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Theorem lsatcvat3 29242
Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22976 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 2283 . 2  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
5 lsatcvat3.w . 2  |-  ( ph  ->  W  e.  LVec )
6 lveclmod 15859 . . . 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 29187 . . . 4  |-  ( ph  ->  Q  e.  S )
11 lsatcvat3.r . . . . 5  |-  ( ph  ->  R  e.  A )
121, 3, 7, 11lsatlssel 29187 . . . 4  |-  ( ph  ->  R  e.  S )
131, 2lsmcl 15836 . . . 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 15722 . . 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 29231 . . . . 5  |-  ( ph  ->  ( -.  R  C_  U 
<->  U (  <oLL  `  W ) ( U  .(+)  R ) ) )
2018, 19mpbid 201 . . . 4  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  R ) )
21 lmodabl 15672 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
227, 21syl 15 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
231lsssssubg 15715 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
247, 23syl 15 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
2524, 10sseldd 3181 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
2624, 12sseldd 3181 . . . . . . . . . 10  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
272lsmcom 15150 . . . . . . . . . 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 5874 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  ( R 
.(+)  Q ) ) )
3024, 8sseldd 3181 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
312lsmass 14979 . . . . . . . . 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 2318 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( ( U  .(+)  R )  .(+)  Q )
)
341, 2lsmcl 15836 . . . . . . . . . 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 3181 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
37 lsatcvat3.l . . . . . . . 8  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
382lsmless2 14971 . . . . . . . 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 3212 . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) ) )
412lsmidm 14973 . . . . . . 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 3214 . . . . 5  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( U  .(+)  R ) )
4424, 14sseldd 3181 . . . . . 6  |-  ( ph  ->  ( Q  .(+)  R )  e.  (SubGrp `  W
) )
452lsmub2 14968 . . . . . . 7  |-  ( ( Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  R  C_  ( Q  .(+)  R ) )
4625, 26, 45syl2anc 642 . . . . . 6  |-  ( ph  ->  R  C_  ( Q  .(+) 
R ) )
472lsmless2 14971 . . . . . 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 3196 . . . 4  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  R ) )
5020, 49breqtrrd 4049 . . 3  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  ( Q 
.(+)  R ) ) )
511, 2, 4, 7, 8, 14, 50lcvexchlem4 29227 . 2  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) ) (  <oLL  `  W ) ( Q  .(+)  R )
)
521, 2, 3, 4, 5, 16, 9, 11, 17, 51lsatcvat2 29241 1  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858  SubGrpcsubg 14615   LSSumclsm 14945   Abelcabel 15090   LModclmod 15627   LSubSpclss 15689   LVecclvec 15855  LSAtomsclsa 29164    <oLL clcv 29208
This theorem is referenced by:  l1cvat  29245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lcv 29209
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