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Theorem lsatcvat3 29864
Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22992 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 2296 . 2  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
5 lsatcvat3.w . 2  |-  ( ph  ->  W  e.  LVec )
6 lveclmod 15875 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
75, 6syl 15 . . 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 29809 . . . 4  |-  ( ph  ->  Q  e.  S )
11 lsatcvat3.r . . . . 5  |-  ( ph  ->  R  e.  A )
121, 3, 7, 11lsatlssel 29809 . . . 4  |-  ( ph  ->  R  e.  S )
131, 2lsmcl 15852 . . . 4  |-  ( ( W  e.  LMod  /\  Q  e.  S  /\  R  e.  S )  ->  ( Q  .(+)  R )  e.  S )
147, 10, 12, 13syl3anc 1182 . . 3  |-  ( ph  ->  ( Q  .(+)  R )  e.  S )
151lssincl 15738 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  ( Q  .(+)  R )  e.  S )  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  S
)
167, 8, 14, 15syl3anc 1182 . 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 29853 . . . . 5  |-  ( ph  ->  ( -.  R  C_  U 
<->  U (  <oLL  `  W ) ( U  .(+)  R ) ) )
2018, 19mpbid 201 . . . 4  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  R ) )
21 lmodabl 15688 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
227, 21syl 15 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
231lsssssubg 15731 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
247, 23syl 15 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
2524, 10sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
2624, 12sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
272lsmcom 15166 . . . . . . . . . 10  |-  ( ( W  e.  Abel  /\  Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W ) )  -> 
( Q  .(+)  R )  =  ( R  .(+)  Q ) )
2822, 25, 26, 27syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( Q  .(+)  R )  =  ( R  .(+)  Q ) )
2928oveq2d 5890 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  ( R 
.(+)  Q ) ) )
3024, 8sseldd 3194 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
312lsmass 14995 . . . . . . . . 9  |-  ( ( U  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )  /\  Q  e.  (SubGrp `  W ) )  -> 
( ( U  .(+)  R )  .(+)  Q )  =  ( U  .(+)  ( R  .(+)  Q )
) )
3230, 26, 25, 31syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( U  .(+)  R )  .(+)  Q )  =  ( U  .(+)  ( R  .(+)  Q )
) )
3329, 32eqtr4d 2331 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( ( U  .(+)  R )  .(+)  Q )
)
341, 2lsmcl 15852 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  R  e.  S )  ->  ( U  .(+)  R )  e.  S )
357, 8, 12, 34syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( U  .(+)  R )  e.  S )
3624, 35sseldd 3194 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
37 lsatcvat3.l . . . . . . . 8  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
382lsmless2 14987 . . . . . . . 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 1182 . . . . . . 7  |-  ( ph  ->  ( ( U  .(+)  R )  .(+)  Q )  C_  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) ) )
4033, 39eqsstrd 3225 . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) ) )
412lsmidm 14989 . . . . . . 7  |-  ( ( U  .(+)  R )  e.  (SubGrp `  W )  ->  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) )  =  ( U  .(+)  R ) )
4236, 41syl 15 . . . . . 6  |-  ( ph  ->  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) )  =  ( U  .(+)  R ) )
4340, 42sseqtrd 3227 . . . . 5  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( U  .(+)  R ) )
4424, 14sseldd 3194 . . . . . 6  |-  ( ph  ->  ( Q  .(+)  R )  e.  (SubGrp `  W
) )
452lsmub2 14984 . . . . . . 7  |-  ( ( Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  R  C_  ( Q  .(+)  R ) )
4625, 26, 45syl2anc 642 . . . . . 6  |-  ( ph  ->  R  C_  ( Q  .(+) 
R ) )
472lsmless2 14987 . . . . . 6  |-  ( ( U  e.  (SubGrp `  W )  /\  ( Q  .(+)  R )  e.  (SubGrp `  W )  /\  R  C_  ( Q 
.(+)  R ) )  -> 
( U  .(+)  R ) 
C_  ( U  .(+)  ( Q  .(+)  R )
) )
4830, 44, 46, 47syl3anc 1182 . . . . 5  |-  ( ph  ->  ( U  .(+)  R ) 
C_  ( U  .(+)  ( Q  .(+)  R )
) )
4943, 48eqssd 3209 . . . 4  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  R ) )
5020, 49breqtrrd 4065 . . 3  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  ( Q 
.(+)  R ) ) )
511, 2, 4, 7, 8, 14, 50lcvexchlem4 29849 . 2  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) ) (  <oLL  `  W ) ( Q  .(+)  R )
)
521, 2, 3, 4, 5, 16, 9, 11, 17, 51lsatcvat2 29863 1  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459    i^i cin 3164    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874  SubGrpcsubg 14631   LSSumclsm 14961   Abelcabel 15106   LModclmod 15643   LSubSpclss 15705   LVecclvec 15871  LSAtomsclsa 29786    <oLL clcv 29830
This theorem is referenced by:  l1cvat  29867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-oppg 14835  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-lcv 29831
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