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Theorem lsatcvatlem 34875
Description: Lemma for lsatcvat 34876. (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 17878 . . . 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 34837 . 2  |-  ( ph  ->  E. x  e.  A  x  C_  U )
10 eqid 2457 . . . . 5  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
1143ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  LVec )
1263ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  LMod )
13 simp2 997 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  A )
141, 3, 12, 13lsatlssel 34823 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  S )
15 lsatcvat.q . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
161, 3, 6, 15lsatlssel 34823 . . . . . . 7  |-  ( ph  ->  Q  e.  S )
17163ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  S )
18 lsatcvat.p . . . . . . 7  |-  .(+)  =  (
LSSum `  W )
191, 18lsmcl 17855 . . . . . 6  |-  ( ( W  e.  LMod  /\  Q  e.  S  /\  x  e.  S )  ->  ( Q  .(+)  x )  e.  S )
2012, 17, 14, 19syl3anc 1228 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
x )  e.  S
)
2173ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  e.  S )
22 lsatcvat.m . . . . . . . . . 10  |-  ( ph  ->  -.  Q  C_  U
)
23223ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  -.  Q  C_  U )
24 sseq1 3520 . . . . . . . . . . . 12  |-  ( x  =  Q  ->  (
x  C_  U  <->  Q  C_  U
) )
2524biimpcd 224 . . . . . . . . . . 11  |-  ( x 
C_  U  ->  (
x  =  Q  ->  Q  C_  U ) )
2625necon3bd 2669 . . . . . . . . . 10  |-  ( x 
C_  U  ->  ( -.  Q  C_  U  ->  x  =/=  Q ) )
27263ad2ant3 1019 . . . . . . . . 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 1017 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  A )
302, 3, 11, 13, 29lsatnem0 34871 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  =/=  Q  <->  ( x  i^i 
Q )  =  {  .0.  } ) )
3128, 30mpbid 210 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  i^i  Q )  =  {  .0.  } )
321, 18, 2, 3, 10, 11, 14, 29lcvp 34866 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( (
x  i^i  Q )  =  {  .0.  }  <->  x (  <oLL  `  W ) ( x 
.(+)  Q ) ) )
3331, 32mpbid 210 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x (  <oLL  `  W ) ( x 
.(+)  Q ) )
34 lmodabl 17683 . . . . . . . 8  |-  ( W  e.  LMod  ->  W  e. 
Abel )
3512, 34syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  Abel )
361lsssssubg 17730 . . . . . . . . 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 3500 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  (SubGrp `  W ) )
3937, 17sseldd 3500 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  (SubGrp `  W ) )
4018lsmcom 16990 . . . . . . 7  |-  ( ( W  e.  Abel  /\  x  e.  (SubGrp `  W )  /\  Q  e.  (SubGrp `  W ) )  -> 
( x  .(+)  Q )  =  ( Q  .(+)  x ) )
4135, 38, 39, 40syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  .(+) 
Q )  =  ( Q  .(+)  x )
)
4233, 41breqtrd 4480 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x (  <oLL  `  W ) ( Q 
.(+)  x ) )
43 simp3 998 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  C_  U
)
44 lsatcvat.l . . . . . . 7  |-  ( ph  ->  U  C.  ( Q  .(+) 
R ) )
45443ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C.  ( Q  .(+)  R ) )
4618lsmub1 16802 . . . . . . . 8  |-  ( ( Q  e.  (SubGrp `  W )  /\  x  e.  (SubGrp `  W )
)  ->  Q  C_  ( Q  .(+)  x ) )
4739, 38, 46syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  C_  ( Q  .(+)  x ) )
48 lsatcvat.r . . . . . . . . 9  |-  ( ph  ->  R  e.  A )
49483ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  A )
5044pssssd 3597 . . . . . . . . . 10  |-  ( ph  ->  U  C_  ( Q  .(+) 
R ) )
51503ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C_  ( Q  .(+)  R ) )
5243, 51sstrd 3509 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  C_  ( Q  .(+)  R ) )
5318, 3, 11, 13, 49, 29, 52, 28lsatexch1 34872 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  C_  ( Q  .(+)  x ) )
541, 3, 6, 48lsatlssel 34823 . . . . . . . . . 10  |-  ( ph  ->  R  e.  S )
55543ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  S )
5637, 55sseldd 3500 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  (SubGrp `  W ) )
5737, 20sseldd 3500 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
x )  e.  (SubGrp `  W ) )
5818lsmlub 16809 . . . . . . . 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 1228 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( ( Q  C_  ( Q  .(+)  x )  /\  R  C_  ( Q  .(+)  x ) )  <->  ( Q  .(+)  R )  C_  ( Q  .(+) 
x ) ) )
6047, 53, 59mpbi2and 921 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
R )  C_  ( Q  .(+)  x ) )
6145, 60psssstrd 3609 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C.  ( Q  .(+)  x ) )
621, 10, 11, 14, 20, 21, 42, 43, 61lcvnbtwn3 34854 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  =  x )
6362, 13eqeltrd 2545 . . 3  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  e.  A )
6463rexlimdv3a 2951 . 2  |-  ( ph  ->  ( E. x  e.  A  x  C_  U  ->  U  e.  A ) )
659, 64mpd 15 1  |-  ( ph  ->  U  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    i^i cin 3470    C_ wss 3471    C. wpss 3472   {csn 4032   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   0gc0g 14856  SubGrpcsubg 16321   LSSumclsm 16780   Abelcabl 16925   LModclmod 17638   LSubSpclss 17704   LVecclvec 17874  LSAtomsclsa 34800    <oLL clcv 34844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-0g 14858  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-oppg 16507  df-lsm 16782  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-oppr 17398  df-dvdsr 17416  df-unit 17417  df-invr 17447  df-drng 17524  df-lmod 17640  df-lss 17705  df-lsp 17744  df-lvec 17875  df-lsatoms 34802  df-lcv 34845
This theorem is referenced by:  lsatcvat  34876
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