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Theorem lsatexch 29526
Description: The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 23837 analog.) (Contributed by NM, 10-Jan-2015.)
Hypotheses
Ref Expression
lsatexch.s  |-  S  =  ( LSubSp `  W )
lsatexch.p  |-  .(+)  =  (
LSSum `  W )
lsatexch.o  |-  .0.  =  ( 0g `  W )
lsatexch.a  |-  A  =  (LSAtoms `  W )
lsatexch.w  |-  ( ph  ->  W  e.  LVec )
lsatexch.u  |-  ( ph  ->  U  e.  S )
lsatexch.q  |-  ( ph  ->  Q  e.  A )
lsatexch.r  |-  ( ph  ->  R  e.  A )
lsatexch.l  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
lsatexch.z  |-  ( ph  ->  ( U  i^i  Q
)  =  {  .0.  } )
Assertion
Ref Expression
lsatexch  |-  ( ph  ->  R  C_  ( U  .(+) 
Q ) )

Proof of Theorem lsatexch
StepHypRef Expression
1 lsatexch.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
2 lveclmod 16133 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
4 lsatexch.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssssubg 15989 . . . . 5  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
63, 5syl 16 . . . 4  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
7 lsatexch.u . . . 4  |-  ( ph  ->  U  e.  S )
86, 7sseldd 3309 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
9 lsatexch.a . . . . 5  |-  A  =  (LSAtoms `  W )
10 lsatexch.r . . . . 5  |-  ( ph  ->  R  e.  A )
114, 9, 3, 10lsatlssel 29480 . . . 4  |-  ( ph  ->  R  e.  S )
126, 11sseldd 3309 . . 3  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
13 lsatexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
1413lsmub2 15246 . . 3  |-  ( ( U  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  R  C_  ( U  .(+)  R ) )
158, 12, 14syl2anc 643 . 2  |-  ( ph  ->  R  C_  ( U  .(+) 
R ) )
16 eqid 2404 . . 3  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
174, 13lsmcl 16110 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  R  e.  S )  ->  ( U  .(+)  R )  e.  S )
183, 7, 11, 17syl3anc 1184 . . 3  |-  ( ph  ->  ( U  .(+)  R )  e.  S )
19 lsatexch.q . . . . 5  |-  ( ph  ->  Q  e.  A )
204, 9, 3, 19lsatlssel 29480 . . . 4  |-  ( ph  ->  Q  e.  S )
214, 13lsmcl 16110 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  Q  e.  S )  ->  ( U  .(+)  Q )  e.  S )
223, 7, 20, 21syl3anc 1184 . . 3  |-  ( ph  ->  ( U  .(+)  Q )  e.  S )
23 lsatexch.z . . . . . . 7  |-  ( ph  ->  ( U  i^i  Q
)  =  {  .0.  } )
24 lsatexch.o . . . . . . . 8  |-  .0.  =  ( 0g `  W )
254, 13, 24, 9, 16, 1, 7, 19lcvp 29523 . . . . . . 7  |-  ( ph  ->  ( ( U  i^i  Q )  =  {  .0.  }  <-> 
U (  <oLL  `  W ) ( U  .(+)  Q ) ) )
2623, 25mpbid 202 . . . . . 6  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  Q ) )
274, 16, 1, 7, 22, 26lcvpss 29507 . . . . 5  |-  ( ph  ->  U  C.  ( U 
.(+)  Q ) )
2813lsmub1 15245 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  U  C_  ( U  .(+)  R ) )
298, 12, 28syl2anc 643 . . . . . 6  |-  ( ph  ->  U  C_  ( U  .(+) 
R ) )
30 lsatexch.l . . . . . 6  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
316, 20sseldd 3309 . . . . . . 7  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
326, 18sseldd 3309 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
3313lsmlub 15252 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  W )  /\  Q  e.  (SubGrp `  W )  /\  ( U  .(+)  R )  e.  (SubGrp `  W
) )  ->  (
( U  C_  ( U  .(+)  R )  /\  Q  C_  ( U  .(+)  R ) )  <->  ( U  .(+) 
Q )  C_  ( U  .(+)  R ) ) )
348, 31, 32, 33syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( U  C_  ( U  .(+)  R )  /\  Q  C_  ( U  .(+)  R ) )  <-> 
( U  .(+)  Q ) 
C_  ( U  .(+)  R ) ) )
3529, 30, 34mpbi2and 888 . . . . 5  |-  ( ph  ->  ( U  .(+)  Q ) 
C_  ( U  .(+)  R ) )
3627, 35psssstrd 3416 . . . 4  |-  ( ph  ->  U  C.  ( U 
.(+)  R ) )
374, 13, 9, 16, 1, 7, 10lcv2 29525 . . . 4  |-  ( ph  ->  ( U  C.  ( U  .(+)  R )  <->  U (  <oLL  `  W ) ( U 
.(+)  R ) ) )
3836, 37mpbid 202 . . 3  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  R ) )
394, 16, 1, 7, 18, 22, 38, 27, 35lcvnbtwn2 29510 . 2  |-  ( ph  ->  ( U  .(+)  Q )  =  ( U  .(+)  R ) )
4015, 39sseqtr4d 3345 1  |-  ( ph  ->  R  C_  ( U  .(+) 
Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3279    C_ wss 3280    C. wpss 3281   {csn 3774   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   0gc0g 13678  SubGrpcsubg 14893   LSSumclsm 15223   LModclmod 15905   LSubSpclss 15963   LVecclvec 16129  LSAtomsclsa 29457    <oLL clcv 29501
This theorem is referenced by:  lsatexch1  29529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-oppg 15097  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459  df-lcv 29502
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