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Theorem lsatexch 28400
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 22921 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 15821 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 17 . . . . 5  |-  ( ph  ->  W  e.  LMod )
4 lsatexch.s . . . . . 6  |-  S  =  ( LSubSp `  W )
54lsssssubg 15677 . . . . 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 3156 . . 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 28354 . . . 4  |-  ( ph  ->  R  e.  S )
126, 11sseldd 3156 . . 3  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
13 lsatexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
1413lsmub2 14930 . . 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 2258 . . 3  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
174, 13lsmcl 15798 . . . 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 28354 . . . 4  |-  ( ph  ->  Q  e.  S )
214, 13lsmcl 15798 . . . 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 28397 . . . . . . 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 28381 . . . . 5  |-  ( ph  ->  U  C.  ( U 
.(+)  Q ) )
2813lsmub1 14929 . . . . . . 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 3156 . . . . . . 7  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
326, 18sseldd 3156 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
3313lsmlub 14936 . . . . . . 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 3257 . . . . 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 28399 . . . 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 28384 . 2  |-  ( ph  ->  ( U  .(+)  Q )  =  ( U  .(+)  R ) )
4115, 40sseqtr4d 3190 1  |-  ( ph  ->  R  C_  ( U  .(+) 
Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    i^i cin 3126    C_ wss 3127    C. wpss 3128   {csn 3614   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   0gc0g 13362  SubGrpcsubg 14577   LSSumclsm 14907   LModclmod 15589   LSubSpclss 15651   LVecclvec 15817  LSAtomsclsa 28331    <oLL clcv 28375
This theorem is referenced by:  lsatexch1  28403
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-0g 13366  df-mre 13450  df-mrc 13451  df-acs 13453  df-mnd 14329  df-submnd 14378  df-grp 14451  df-minusg 14452  df-sbg 14453  df-subg 14580  df-cntz 14755  df-oppg 14781  df-lsm 14909  df-cmn 15053  df-abl 15054  df-mgp 15288  df-ring 15302  df-ur 15304  df-oppr 15367  df-dvdsr 15385  df-unit 15386  df-invr 15416  df-drng 15476  df-lmod 15591  df-lss 15652  df-lsp 15691  df-lvec 15818  df-lsatoms 28333  df-lcv 28376
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