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Theorem lsatexch 28500
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 22953 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 15853 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 17 . . . . 5  |-  ( ph  ->  W  e.  LMod )
4 lsatexch.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssssubg 15709 . . . . 5  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
63, 5syl 17 . . . 4  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
7 lsatexch.u . . . 4  |-  ( ph  ->  U  e.  S )
86, 7sseldd 3182 . . 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 28454 . . . 4  |-  ( ph  ->  R  e.  S )
126, 11sseldd 3182 . . 3  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
13 lsatexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
1413lsmub2 14962 . . 3  |-  ( ( U  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  R  C_  ( U  .(+)  R ) )
158, 12, 14syl2anc 645 . 2  |-  ( ph  ->  R  C_  ( U  .(+) 
R ) )
16 eqid 2284 . . 3  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
174, 13lsmcl 15830 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  R  e.  S )  ->  ( U  .(+)  R )  e.  S )
183, 7, 11, 17syl3anc 1187 . . 3  |-  ( ph  ->  ( U  .(+)  R )  e.  S )
19 lsatexch.q . . . . 5  |-  ( ph  ->  Q  e.  A )
204, 9, 3, 19lsatlssel 28454 . . . 4  |-  ( ph  ->  Q  e.  S )
214, 13lsmcl 15830 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  Q  e.  S )  ->  ( U  .(+)  Q )  e.  S )
223, 7, 20, 21syl3anc 1187 . . 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 28497 . . . . . . 7  |-  ( ph  ->  ( ( U  i^i  Q )  =  {  .0.  }  <-> 
U (  <oLL  `  W ) ( U  .(+)  Q ) ) )
2623, 25mpbid 203 . . . . . 6  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  Q ) )
274, 16, 1, 7, 22, 26lcvpss 28481 . . . . 5  |-  ( ph  ->  U  C.  ( U 
.(+)  Q ) )
2813lsmub1 14961 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  U  C_  ( U  .(+)  R ) )
298, 12, 28syl2anc 645 . . . . . 6  |-  ( ph  ->  U  C_  ( U  .(+) 
R ) )
30 lsatexch.l . . . . . 6  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
316, 20sseldd 3182 . . . . . . 7  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
326, 18sseldd 3182 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
3313lsmlub 14968 . . . . . . 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 1187 . . . . . 6  |-  ( ph  ->  ( ( U  C_  ( U  .(+)  R )  /\  Q  C_  ( U  .(+)  R ) )  <-> 
( U  .(+)  Q ) 
C_  ( U  .(+)  R ) ) )
3529, 30, 34mpbi2and 892 . . . . 5  |-  ( ph  ->  ( U  .(+)  Q ) 
C_  ( U  .(+)  R ) )
36 psssstr 3283 . . . . 5  |-  ( ( U  C.  ( U 
.(+)  Q )  /\  ( U  .(+)  Q )  C_  ( U  .(+)  R ) )  ->  U  C.  ( U  .(+)  R ) )
3727, 35, 36syl2anc 645 . . . 4  |-  ( ph  ->  U  C.  ( U 
.(+)  R ) )
384, 13, 9, 16, 1, 7, 10lcv2 28499 . . . 4  |-  ( ph  ->  ( U  C.  ( U  .(+)  R )  <->  U (  <oLL  `  W ) ( U 
.(+)  R ) ) )
3937, 38mpbid 203 . . 3  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  R ) )
404, 16, 1, 7, 18, 22, 39, 27, 35lcvnbtwn2 28484 . 2  |-  ( ph  ->  ( U  .(+)  Q )  =  ( U  .(+)  R ) )
4115, 40sseqtr4d 3216 1  |-  ( ph  ->  R  C_  ( U  .(+) 
Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1628    e. wcel 1688    i^i cin 3152    C_ wss 3153    C. wpss 3154   {csn 3641   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   0gc0g 13394  SubGrpcsubg 14609   LSSumclsm 14939   LModclmod 15621   LSubSpclss 15683   LVecclvec 15849  LSAtomsclsa 28431    <oLL clcv 28475
This theorem is referenced by:  lsatexch1  28503
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-tpos 6195  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-3 9800  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-0g 13398  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-grp 14483  df-minusg 14484  df-sbg 14485  df-subg 14612  df-cntz 14787  df-oppg 14813  df-lsm 14941  df-cmn 15085  df-abl 15086  df-mgp 15320  df-rng 15334  df-ur 15336  df-oppr 15399  df-dvdsr 15417  df-unit 15418  df-invr 15448  df-drng 15508  df-lmod 15623  df-lss 15684  df-lsp 15723  df-lvec 15850  df-lsatoms 28433  df-lcv 28476
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