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Theorem lsatcvatlem 29861
Description: Lemma for lsatcvat 29862. (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 15875 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 15 . . 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 29823 . 2  |-  ( ph  ->  E. x  e.  A  x  C_  U )
10 eqid 2296 . . . . 5  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
1143ad2ant1 976 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  LVec )
1263ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  LMod )
13 simp2 956 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  A )
141, 3, 12, 13lsatlssel 29809 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  S )
15 lsatcvat.q . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
161, 3, 6, 15lsatlssel 29809 . . . . . . 7  |-  ( ph  ->  Q  e.  S )
17163ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  S )
18 lsatcvat.p . . . . . . 7  |-  .(+)  =  (
LSSum `  W )
191, 18lsmcl 15852 . . . . . 6  |-  ( ( W  e.  LMod  /\  Q  e.  S  /\  x  e.  S )  ->  ( Q  .(+)  x )  e.  S )
2012, 17, 14, 19syl3anc 1182 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
x )  e.  S
)
2173ad2ant1 976 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  e.  S )
22 lsatcvat.m . . . . . . . . . 10  |-  ( ph  ->  -.  Q  C_  U
)
23223ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  -.  Q  C_  U )
24 sseq1 3212 . . . . . . . . . . . 12  |-  ( x  =  Q  ->  (
x  C_  U  <->  Q  C_  U
) )
2524biimpcd 215 . . . . . . . . . . 11  |-  ( x 
C_  U  ->  (
x  =  Q  ->  Q  C_  U ) )
2625necon3bd 2496 . . . . . . . . . 10  |-  ( x 
C_  U  ->  ( -.  Q  C_  U  ->  x  =/=  Q ) )
27263ad2ant3 978 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( -.  Q  C_  U  ->  x  =/=  Q ) )
2823, 27mpd 14 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  =/=  Q )
29153ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  A )
302, 3, 11, 13, 29lsatnem0 29857 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  =/=  Q  <->  ( x  i^i 
Q )  =  {  .0.  } ) )
3128, 30mpbid 201 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  i^i  Q )  =  {  .0.  } )
321, 18, 2, 3, 10, 11, 14, 29lcvp 29852 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( (
x  i^i  Q )  =  {  .0.  }  <->  x (  <oLL  `  W ) ( x 
.(+)  Q ) ) )
3331, 32mpbid 201 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x (  <oLL  `  W ) ( x 
.(+)  Q ) )
34 lmodabl 15688 . . . . . . . 8  |-  ( W  e.  LMod  ->  W  e. 
Abel )
3512, 34syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  Abel )
361lsssssubg 15731 . . . . . . . . 9  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
3712, 36syl 15 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  S  C_  (SubGrp `  W ) )
3837, 14sseldd 3194 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  (SubGrp `  W ) )
3937, 17sseldd 3194 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  (SubGrp `  W ) )
4018lsmcom 15166 . . . . . . 7  |-  ( ( W  e.  Abel  /\  x  e.  (SubGrp `  W )  /\  Q  e.  (SubGrp `  W ) )  -> 
( x  .(+)  Q )  =  ( Q  .(+)  x ) )
4135, 38, 39, 40syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  .(+) 
Q )  =  ( Q  .(+)  x )
)
4233, 41breqtrd 4063 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x (  <oLL  `  W ) ( Q 
.(+)  x ) )
43 simp3 957 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  C_  U
)
44 lsatcvat.l . . . . . . 7  |-  ( ph  ->  U  C.  ( Q 
.(+)  R ) )
45443ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C.  ( Q  .(+)  R ) )
4618lsmub1 14983 . . . . . . . . . 10  |-  ( ( Q  e.  (SubGrp `  W )  /\  x  e.  (SubGrp `  W )
)  ->  Q  C_  ( Q  .(+)  x ) )
4739, 38, 46syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  C_  ( Q  .(+)  x ) )
48 lsatcvat.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  A )
49483ad2ant1 976 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  A )
5044pssssd 3286 . . . . . . . . . . . 12  |-  ( ph  ->  U  C_  ( Q  .(+) 
R ) )
51503ad2ant1 976 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C_  ( Q  .(+)  R ) )
5243, 51sstrd 3202 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  C_  ( Q  .(+)  R ) )
5318, 3, 11, 13, 49, 29, 52, 28lsatexch1 29858 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  C_  ( Q  .(+)  x ) )
541, 3, 6, 48lsatlssel 29809 . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  S )
55543ad2ant1 976 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  S )
5637, 55sseldd 3194 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  (SubGrp `  W ) )
5737, 20sseldd 3194 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
x )  e.  (SubGrp `  W ) )
5818lsmlub 14990 . . . . . . . . . 10  |-  ( ( 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 1182 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( ( Q  C_  ( Q  .(+)  x )  /\  R  C_  ( Q  .(+)  x ) )  <->  ( Q  .(+)  R )  C_  ( Q  .(+) 
x ) ) )
6047, 53, 59mpbi2and 887 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
R )  C_  ( Q  .(+)  x ) )
616, 36syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
6261, 16sseldd 3194 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
6361, 54sseldd 3194 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  (SubGrp `  W ) )
6418lsmub1 14983 . . . . . . . . . . 11  |-  ( ( Q  e.  (SubGrp `  W )  /\  R  e.  (SubGrp `  W )
)  ->  Q  C_  ( Q  .(+)  R ) )
6562, 63, 64syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  Q  C_  ( Q  .(+) 
R ) )
66653ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  C_  ( Q  .(+)  R ) )
671, 18lsmcl 15852 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  Q  e.  S  /\  R  e.  S )  ->  ( Q  .(+)  R )  e.  S )
686, 16, 54, 67syl3anc 1182 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  .(+)  R )  e.  S )
69683ad2ant1 976 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
R )  e.  S
)
7037, 69sseldd 3194 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
R )  e.  (SubGrp `  W ) )
7118lsmlub 14990 . . . . . . . . . 10  |-  ( ( Q  e.  (SubGrp `  W )  /\  x  e.  (SubGrp `  W )  /\  ( Q  .(+)  R )  e.  (SubGrp `  W
) )  ->  (
( Q  C_  ( Q  .(+)  R )  /\  x  C_  ( Q  .(+)  R ) )  <->  ( Q  .(+) 
x )  C_  ( Q  .(+)  R ) ) )
7239, 38, 70, 71syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( ( Q  C_  ( Q  .(+)  R )  /\  x  C_  ( Q  .(+)  R ) )  <->  ( Q  .(+)  x )  C_  ( Q  .(+) 
R ) ) )
7366, 52, 72mpbi2and 887 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
x )  C_  ( Q  .(+)  R ) )
7460, 73eqssd 3209 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
R )  =  ( Q  .(+)  x )
)
7574psseq2d 3282 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( U  C.  ( Q  .(+)  R )  <-> 
U  C.  ( Q  .(+) 
x ) ) )
7645, 75mpbid 201 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C.  ( Q  .(+)  x ) )
771, 10, 11, 14, 20, 21, 42, 43, 76lcvnbtwn3 29840 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  =  x )
7877, 13eqeltrd 2370 . . 3  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  e.  A )
7978rexlimdv3a 2682 . 2  |-  ( ph  ->  ( E. x  e.  A  x  C_  U  ->  U  e.  A ) )
809, 79mpd 14 1  |-  ( ph  ->  U  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    i^i cin 3164    C_ wss 3165    C. wpss 3166   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   0gc0g 13416  SubGrpcsubg 14631   LSSumclsm 14961   Abelcabel 15106   LModclmod 15643   LSubSpclss 15705   LVecclvec 15871  LSAtomsclsa 29786    <oLL clcv 29830
This theorem is referenced by:  lsatcvat  29862
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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