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Theorem lcfrlem38 32305
Description: Lemma for lcfr 32310. Combine lcfrlem27 32294 and lcfrlem37 32304. (Contributed by NM, 11-Mar-2015.)
Hypotheses
Ref Expression
lcfrlem38.h  |-  H  =  ( LHyp `  K
)
lcfrlem38.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lcfrlem38.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcfrlem38.p  |-  .+  =  ( +g  `  U )
lcfrlem38.f  |-  F  =  (LFnl `  U )
lcfrlem38.l  |-  L  =  (LKer `  U )
lcfrlem38.d  |-  D  =  (LDual `  U )
lcfrlem38.q  |-  Q  =  ( LSubSp `  D )
lcfrlem38.c  |-  C  =  { f  e.  (LFnl `  U )  |  ( 
._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( L `  f ) }
lcfrlem38.e  |-  E  = 
U_ g  e.  G  (  ._|_  `  ( L `  g ) )
lcfrlem38.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lcfrlem38.g  |-  ( ph  ->  G  e.  Q )
lcfrlem38.gs  |-  ( ph  ->  G  C_  C )
lcfrlem38.xe  |-  ( ph  ->  X  e.  E )
lcfrlem38.ye  |-  ( ph  ->  Y  e.  E )
lcfrlem38.z  |-  .0.  =  ( 0g `  U )
lcfrlem38.x  |-  ( ph  ->  X  =/=  .0.  )
lcfrlem38.y  |-  ( ph  ->  Y  =/=  .0.  )
lcfrlem38.sp  |-  N  =  ( LSpan `  U )
lcfrlem38.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
lcfrlem38.b  |-  B  =  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  { ( X  .+  Y ) } ) )
lcfrlem38.i  |-  ( ph  ->  I  e.  B )
lcfrlem38.n  |-  ( ph  ->  I  =/=  .0.  )
lcfrlem38.v  |-  V  =  ( Base `  U
)
lcfrlem38.t  |-  .x.  =  ( .s `  U )
lcfrlem38.s  |-  S  =  (Scalar `  U )
lcfrlem38.r  |-  R  =  ( Base `  S
)
lcfrlem38.j  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
Assertion
Ref Expression
lcfrlem38  |-  ( ph  ->  ( X  .+  Y
)  e.  E )
Distinct variable groups:    g, k, D    g, G, k    g, I, k    f, g, k, J    f, L, g, k    v, f, w, x,  ._|_ , g, k    .+ , f, g, k, v, w, x    R, f, k, v, x    S, g, k    .x. , f, k, v, w, x    U, f, g, k, v, w, x    f, V, g, v, x    f, X, g, k, v, w, x    f, Y, g, k, v, w, x    .0. , f, g, k, x    ph, g, k
Allowed substitution hints:    ph( x, w, v, f)    B( x, w, v, f, g, k)    C( x, w, v, f, g, k)    D( x, w, v, f)    Q( x, w, v, f, g, k)    R( w, g)    S( x, w, v, f)    .x. ( g)    E( x, w, v, f, g, k)    F( x, w, v, f, g, k)    G( x, w, v, f)    H( x, w, v, f, g, k)    I( x, w, v, f)    J( x, w, v)    K( x, w, v, f, g, k)    L( x, w, v)    N( x, w, v, f, g, k)    V( w, k)    W( x, w, v, f, g, k)    .0. ( w, v)

Proof of Theorem lcfrlem38
StepHypRef Expression
1 lcfrlem38.h . . 3  |-  H  =  ( LHyp `  K
)
2 lcfrlem38.o . . 3  |-  ._|_  =  ( ( ocH `  K
) `  W )
3 lcfrlem38.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
4 lcfrlem38.v . . 3  |-  V  =  ( Base `  U
)
5 lcfrlem38.p . . 3  |-  .+  =  ( +g  `  U )
6 lcfrlem38.z . . 3  |-  .0.  =  ( 0g `  U )
7 lcfrlem38.sp . . 3  |-  N  =  ( LSpan `  U )
8 eqid 2435 . . 3  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
9 lcfrlem38.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
109adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 lcfrlem38.l . . . . . 6  |-  L  =  (LKer `  U )
12 lcfrlem38.d . . . . . 6  |-  D  =  (LDual `  U )
13 lcfrlem38.q . . . . . 6  |-  Q  =  ( LSubSp `  D )
14 lcfrlem38.e . . . . . 6  |-  E  = 
U_ g  e.  G  (  ._|_  `  ( L `  g ) )
15 lcfrlem38.g . . . . . 6  |-  ( ph  ->  G  e.  Q )
16 lcfrlem38.xe . . . . . 6  |-  ( ph  ->  X  e.  E )
171, 2, 3, 4, 11, 12, 13, 14, 9, 15, 16lcfrlem4 32270 . . . . 5  |-  ( ph  ->  X  e.  V )
18 lcfrlem38.x . . . . 5  |-  ( ph  ->  X  =/=  .0.  )
19 eldifsn 3919 . . . . 5  |-  ( X  e.  ( V  \  {  .0.  } )  <->  ( X  e.  V  /\  X  =/= 
.0.  ) )
2017, 18, 19sylanbrc 646 . . . 4  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2120adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  X  e.  ( V  \  {  .0.  } ) )
22 lcfrlem38.ye . . . . . 6  |-  ( ph  ->  Y  e.  E )
231, 2, 3, 4, 11, 12, 13, 14, 9, 15, 22lcfrlem4 32270 . . . . 5  |-  ( ph  ->  Y  e.  V )
24 lcfrlem38.y . . . . 5  |-  ( ph  ->  Y  =/=  .0.  )
25 eldifsn 3919 . . . . 5  |-  ( Y  e.  ( V  \  {  .0.  } )  <->  ( Y  e.  V  /\  Y  =/= 
.0.  ) )
2623, 24, 25sylanbrc 646 . . . 4  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2726adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  Y  e.  ( V  \  {  .0.  } ) )
28 lcfrlem38.ne . . . 4  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
2928adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
30 lcfrlem38.b . . 3  |-  B  =  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  { ( X  .+  Y ) } ) )
31 lcfrlem38.t . . 3  |-  .x.  =  ( .s `  U )
32 lcfrlem38.s . . 3  |-  S  =  (Scalar `  U )
33 eqid 2435 . . 3  |-  ( 0g
`  S )  =  ( 0g `  S
)
34 lcfrlem38.r . . 3  |-  R  =  ( Base `  S
)
35 lcfrlem38.j . . 3  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
36 lcfrlem38.i . . . 4  |-  ( ph  ->  I  e.  B )
3736adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  I  e.  B )
38 simpr 448 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )
39 lcfrlem38.n . . . 4  |-  ( ph  ->  I  =/=  .0.  )
4039adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  I  =/=  .0.  )
4115, 13syl6eleq 2525 . . . 4  |-  ( ph  ->  G  e.  ( LSubSp `  D ) )
4241adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  G  e.  ( LSubSp `  D )
)
43 lcfrlem38.gs . . . . 5  |-  ( ph  ->  G  C_  C )
44 lcfrlem38.c . . . . 5  |-  C  =  { f  e.  (LFnl `  U )  |  ( 
._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( L `  f ) }
4543, 44syl6sseq 3386 . . . 4  |-  ( ph  ->  G  C_  { f  e.  (LFnl `  U )  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) } )
4645adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  G  C_  { f  e.  (LFnl `  U
)  |  (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f ) } )
4716adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  X  e.  E )
4822adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  Y  e.  E )
491, 2, 3, 4, 5, 6, 7, 8, 10, 21, 27, 29, 30, 31, 32, 33, 34, 35, 37, 11, 12, 38, 40, 42, 46, 14, 47, 48lcfrlem27 32294 . 2  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =  ( 0g `  S ) )  ->  ( X  .+  Y )  e.  E
)
509adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
5120adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  X  e.  ( V  \  {  .0.  } ) )
5226adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  Y  e.  ( V  \  {  .0.  } ) )
5328adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
5436adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  I  e.  B
)
55 simpr 448 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  ( ( J `
 Y ) `  I )  =/=  ( 0g `  S ) )
56 eqid 2435 . . 3  |-  ( invr `  S )  =  (
invr `  S )
57 eqid 2435 . . 3  |-  ( -g `  D )  =  (
-g `  D )
58 eqid 2435 . . 3  |-  ( ( J `  X ) ( -g `  D
) ( ( ( ( invr `  S
) `  ( ( J `  Y ) `  I ) ) ( .r `  S ) ( ( J `  X ) `  I
) ) ( .s
`  D ) ( J `  Y ) ) )  =  ( ( J `  X
) ( -g `  D
) ( ( ( ( invr `  S
) `  ( ( J `  Y ) `  I ) ) ( .r `  S ) ( ( J `  X ) `  I
) ) ( .s
`  D ) ( J `  Y ) ) )
5941adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  G  e.  (
LSubSp `  D ) )
6045adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  G  C_  { f  e.  (LFnl `  U
)  |  (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f ) } )
6116adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  X  e.  E
)
6222adantr 452 . . 3  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  Y  e.  E
)
631, 2, 3, 4, 5, 6, 7, 8, 50, 51, 52, 53, 30, 31, 32, 33, 34, 35, 54, 11, 12, 55, 56, 57, 58, 59, 60, 14, 61, 62lcfrlem37 32304 . 2  |-  ( (
ph  /\  ( ( J `  Y ) `  I )  =/=  ( 0g `  S ) )  ->  ( X  .+  Y )  e.  E
)
6449, 63pm2.61dane 2676 1  |-  ( ph  ->  ( X  .+  Y
)  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   {crab 2701    \ cdif 3309    i^i cin 3311    C_ wss 3312   {csn 3806   {cpr 3807   U_ciun 4085    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   iota_crio 6534   Basecbs 13461   +g cplusg 13521   .rcmulr 13522  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   -gcsg 14680   invrcinvr 15768   LSubSpclss 16000   LSpanclspn 16039  LSAtomsclsa 29699  LFnlclfn 29782  LKerclk 29810  LDualcld 29848   HLchlt 30075   LHypclh 30708   DVecHcdvh 31803   ocHcoch 32072
This theorem is referenced by:  lcfrlem39  32306
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 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 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-iin 4088  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-undef 6535  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 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-oppg 15134  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29701  df-lshyp 29702  df-lcv 29744  df-lfl 29783  df-lkr 29811  df-ldual 29849  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-llines 30222  df-lplanes 30223  df-lvols 30224  df-lines 30225  df-psubsp 30227  df-pmap 30228  df-padd 30520  df-lhyp 30712  df-laut 30713  df-ldil 30828  df-ltrn 30829  df-trl 30883  df-tgrp 31467  df-tendo 31479  df-edring 31481  df-dveca 31727  df-disoa 31754  df-dvech 31804  df-dib 31864  df-dic 31898  df-dih 31954  df-doch 32073  df-djh 32120
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