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Theorem lsatexch 29233
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 22961 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 15859 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 15 . . . . 5  |-  ( ph  ->  W  e.  LMod )
4 lsatexch.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssssubg 15715 . . . . 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 3181 . . 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 29187 . . . 4  |-  ( ph  ->  R  e.  S )
126, 11sseldd 3181 . . 3  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
13 lsatexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
1413lsmub2 14968 . . 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 2283 . . 3  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
174, 13lsmcl 15836 . . . 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 29187 . . . 4  |-  ( ph  ->  Q  e.  S )
214, 13lsmcl 15836 . . . 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 29230 . . . . . . 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 29214 . . . . 5  |-  ( ph  ->  U  C.  ( U 
.(+)  Q ) )
2813lsmub1 14967 . . . . . . 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 3181 . . . . . . 7  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
326, 18sseldd 3181 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
3313lsmlub 14974 . . . . . . 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 3282 . . . . 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 29232 . . . 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 29217 . 2  |-  ( ph  ->  ( U  .(+)  Q )  =  ( U  .(+)  R ) )
4115, 40sseqtr4d 3215 1  |-  ( ph  ->  R  C_  ( U  .(+) 
Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152    C. wpss 3153   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   0gc0g 13400  SubGrpcsubg 14615   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LVecclvec 15855  LSAtomsclsa 29164    <oLL clcv 29208
This theorem is referenced by:  lsatexch1  29236
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-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-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  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-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lcv 29209
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