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Theorem lsatcvat3 29539
Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23856 analog.) (Contributed by NM, 11-Jan-2015.)
Hypotheses
Ref Expression
lsatcvat3.s  |-  S  =  ( LSubSp `  W )
lsatcvat3.p  |-  .(+)  =  (
LSSum `  W )
lsatcvat3.a  |-  A  =  (LSAtoms `  W )
lsatcvat3.w  |-  ( ph  ->  W  e.  LVec )
lsatcvat3.u  |-  ( ph  ->  U  e.  S )
lsatcvat3.q  |-  ( ph  ->  Q  e.  A )
lsatcvat3.r  |-  ( ph  ->  R  e.  A )
lsatcvat3.n  |-  ( ph  ->  Q  =/=  R )
lsatcvat3.m  |-  ( ph  ->  -.  R  C_  U
)
lsatcvat3.l  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
Assertion
Ref Expression
lsatcvat3  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  A )

Proof of Theorem lsatcvat3
StepHypRef Expression
1 lsatcvat3.s . 2  |-  S  =  ( LSubSp `  W )
2 lsatcvat3.p . 2  |-  .(+)  =  (
LSSum `  W )
3 lsatcvat3.a . 2  |-  A  =  (LSAtoms `  W )
4 eqid 2408 . 2  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
5 lsatcvat3.w . 2  |-  ( ph  ->  W  e.  LVec )
6 lveclmod 16137 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
75, 6syl 16 . . 3  |-  ( ph  ->  W  e.  LMod )
8 lsatcvat3.u . . 3  |-  ( ph  ->  U  e.  S )
9 lsatcvat3.q . . . . 5  |-  ( ph  ->  Q  e.  A )
101, 3, 7, 9lsatlssel 29484 . . . 4  |-  ( ph  ->  Q  e.  S )
11 lsatcvat3.r . . . . 5  |-  ( ph  ->  R  e.  A )
121, 3, 7, 11lsatlssel 29484 . . . 4  |-  ( ph  ->  R  e.  S )
131, 2lsmcl 16114 . . . 4  |-  ( ( W  e.  LMod  /\  Q  e.  S  /\  R  e.  S )  ->  ( Q  .(+)  R )  e.  S )
147, 10, 12, 13syl3anc 1184 . . 3  |-  ( ph  ->  ( Q  .(+)  R )  e.  S )
151lssincl 16000 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  ( Q  .(+)  R )  e.  S )  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  S
)
167, 8, 14, 15syl3anc 1184 . 2  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  S )
17 lsatcvat3.n . 2  |-  ( ph  ->  Q  =/=  R )
18 lsatcvat3.m . . . . 5  |-  ( ph  ->  -.  R  C_  U
)
191, 2, 3, 4, 5, 8, 11lcv1 29528 . . . . 5  |-  ( ph  ->  ( -.  R  C_  U 
<->  U (  <oLL  `  W ) ( U  .(+)  R ) ) )
2018, 19mpbid 202 . . . 4  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  R ) )
21 lmodabl 15950 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
227, 21syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
231lsssssubg 15993 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
247, 23syl 16 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
2524, 10sseldd 3313 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
2624, 12sseldd 3313 . . . . . . . . . 10  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
272lsmcom 15432 . . . . . . . . . 10  |-  ( ( W  e.  Abel  /\  Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W ) )  -> 
( Q  .(+)  R )  =  ( R  .(+)  Q ) )
2822, 25, 26, 27syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( Q  .(+)  R )  =  ( R  .(+)  Q ) )
2928oveq2d 6060 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  ( R 
.(+)  Q ) ) )
3024, 8sseldd 3313 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
312lsmass 15261 . . . . . . . . 9  |-  ( ( U  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )  /\  Q  e.  (SubGrp `  W ) )  -> 
( ( U  .(+)  R )  .(+)  Q )  =  ( U  .(+)  ( R  .(+)  Q )
) )
3230, 26, 25, 31syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( ( U  .(+)  R )  .(+)  Q )  =  ( U  .(+)  ( R  .(+)  Q )
) )
3329, 32eqtr4d 2443 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( ( U  .(+)  R )  .(+)  Q )
)
341, 2lsmcl 16114 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  R  e.  S )  ->  ( U  .(+)  R )  e.  S )
357, 8, 12, 34syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( U  .(+)  R )  e.  S )
3624, 35sseldd 3313 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
37 lsatcvat3.l . . . . . . . 8  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
382lsmless2 15253 . . . . . . . 8  |-  ( ( ( U  .(+)  R )  e.  (SubGrp `  W
)  /\  ( U  .(+) 
R )  e.  (SubGrp `  W )  /\  Q  C_  ( U  .(+)  R ) )  ->  ( ( U  .(+)  R )  .(+)  Q )  C_  ( ( U  .(+)  R )  .(+)  ( U  .(+)  R )
) )
3936, 36, 37, 38syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( U  .(+)  R )  .(+)  Q )  C_  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) ) )
4033, 39eqsstrd 3346 . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) ) )
412lsmidm 15255 . . . . . . 7  |-  ( ( U  .(+)  R )  e.  (SubGrp `  W )  ->  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) )  =  ( U  .(+)  R ) )
4236, 41syl 16 . . . . . 6  |-  ( ph  ->  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) )  =  ( U  .(+)  R ) )
4340, 42sseqtrd 3348 . . . . 5  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( U  .(+)  R ) )
4424, 14sseldd 3313 . . . . . 6  |-  ( ph  ->  ( Q  .(+)  R )  e.  (SubGrp `  W
) )
452lsmub2 15250 . . . . . . 7  |-  ( ( Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  R  C_  ( Q  .(+)  R ) )
4625, 26, 45syl2anc 643 . . . . . 6  |-  ( ph  ->  R  C_  ( Q  .(+) 
R ) )
472lsmless2 15253 . . . . . 6  |-  ( ( U  e.  (SubGrp `  W )  /\  ( Q  .(+)  R )  e.  (SubGrp `  W )  /\  R  C_  ( Q 
.(+)  R ) )  -> 
( U  .(+)  R ) 
C_  ( U  .(+)  ( Q  .(+)  R )
) )
4830, 44, 46, 47syl3anc 1184 . . . . 5  |-  ( ph  ->  ( U  .(+)  R ) 
C_  ( U  .(+)  ( Q  .(+)  R )
) )
4943, 48eqssd 3329 . . . 4  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  R ) )
5020, 49breqtrrd 4202 . . 3  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  ( Q 
.(+)  R ) ) )
511, 2, 4, 7, 8, 14, 50lcvexchlem4 29524 . 2  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) ) (  <oLL  `  W ) ( Q  .(+)  R )
)
521, 2, 3, 4, 5, 16, 9, 11, 17, 51lsatcvat2 29538 1  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2571    i^i cin 3283    C_ wss 3284   class class class wbr 4176   ` cfv 5417  (class class class)co 6044  SubGrpcsubg 14897   LSSumclsm 15227   Abelcabel 15372   LModclmod 15909   LSubSpclss 15967   LVecclvec 16133  LSAtomsclsa 29461    <oLL clcv 29505
This theorem is referenced by:  l1cvat  29542
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-tpos 6442  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-0g 13686  df-mre 13770  df-mrc 13771  df-acs 13773  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-oppg 15101  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lsatoms 29463  df-lcv 29506
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