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Theorem lcfrlem35 32375
Description: Lemma for lcfr 32383. (Contributed by NM, 2-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 )
lcfrlem28.jn  |-  ( ph  ->  ( ( J `  Y ) `  I
)  =/=  Q )
lcfrlem29.i  |-  F  =  ( invr `  S
)
lcfrlem30.m  |-  .-  =  ( -g `  D )
lcfrlem30.c  |-  C  =  ( ( J `  X )  .-  (
( ( F `  ( ( J `  Y ) `  I
) ) ( .r
`  S ) ( ( J `  X
) `  I )
) ( .s `  D ) ( J `
 Y ) ) )
Assertion
Ref Expression
lcfrlem35  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  =  ( L `  C ) )
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)    C( 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)    F( 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)    .- ( x, w, v, k)    N( x, w, v, k)    V( w, k)    W( x, w, v, k)    .0. ( w, v, k)

Proof of Theorem lcfrlem35
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 2436 . . . 4  |-  ( LSSum `  U )  =  (
LSSum `  U )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14lcfrlem23 32363 . . 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 32364 . . . . 5  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  =  ( ( L `  ( J `  X ) )  i^i  ( L `
 ( J `  Y ) ) ) )
24 eqid 2436 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
25 lcfrlem29.i . . . . . 6  |-  F  =  ( invr `  S
)
26 eqid 2436 . . . . . 6  |-  (LFnl `  U )  =  (LFnl `  U )
27 lcfrlem25.d . . . . . 6  |-  D  =  (LDual `  U )
28 eqid 2436 . . . . . 6  |-  ( .s
`  D )  =  ( .s `  D
)
29 lcfrlem30.m . . . . . 6  |-  .-  =  ( -g `  D )
301, 3, 9dvhlvec 31907 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
31 eqid 2436 . . . . . . 7  |-  ( 0g
`  D )  =  ( 0g `  D
)
32 eqid 2436 . . . . . . 7  |-  { f  e.  (LFnl `  U
)  |  (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f ) }  =  { f  e.  (LFnl `  U )  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
331, 2, 3, 4, 5, 16, 17, 19, 6, 26, 22, 27, 31, 32, 20, 9, 10lcfrlem10 32350 . . . . . 6  |-  ( ph  ->  ( J `  X
)  e.  (LFnl `  U ) )
341, 2, 3, 4, 5, 16, 17, 19, 6, 26, 22, 27, 31, 32, 20, 9, 11lcfrlem10 32350 . . . . . 6  |-  ( ph  ->  ( J `  Y
)  e.  (LFnl `  U ) )
35 eqid 2436 . . . . . . . 8  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
361, 3, 9dvhlmod 31908 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
371, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13lcfrlem22 32362 . . . . . . . 8  |-  ( ph  ->  B  e.  A )
3835, 8, 36, 37lsatlssel 29795 . . . . . . 7  |-  ( ph  ->  B  e.  ( LSubSp `  U ) )
394, 35lssel 16014 . . . . . . 7  |-  ( ( B  e.  ( LSubSp `  U )  /\  I  e.  B )  ->  I  e.  V )
4038, 21, 39syl2anc 643 . . . . . 6  |-  ( ph  ->  I  e.  V )
41 lcfrlem28.jn . . . . . 6  |-  ( ph  ->  ( ( J `  Y ) `  I
)  =/=  Q )
42 lcfrlem30.c . . . . . 6  |-  C  =  ( ( J `  X )  .-  (
( ( F `  ( ( J `  Y ) `  I
) ) ( .r
`  S ) ( ( J `  X
) `  I )
) ( .s `  D ) ( J `
 Y ) ) )
434, 17, 24, 18, 25, 26, 27, 28, 29, 30, 33, 34, 40, 41, 42, 22lcfrlem2 32341 . . . . 5  |-  ( ph  ->  ( ( L `  ( J `  X ) )  i^i  ( L `
 ( J `  Y ) ) ) 
C_  ( L `  C ) )
4423, 43eqsstrd 3382 . . . 4  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  C_  ( L `  C ) )
451, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 27, 41lcfrlem28 32368 . . . . . 6  |-  ( ph  ->  I  =/=  .0.  )
466, 7, 8, 30, 37, 21, 45lsatel 29803 . . . . 5  |-  ( ph  ->  B  =  ( N `
 { I }
) )
471, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 27, 41, 25, 29, 42lcfrlem30 32370 . . . . . . 7  |-  ( ph  ->  C  e.  (LFnl `  U ) )
4826, 22, 35lkrlss 29893 . . . . . . 7  |-  ( ( U  e.  LMod  /\  C  e.  (LFnl `  U )
)  ->  ( L `  C )  e.  (
LSubSp `  U ) )
4936, 47, 48syl2anc 643 . . . . . 6  |-  ( ph  ->  ( L `  C
)  e.  ( LSubSp `  U ) )
504, 17, 24, 18, 25, 26, 27, 28, 29, 30, 33, 34, 40, 41, 42, 22lcfrlem3 32342 . . . . . 6  |-  ( ph  ->  I  e.  ( L `
 C ) )
5135, 7, 36, 49, 50lspsnel5a 16072 . . . . 5  |-  ( ph  ->  ( N `  {
I } )  C_  ( L `  C ) )
5246, 51eqsstrd 3382 . . . 4  |-  ( ph  ->  B  C_  ( L `  C ) )
5335lsssssubg 16034 . . . . . . 7  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
5436, 53syl 16 . . . . . 6  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
5510eldifad 3332 . . . . . . . 8  |-  ( ph  ->  X  e.  V )
5611eldifad 3332 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
57 prssi 3954 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  { X ,  Y }  C_  V )
5855, 56, 57syl2anc 643 . . . . . . 7  |-  ( ph  ->  { X ,  Y }  C_  V )
591, 3, 4, 35, 2dochlss 32152 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { X ,  Y }  C_  V )  ->  (  ._|_  `  { X ,  Y }
)  e.  ( LSubSp `  U ) )
609, 58, 59syl2anc 643 . . . . . 6  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  e.  ( LSubSp `  U )
)
6154, 60sseldd 3349 . . . . 5  |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  e.  (SubGrp `  U )
)
6254, 38sseldd 3349 . . . . 5  |-  ( ph  ->  B  e.  (SubGrp `  U ) )
6354, 49sseldd 3349 . . . . 5  |-  ( ph  ->  ( L `  C
)  e.  (SubGrp `  U ) )
6414lsmlub 15297 . . . . 5  |-  ( ( (  ._|_  `  { X ,  Y } )  e.  (SubGrp `  U )  /\  B  e.  (SubGrp `  U )  /\  ( L `  C )  e.  (SubGrp `  U )
)  ->  ( (
(  ._|_  `  { X ,  Y } )  C_  ( L `  C )  /\  B  C_  ( L `  C )
)  <->  ( (  ._|_  `  { X ,  Y } ) ( LSSum `  U ) B ) 
C_  ( L `  C ) ) )
6561, 62, 63, 64syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( (  ._|_  `  { X ,  Y } )  C_  ( L `  C )  /\  B  C_  ( L `
 C ) )  <-> 
( (  ._|_  `  { X ,  Y }
) ( LSSum `  U
) B )  C_  ( L `  C ) ) )
6644, 52, 65mpbi2and 888 . . 3  |-  ( ph  ->  ( (  ._|_  `  { X ,  Y }
) ( LSSum `  U
) B )  C_  ( L `  C ) )
6715, 66eqsstr3d 3383 . 2  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  C_  ( L `  C )
)
68 eqid 2436 . . 3  |-  (LSHyp `  U )  =  (LSHyp `  U )
691, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12lcfrlem17 32357 . . . 4  |-  ( ph  ->  ( X  .+  Y
)  e.  ( V 
\  {  .0.  }
) )
701, 2, 3, 4, 6, 68, 9, 69dochsnshp 32251 . . 3  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  e.  (LSHyp `  U ) )
711, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 27, 41, 25, 29, 42lcfrlem34 32374 . . . 4  |-  ( ph  ->  C  =/=  ( 0g
`  D ) )
7268, 26, 22, 27, 31, 30, 47lduallkr3 29960 . . . 4  |-  ( ph  ->  ( ( L `  C )  e.  (LSHyp `  U )  <->  C  =/=  ( 0g `  D ) ) )
7371, 72mpbird 224 . . 3  |-  ( ph  ->  ( L `  C
)  e.  (LSHyp `  U ) )
7468, 30, 70, 73lshpcmp 29786 . 2  |-  ( ph  ->  ( (  ._|_  `  {
( X  .+  Y
) } )  C_  ( L `  C )  <-> 
(  ._|_  `  { ( X  .+  Y ) } )  =  ( L `
 C ) ) )
7567, 74mpbid 202 1  |-  ( ph  ->  (  ._|_  `  { ( X  .+  Y ) } )  =  ( L `  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709    \ cdif 3317    i^i cin 3319    C_ wss 3320   {csn 3814   {cpr 3815    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   -gcsg 14688  SubGrpcsubg 14938   LSSumclsm 15268   invrcinvr 15776   LModclmod 15950   LSubSpclss 16008   LSpanclspn 16047  LSAtomsclsa 29772  LSHypclsh 29773  LFnlclfn 29855  LKerclk 29883  LDualcld 29921   HLchlt 30148   LHypclh 30781   DVecHcdvh 31876   ocHcoch 32145
This theorem is referenced by:  lcfrlem36  32376
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-undef 6543  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-0g 13727  df-mre 13811  df-mrc 13812  df-acs 13814  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-oppg 15142  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-drng 15837  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lvec 16175  df-lsatoms 29774  df-lshyp 29775  df-lcv 29817  df-lfl 29856  df-lkr 29884  df-ldual 29922  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tgrp 31540  df-tendo 31552  df-edring 31554  df-dveca 31800  df-disoa 31827  df-dvech 31877  df-dib 31937  df-dic 31971  df-dih 32027  df-doch 32146  df-djh 32193
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