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Theorem lsatcvat3 29850
Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23899 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 2436 . 2  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
5 lsatcvat3.w . 2  |-  ( ph  ->  W  e.  LVec )
6 lveclmod 16178 . . . 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 29795 . . . 4  |-  ( ph  ->  Q  e.  S )
11 lsatcvat3.r . . . . 5  |-  ( ph  ->  R  e.  A )
121, 3, 7, 11lsatlssel 29795 . . . 4  |-  ( ph  ->  R  e.  S )
131, 2lsmcl 16155 . . . 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 16041 . . 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 29839 . . . . 5  |-  ( ph  ->  ( -.  R  C_  U 
<->  U (  <oLL  `  W ) ( U  .(+)  R ) ) )
2018, 19mpbid 202 . . . 4  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  R ) )
21 lmodabl 15991 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
227, 21syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
231lsssssubg 16034 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
247, 23syl 16 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
2524, 10sseldd 3349 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
2624, 12sseldd 3349 . . . . . . . . . 10  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
272lsmcom 15473 . . . . . . . . . 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 6097 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  ( R 
.(+)  Q ) ) )
3024, 8sseldd 3349 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
312lsmass 15302 . . . . . . . . 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 2471 . . . . . . 7  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( ( U  .(+)  R )  .(+)  Q )
)
341, 2lsmcl 16155 . . . . . . . . . 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 3349 . . . . . . . 8  |-  ( ph  ->  ( U  .(+)  R )  e.  (SubGrp `  W
) )
37 lsatcvat3.l . . . . . . . 8  |-  ( ph  ->  Q  C_  ( U  .(+) 
R ) )
382lsmless2 15294 . . . . . . . 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 3382 . . . . . 6  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( ( U  .(+)  R )  .(+)  ( U  .(+) 
R ) ) )
412lsmidm 15296 . . . . . . 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 3384 . . . . 5  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  C_  ( U  .(+)  R ) )
4424, 14sseldd 3349 . . . . . 6  |-  ( ph  ->  ( Q  .(+)  R )  e.  (SubGrp `  W
) )
452lsmub2 15291 . . . . . . 7  |-  ( ( Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  R  C_  ( Q  .(+)  R ) )
4625, 26, 45syl2anc 643 . . . . . 6  |-  ( ph  ->  R  C_  ( Q  .(+) 
R ) )
472lsmless2 15294 . . . . . 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 3365 . . . 4  |-  ( ph  ->  ( U  .(+)  ( Q 
.(+)  R ) )  =  ( U  .(+)  R ) )
5020, 49breqtrrd 4238 . . 3  |-  ( ph  ->  U (  <oLL  `  W ) ( U  .(+)  ( Q 
.(+)  R ) ) )
511, 2, 4, 7, 8, 14, 50lcvexchlem4 29835 . 2  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) ) (  <oLL  `  W ) ( Q  .(+)  R )
)
521, 2, 3, 4, 5, 16, 9, 11, 17, 51lsatcvat2 29849 1  |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599    i^i cin 3319    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081  SubGrpcsubg 14938   LSSumclsm 15268   Abelcabel 15413   LModclmod 15950   LSubSpclss 16008   LVecclvec 16174  LSAtomsclsa 29772    <oLL clcv 29816
This theorem is referenced by:  l1cvat  29853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-oppg 15142  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lvec 16175  df-lsatoms 29774  df-lcv 29817
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