Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcfrlem37 Unicode version

Theorem lcfrlem37 31142
Description: Lemma for lcfr 31148. (Contributed by NM, 8-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 ) ) )
lcfrlem37.g  |-  ( ph  ->  G  e.  ( LSubSp `  D ) )
lcfrlem37.gs  |-  ( ph  ->  G  C_  { f  e.  (LFnl `  U )  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) } )
lcfrlem37.e  |-  E  = 
U_ g  e.  G  (  ._|_  `  ( L `  g ) )
lcfrlem37.xe  |-  ( ph  ->  X  e.  E )
lcfrlem37.ye  |-  ( ph  ->  Y  e.  E )
Assertion
Ref Expression
lcfrlem37  |-  ( ph  ->  ( X  .+  Y
)  e.  E )
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.    f, J    f, L    ._|_ ,
f    .+ , f    R, f    .x. , f    U, f    f, V   
f, X    f, Y, k, v, w, x, g    C, g, k    D, g, k    g, G, k   
g, I, k    f,
g, J, k    g, L, k    ._|_ , g    .+ , g    Q, g, k    U, k   
g, V    g, X    g, Y    ph, g, k    v,
g, w, x
Allowed substitution hints:    ph( x, w, v, f)    A( x, w, v, f, g, k)    B( x, w, v, f, g, k)    C( x, w, v, f)    D( x, w, v, f)    Q( x, w, v, f)    R( w, g)    S( x, w, v, f, g)    .x. ( g)    U( x, w, v, 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)    .- ( x, w, v, f, g, k)    N( x, w, v, f, g, k)    V( w, k)    W( x, w, v, f, g, k)    .0. ( w, v, f, g, k)

Proof of Theorem lcfrlem37
StepHypRef Expression
1 lcfrlem30.c . . . . 5  |-  C  =  ( ( J `  X )  .-  (
( ( F `  ( ( J `  Y ) `  I
) ) ( .r
`  S ) ( ( J `  X
) `  I )
) ( .s `  D ) ( J `
 Y ) ) )
2 lcfrlem25.d . . . . . 6  |-  D  =  (LDual `  U )
3 lcfrlem30.m . . . . . 6  |-  .-  =  ( -g `  D )
4 eqid 2283 . . . . . 6  |-  ( LSubSp `  D )  =  (
LSubSp `  D )
5 lcfrlem17.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
6 lcfrlem17.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
7 lcfrlem17.k . . . . . . 7  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
85, 6, 7dvhlmod 30673 . . . . . 6  |-  ( ph  ->  U  e.  LMod )
9 lcfrlem37.g . . . . . 6  |-  ( ph  ->  G  e.  ( LSubSp `  D ) )
10 lcfrlem17.o . . . . . . 7  |-  ._|_  =  ( ( ocH `  K
) `  W )
11 lcfrlem17.v . . . . . . 7  |-  V  =  ( Base `  U
)
12 lcfrlem17.p . . . . . . 7  |-  .+  =  ( +g  `  U )
13 lcfrlem24.t . . . . . . 7  |-  .x.  =  ( .s `  U )
14 lcfrlem24.s . . . . . . 7  |-  S  =  (Scalar `  U )
15 lcfrlem24.r . . . . . . 7  |-  R  =  ( Base `  S
)
16 lcfrlem17.z . . . . . . 7  |-  .0.  =  ( 0g `  U )
17 eqid 2283 . . . . . . 7  |-  (LFnl `  U )  =  (LFnl `  U )
18 lcfrlem24.l . . . . . . 7  |-  L  =  (LKer `  U )
19 eqid 2283 . . . . . . 7  |-  ( 0g
`  D )  =  ( 0g `  D
)
20 eqid 2283 . . . . . . 7  |-  { f  e.  (LFnl `  U
)  |  (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f ) }  =  { f  e.  (LFnl `  U )  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
21 lcfrlem24.j . . . . . . 7  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
22 lcfrlem37.gs . . . . . . 7  |-  ( ph  ->  G  C_  { f  e.  (LFnl `  U )  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) } )
23 lcfrlem37.e . . . . . . 7  |-  E  = 
U_ g  e.  G  (  ._|_  `  ( L `  g ) )
24 lcfrlem37.xe . . . . . . . 8  |-  ( ph  ->  X  e.  E )
25 lcfrlem17.x . . . . . . . . 9  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
26 eldifsni 3750 . . . . . . . . 9  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  =/=  .0.  )
2725, 26syl 15 . . . . . . . 8  |-  ( ph  ->  X  =/=  .0.  )
28 eldifsn 3749 . . . . . . . 8  |-  ( X  e.  ( E  \  {  .0.  } )  <->  ( X  e.  E  /\  X  =/= 
.0.  ) )
2924, 27, 28sylanbrc 645 . . . . . . 7  |-  ( ph  ->  X  e.  ( E 
\  {  .0.  }
) )
305, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 29lcfrlem16 31121 . . . . . 6  |-  ( ph  ->  ( J `  X
)  e.  G )
31 eqid 2283 . . . . . . 7  |-  ( .s
`  D )  =  ( .s `  D
)
32 lcfrlem17.n . . . . . . . 8  |-  N  =  ( LSpan `  U )
33 lcfrlem17.a . . . . . . . 8  |-  A  =  (LSAtoms `  U )
34 lcfrlem17.y . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
35 lcfrlem17.ne . . . . . . . 8  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
36 lcfrlem22.b . . . . . . . 8  |-  B  =  ( ( N `  { X ,  Y }
)  i^i  (  ._|_  `  { ( X  .+  Y ) } ) )
37 lcfrlem24.q . . . . . . . 8  |-  Q  =  ( 0g `  S
)
38 lcfrlem24.ib . . . . . . . 8  |-  ( ph  ->  I  e.  B )
39 lcfrlem28.jn . . . . . . . 8  |-  ( ph  ->  ( ( J `  Y ) `  I
)  =/=  Q )
40 lcfrlem29.i . . . . . . . 8  |-  F  =  ( invr `  S
)
415, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40lcfrlem29 31134 . . . . . . 7  |-  ( ph  ->  ( ( F `  ( ( J `  Y ) `  I
) ) ( .r
`  S ) ( ( J `  X
) `  I )
)  e.  R )
42 lcfrlem37.ye . . . . . . . . 9  |-  ( ph  ->  Y  e.  E )
43 eldifsni 3750 . . . . . . . . . 10  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
4434, 43syl 15 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  .0.  )
45 eldifsn 3749 . . . . . . . . 9  |-  ( Y  e.  ( E  \  {  .0.  } )  <->  ( Y  e.  E  /\  Y  =/= 
.0.  ) )
4642, 44, 45sylanbrc 645 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( E 
\  {  .0.  }
) )
475, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 46lcfrlem16 31121 . . . . . . 7  |-  ( ph  ->  ( J `  Y
)  e.  G )
4814, 15, 2, 31, 4, 8, 9, 41, 47ldualssvscl 28721 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 ( ( J `
 Y ) `  I ) ) ( .r `  S ) ( ( J `  X ) `  I
) ) ( .s
`  D ) ( J `  Y ) )  e.  G )
492, 3, 4, 8, 9, 30, 48ldualssvsubcl 28722 . . . . 5  |-  ( ph  ->  ( ( J `  X )  .-  (
( ( F `  ( ( J `  Y ) `  I
) ) ( .r
`  S ) ( ( J `  X
) `  I )
) ( .s `  D ) ( J `
 Y ) ) )  e.  G )
501, 49syl5eqel 2367 . . . 4  |-  ( ph  ->  C  e.  G )
515, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40, 3, 1lcfrlem36 31141 . . . 4  |-  ( ph  ->  ( X  .+  Y
)  e.  (  ._|_  `  ( L `  C
) ) )
52 fveq2 5525 . . . . . . 7  |-  ( g  =  C  ->  ( L `  g )  =  ( L `  C ) )
5352fveq2d 5529 . . . . . 6  |-  ( g  =  C  ->  (  ._|_  `  ( L `  g ) )  =  (  ._|_  `  ( L `
 C ) ) )
5453eleq2d 2350 . . . . 5  |-  ( g  =  C  ->  (
( X  .+  Y
)  e.  (  ._|_  `  ( L `  g
) )  <->  ( X  .+  Y )  e.  ( 
._|_  `  ( L `  C ) ) ) )
5554rspcev 2884 . . . 4  |-  ( ( C  e.  G  /\  ( X  .+  Y )  e.  (  ._|_  `  ( L `  C )
) )  ->  E. g  e.  G  ( X  .+  Y )  e.  ( 
._|_  `  ( L `  g ) ) )
5650, 51, 55syl2anc 642 . . 3  |-  ( ph  ->  E. g  e.  G  ( X  .+  Y )  e.  (  ._|_  `  ( L `  g )
) )
57 eliun 3909 . . 3  |-  ( ( X  .+  Y )  e.  U_ g  e.  G  (  ._|_  `  ( L `  g )
)  <->  E. g  e.  G  ( X  .+  Y )  e.  (  ._|_  `  ( L `  g )
) )
5856, 57sylibr 203 . 2  |-  ( ph  ->  ( X  .+  Y
)  e.  U_ g  e.  G  (  ._|_  `  ( L `  g
) ) )
5958, 23syl6eleqr 2374 1  |-  ( ph  ->  ( X  .+  Y
)  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   {cpr 3641   U_ciun 3905    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   -gcsg 14365   invrcinvr 15453   LSubSpclss 15689   LSpanclspn 15728  LSAtomsclsa 28537  LFnlclfn 28620  LKerclk 28648  LDualcld 28686   HLchlt 28913   LHypclh 29546   DVecHcdvh 30641   ocHcoch 30910
This theorem is referenced by:  lcfrlem38  31143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 28539  df-lshyp 28540  df-lcv 28582  df-lfl 28621  df-lkr 28649  df-ldual 28687  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-llines 29060  df-lplanes 29061  df-lvols 29062  df-lines 29063  df-psubsp 29065  df-pmap 29066  df-padd 29358  df-lhyp 29550  df-laut 29551  df-ldil 29666  df-ltrn 29667  df-trl 29721  df-tgrp 30305  df-tendo 30317  df-edring 30319  df-dveca 30565  df-disoa 30592  df-dvech 30642  df-dib 30702  df-dic 30736  df-dih 30792  df-doch 30911  df-djh 30958
  Copyright terms: Public domain W3C validator