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Theorem lsatcvatlem 29849
Description: Lemma for lsatcvat 29850. (Contributed by NM, 10-Jan-2015.)
Hypotheses
Ref Expression
lsatcvat.o  |-  .0.  =  ( 0g `  W )
lsatcvat.s  |-  S  =  ( LSubSp `  W )
lsatcvat.p  |-  .(+)  =  (
LSSum `  W )
lsatcvat.a  |-  A  =  (LSAtoms `  W )
lsatcvat.w  |-  ( ph  ->  W  e.  LVec )
lsatcvat.u  |-  ( ph  ->  U  e.  S )
lsatcvat.q  |-  ( ph  ->  Q  e.  A )
lsatcvat.r  |-  ( ph  ->  R  e.  A )
lsatcvat.n  |-  ( ph  ->  U  =/=  {  .0.  } )
lsatcvat.l  |-  ( ph  ->  U  C.  ( Q 
.(+)  R ) )
lsatcvat.m  |-  ( ph  ->  -.  Q  C_  U
)
Assertion
Ref Expression
lsatcvatlem  |-  ( ph  ->  U  e.  A )

Proof of Theorem lsatcvatlem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lsatcvat.s . . 3  |-  S  =  ( LSubSp `  W )
2 lsatcvat.o . . 3  |-  .0.  =  ( 0g `  W )
3 lsatcvat.a . . 3  |-  A  =  (LSAtoms `  W )
4 lsatcvat.w . . . 4  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 16180 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 16 . . 3  |-  ( ph  ->  W  e.  LMod )
7 lsatcvat.u . . 3  |-  ( ph  ->  U  e.  S )
8 lsatcvat.n . . 3  |-  ( ph  ->  U  =/=  {  .0.  } )
91, 2, 3, 6, 7, 8lssatomic 29811 . 2  |-  ( ph  ->  E. x  e.  A  x  C_  U )
10 eqid 2438 . . . . 5  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
1143ad2ant1 979 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  LVec )
1263ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  LMod )
13 simp2 959 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  A )
141, 3, 12, 13lsatlssel 29797 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  S )
15 lsatcvat.q . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
161, 3, 6, 15lsatlssel 29797 . . . . . . 7  |-  ( ph  ->  Q  e.  S )
17163ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  S )
18 lsatcvat.p . . . . . . 7  |-  .(+)  =  (
LSSum `  W )
191, 18lsmcl 16157 . . . . . 6  |-  ( ( W  e.  LMod  /\  Q  e.  S  /\  x  e.  S )  ->  ( Q  .(+)  x )  e.  S )
2012, 17, 14, 19syl3anc 1185 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
x )  e.  S
)
2173ad2ant1 979 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  e.  S )
22 lsatcvat.m . . . . . . . . . 10  |-  ( ph  ->  -.  Q  C_  U
)
23223ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  -.  Q  C_  U )
24 sseq1 3371 . . . . . . . . . . . 12  |-  ( x  =  Q  ->  (
x  C_  U  <->  Q  C_  U
) )
2524biimpcd 217 . . . . . . . . . . 11  |-  ( x 
C_  U  ->  (
x  =  Q  ->  Q  C_  U ) )
2625necon3bd 2640 . . . . . . . . . 10  |-  ( x 
C_  U  ->  ( -.  Q  C_  U  ->  x  =/=  Q ) )
27263ad2ant3 981 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( -.  Q  C_  U  ->  x  =/=  Q ) )
2823, 27mpd 15 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  =/=  Q )
29153ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  A )
302, 3, 11, 13, 29lsatnem0 29845 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  =/=  Q  <->  ( x  i^i 
Q )  =  {  .0.  } ) )
3128, 30mpbid 203 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  i^i  Q )  =  {  .0.  } )
321, 18, 2, 3, 10, 11, 14, 29lcvp 29840 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( (
x  i^i  Q )  =  {  .0.  }  <->  x (  <oLL  `  W ) ( x 
.(+)  Q ) ) )
3331, 32mpbid 203 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x (  <oLL  `  W ) ( x 
.(+)  Q ) )
34 lmodabl 15993 . . . . . . . 8  |-  ( W  e.  LMod  ->  W  e. 
Abel )
3512, 34syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  Abel )
361lsssssubg 16036 . . . . . . . . 9  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
3712, 36syl 16 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  S  C_  (SubGrp `  W ) )
3837, 14sseldd 3351 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  (SubGrp `  W ) )
3937, 17sseldd 3351 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  (SubGrp `  W ) )
4018lsmcom 15475 . . . . . . 7  |-  ( ( W  e.  Abel  /\  x  e.  (SubGrp `  W )  /\  Q  e.  (SubGrp `  W ) )  -> 
( x  .(+)  Q )  =  ( Q  .(+)  x ) )
4135, 38, 39, 40syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  .(+) 
Q )  =  ( Q  .(+)  x )
)
4233, 41breqtrd 4238 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x (  <oLL  `  W ) ( Q 
.(+)  x ) )
43 simp3 960 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  C_  U
)
44 lsatcvat.l . . . . . . 7  |-  ( ph  ->  U  C.  ( Q 
.(+)  R ) )
45443ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C.  ( Q  .(+)  R ) )
4618lsmub1 15292 . . . . . . . 8  |-  ( ( Q  e.  (SubGrp `  W )  /\  x  e.  (SubGrp `  W )
)  ->  Q  C_  ( Q  .(+)  x ) )
4739, 38, 46syl2anc 644 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  C_  ( Q  .(+)  x ) )
48 lsatcvat.r . . . . . . . . 9  |-  ( ph  ->  R  e.  A )
49483ad2ant1 979 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  A )
5044pssssd 3446 . . . . . . . . . 10  |-  ( ph  ->  U  C_  ( Q  .(+) 
R ) )
51503ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C_  ( Q  .(+)  R ) )
5243, 51sstrd 3360 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  C_  ( Q  .(+)  R ) )
5318, 3, 11, 13, 49, 29, 52, 28lsatexch1 29846 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  C_  ( Q  .(+)  x ) )
541, 3, 6, 48lsatlssel 29797 . . . . . . . . . 10  |-  ( ph  ->  R  e.  S )
55543ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  S )
5637, 55sseldd 3351 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  (SubGrp `  W ) )
5737, 20sseldd 3351 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
x )  e.  (SubGrp `  W ) )
5818lsmlub 15299 . . . . . . . 8  |-  ( ( Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )  /\  ( Q  .(+)  x )  e.  (SubGrp `  W
) )  ->  (
( Q  C_  ( Q  .(+)  x )  /\  R  C_  ( Q  .(+)  x ) )  <->  ( Q  .(+) 
R )  C_  ( Q  .(+)  x ) ) )
5939, 56, 57, 58syl3anc 1185 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( ( Q  C_  ( Q  .(+)  x )  /\  R  C_  ( Q  .(+)  x ) )  <->  ( Q  .(+)  R )  C_  ( Q  .(+) 
x ) ) )
6047, 53, 59mpbi2and 889 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
R )  C_  ( Q  .(+)  x ) )
6145, 60psssstrd 3458 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C.  ( Q  .(+)  x ) )
621, 10, 11, 14, 20, 21, 42, 43, 61lcvnbtwn3 29828 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  =  x )
6362, 13eqeltrd 2512 . . 3  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  e.  A )
6463rexlimdv3a 2834 . 2  |-  ( ph  ->  ( E. x  e.  A  x  C_  U  ->  U  e.  A ) )
659, 64mpd 15 1  |-  ( ph  ->  U  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    i^i cin 3321    C_ wss 3322    C. wpss 3323   {csn 3816   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   0gc0g 13725  SubGrpcsubg 14940   LSSumclsm 15270   Abelcabel 15415   LModclmod 15952   LSubSpclss 16010   LVecclvec 16176  LSAtomsclsa 29774    <oLL clcv 29818
This theorem is referenced by:  lsatcvat  29850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-0g 13729  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-oppg 15144  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-drng 15839  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lvec 16177  df-lsatoms 29776  df-lcv 29819
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