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Theorem lsatexch 29855
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 22977 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 15875 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 15 . . . . 5  |-  ( ph  ->  W  e.  LMod )
4 lsatexch.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssssubg 15731 . . . . 5  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
63, 5syl 15 . . . 4  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
7 lsatexch.u . . . 4  |-  ( ph  ->  U  e.  S )
86, 7sseldd 3194 . . 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 29809 . . . 4  |-  ( ph  ->  R  e.  S )
126, 11sseldd 3194 . . 3  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
13 lsatexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
1413lsmub2 14984 . . 3  |-  ( ( U  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  R  C_  ( U  .(+)  R ) )
158, 12, 14syl2anc 642 . 2  |-  ( ph  ->  R  C_  ( U  .(+) 
R ) )
16 eqid 2296 . . 3  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
174, 13lsmcl 15852 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  R  e.  S )  ->  ( U  .(+)  R )  e.  S )
183, 7, 11, 17syl3anc 1182 . . 3  |-  ( ph  ->  ( U  .(+)  R )  e.  S )
19 lsatexch.q . . . . 5  |-  ( ph  ->  Q  e.  A )
204, 9, 3, 19lsatlssel 29809 . . . 4  |-  ( ph  ->  Q  e.  S )
214, 13lsmcl 15852 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  Q  e.  S )  ->  ( U  .(+)  Q )  e.  S )
223, 7, 20, 21syl3anc 1182 . . 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 29852 . . . . . . 7  |-  ( ph  ->  ( ( U  i^i  Q )  =  {  .0.  }  <-> 
U (  <oLL  `  W ) ( U  .(+)  Q ) ) )
2623, 25mpbid 201 . . . . . 6  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  Q ) )
274, 16, 1, 7, 22, 26lcvpss 29836 . . . . 5  |-  ( ph  ->  U  C.  ( U 
.(+)  Q ) )
2813lsmub1 14983 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  U  C_  ( U  .(+)  R ) )
298, 12, 28syl2anc 642 . . . . . 6  |-  ( ph  ->  U  C_  ( U  .(+) 
R ) )
30 lsatexch.l . . . . . 6  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
316, 20sseldd 3194 . . . . . . 7  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
326, 18sseldd 3194 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
3313lsmlub 14990 . . . . . . 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 1182 . . . . . 6  |-  ( ph  ->  ( ( U  C_  ( U  .(+)  R )  /\  Q  C_  ( U  .(+)  R ) )  <-> 
( U  .(+)  Q ) 
C_  ( U  .(+)  R ) ) )
3529, 30, 34mpbi2and 887 . . . . 5  |-  ( ph  ->  ( U  .(+)  Q ) 
C_  ( U  .(+)  R ) )
36 psssstr 3295 . . . . 5  |-  ( ( U  C.  ( U 
.(+)  Q )  /\  ( U  .(+)  Q )  C_  ( U  .(+)  R ) )  ->  U  C.  ( U  .(+)  R ) )
3727, 35, 36syl2anc 642 . . . 4  |-  ( ph  ->  U  C.  ( U 
.(+)  R ) )
384, 13, 9, 16, 1, 7, 10lcv2 29854 . . . 4  |-  ( ph  ->  ( U  C.  ( U  .(+)  R )  <->  U (  <oLL  `  W ) ( U 
.(+)  R ) ) )
3937, 38mpbid 201 . . 3  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  R ) )
404, 16, 1, 7, 18, 22, 39, 27, 35lcvnbtwn2 29839 . 2  |-  ( ph  ->  ( U  .(+)  Q )  =  ( U  .(+)  R ) )
4115, 40sseqtr4d 3228 1  |-  ( ph  ->  R  C_  ( U  .(+) 
Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165    C. wpss 3166   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   0gc0g 13416  SubGrpcsubg 14631   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LVecclvec 15871  LSAtomsclsa 29786    <oLL clcv 29830
This theorem is referenced by:  lsatexch1  29858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-oppg 14835  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-lcv 29831
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