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Theorem lsatcvat3 27931
Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22806 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 2253 . 2  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
5 lsatcvat3.w . 2  |-  ( ph  ->  W  e.  LVec )
6 lveclmod 15694 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
75, 6syl 17 . . 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 27876 . . . 4  |-  ( ph  ->  Q  e.  S )
11 lsatcvat3.r . . . . 5  |-  ( ph  ->  R  e.  A )
121, 3, 7, 11lsatlssel 27876 . . . 4  |-  ( ph  ->  R  e.  S )
131, 2lsmcl 15671 . . . 4  |-  ( ( W  e.  LMod  /\  Q  e.  S  /\  R  e.  S )  ->  ( Q  .(+)  R )  e.  S )
147, 10, 12, 13syl3anc 1187 . . 3  |-  ( ph  ->  ( Q  .(+)  R )  e.  S )
151lssincl 15557 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  ( Q  .(+)  R )  e.  S )  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  S
)
167, 8, 14, 15syl3anc 1187 . 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 27920 . . . . 5  |-  ( ph  ->  ( -.  R  C_  U 
<->  U (  <oLL  `  W ) ( U  .(+)  R ) ) )
2018, 19mpbid 203 . . . 4  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  R ) )
21 lmodabl 15507 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
227, 21syl 17 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
231lsssssubg 15550 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
247, 23syl 17 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
2524, 10sseldd 3104 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
2624, 12sseldd 3104 . . . . . . . . . 10  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
272lsmcom 14985 . . . . . . . . . 10  |-  ( ( W  e.  Abel  /\  Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W ) )  -> 
( Q  .(+)  R )  =  ( R  .(+)  Q ) )
2822, 25, 26, 27syl3anc 1187 . . . . . . . . 9  |-  ( ph  ->  ( Q  .(+)  R )  =  ( R  .(+)  Q ) )
2928oveq2d 5726 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  ( R 
.(+)  Q ) ) )
3024, 8sseldd 3104 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
312lsmass 14814 . . . . . . . . 9  |-  ( ( U  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )  /\  Q  e.  (SubGrp `  W ) )  -> 
( ( U  .(+)  R )  .(+)  Q )  =  ( U  .(+)  ( R  .(+)  Q )
) )
3230, 26, 25, 31syl3anc 1187 . . . . . . . 8  |-  ( ph  ->  ( ( U  .(+)  R )  .(+)  Q )  =  ( U  .(+)  ( R  .(+)  Q )
) )
3329, 32eqtr4d 2288 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( ( U  .(+)  R )  .(+)  Q )
)
341, 2lsmcl 15671 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  R  e.  S )  ->  ( U  .(+)  R )  e.  S )
357, 8, 12, 34syl3anc 1187 . . . . . . . . 9  |-  ( ph  ->  ( U  .(+)  R )  e.  S )
3624, 35sseldd 3104 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
37 lsatcvat3.l . . . . . . . 8  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
382lsmless2 14806 . . . . . . . 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 1187 . . . . . . 7  |-  ( ph  ->  ( ( U  .(+)  R )  .(+)  Q )  C_  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) ) )
4033, 39eqsstrd 3133 . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) ) )
412lsmidm 14808 . . . . . . 7  |-  ( ( U  .(+)  R )  e.  (SubGrp `  W )  ->  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) )  =  ( U  .(+)  R ) )
4236, 41syl 17 . . . . . 6  |-  ( ph  ->  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) )  =  ( U  .(+)  R ) )
4340, 42sseqtrd 3135 . . . . 5  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( U  .(+)  R ) )
4424, 14sseldd 3104 . . . . . 6  |-  ( ph  ->  ( Q  .(+)  R )  e.  (SubGrp `  W
) )
452lsmub2 14803 . . . . . . 7  |-  ( ( Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  R  C_  ( Q  .(+)  R ) )
4625, 26, 45syl2anc 645 . . . . . 6  |-  ( ph  ->  R  C_  ( Q  .(+) 
R ) )
472lsmless2 14806 . . . . . 6  |-  ( ( U  e.  (SubGrp `  W )  /\  ( Q  .(+)  R )  e.  (SubGrp `  W )  /\  R  C_  ( Q 
.(+)  R ) )  -> 
( U  .(+)  R ) 
C_  ( U  .(+)  ( Q  .(+)  R )
) )
4830, 44, 46, 47syl3anc 1187 . . . . 5  |-  ( ph  ->  ( U  .(+)  R ) 
C_  ( U  .(+)  ( Q  .(+)  R )
) )
4943, 48eqssd 3117 . . . 4  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  R ) )
5020, 49breqtrrd 3946 . . 3  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  ( Q 
.(+)  R ) ) )
511, 2, 4, 7, 8, 14, 50lcvexchlem4 27916 . 2  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) ) (  <oLL  `  W ) ( Q  .(+)  R )
)
521, 2, 3, 4, 5, 16, 9, 11, 17, 51lsatcvat2 27930 1  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1619    e. wcel 1621    =/= wne 2412    i^i cin 3077    C_ wss 3078   class class class wbr 3920   ` cfv 4592  (class class class)co 5710  SubGrpcsubg 14450   LSSumclsm 14780   Abelcabel 14925   LModclmod 15462   LSubSpclss 15524   LVecclvec 15690  LSAtomsclsa 27853    <oLL clcv 27897
This theorem is referenced by:  l1cvat  27934
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-0g 13278  df-mre 13361  df-mrc 13362  df-acs 13363  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-cntz 14628  df-oppg 14654  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-lsatoms 27855  df-lcv 27898
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