  ->  ( (  ._|_  `  {
( X  .+  Y
) } )  C_  ( L `  ( J `
 Y ) )  <-> 
(  ._|_  `  { ( X  .+  Y ) } )  =  ( L `
 ( J `  Y ) ) ) )
7869, 77mpbid 210 1  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  =  ( L `  ( J `
 Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818    \ cdif 3473    i^i cin 3475    C_ wss 3476   {csn 4027   {cpr 4029    |-> cmpt 4505   ` cfv 5588   iota_crio 6245  (class class class)co 6285   Basecbs 14493   +g cplusg 14558  Scalarcsca 14561   .scvsca 14562   0gc0g 14698  SubGrpcsubg 16009   LSSumclsm 16469   LModclmod 17324   LSubSpclss 17390   LSpanclspn 17429  LSAtomsclsa 33988  LSHypclsh 33989  LFnlclfn 34071  LKerclk 34099  LDualcld 34137   HLchlt 34364   LHypclh 34997   DVecHcdvh 36092   ocHcoch 36361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-riotaBAD 33973
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6956  df-undef 7003  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-0g 14700  df-mre 14844  df-mrc 14845  df-acs 14847  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-p1 15530  df-lat 15536  df-clat 15598  df-mnd 15735  df-submnd 15790  df-grp 15871  df-minusg 15872  df-sbg 15873  df-subg 16012  df-cntz 16169  df-oppg 16195  df-lsm 16471  df-cmn 16615  df-abl 16616  df-mgp 16956  df-ur 16968  df-rng 17014  df-oppr 17085  df-dvdsr 17103  df-unit 17104  df-invr 17134  df-dvr 17145  df-drng 17210  df-lmod 17326  df-lss 17391  df-lsp 17430  df-lvec 17561  df-lsatoms 33990  df-lshyp 33991  df-lcv 34033  df-lfl 34072  df-lkr 34100  df-ldual 34138  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512  df-lvols 34513  df-lines 34514  df-psubsp 34516  df-pmap 34517  df-padd 34809  df-lhyp 35001  df-laut 35002  df-ldil 35117  df-ltrn 35118  df-trl 35172  df-tgrp 35756  df-tendo 35768  df-edring 35770  df-dveca 36016  df-disoa 36043  df-dvech 36093  df-dib 36153  df-dic 36187  df-dih 36243  df-doch 36362  df-djh 36409
This theorem is referenced by:  lcfrlem26  36582
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