Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lsatcvatlem Structured version   Unicode version

Theorem lsatcvatlem 32534
Description: Lemma for lsatcvat 32535. (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 17164 . . . 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 32496 . 2  |-  ( ph  ->  E. x  e.  A  x  C_  U )
10 eqid 2438 . . . . 5  |-  (  <oLL  `  W
)  =  (  <oLL  `  W
)
1143ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  LVec )
1263ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  LMod )
13 simp2 989 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  A )
141, 3, 12, 13lsatlssel 32482 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  S )
15 lsatcvat.q . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
161, 3, 6, 15lsatlssel 32482 . . . . . . 7  |-  ( ph  ->  Q  e.  S )
17163ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  S )
18 lsatcvat.p . . . . . . 7  |-  .(+)  =  (
LSSum `  W )
191, 18lsmcl 17141 . . . . . 6  |-  ( ( W  e.  LMod  /\  Q  e.  S  /\  x  e.  S )  ->  ( Q  .(+)  x )  e.  S )
2012, 17, 14, 19syl3anc 1218 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
x )  e.  S
)
2173ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  e.  S )
22 lsatcvat.m . . . . . . . . . 10  |-  ( ph  ->  -.  Q  C_  U
)
23223ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  -.  Q  C_  U )
24 sseq1 3372 . . . . . . . . . . . 12  |-  ( x  =  Q  ->  (
x  C_  U  <->  Q  C_  U
) )
2524biimpcd 224 . . . . . . . . . . 11  |-  ( x 
C_  U  ->  (
x  =  Q  ->  Q  C_  U ) )
2625necon3bd 2640 . . . . . . . . . 10  |-  ( x 
C_  U  ->  ( -.  Q  C_  U  ->  x  =/=  Q ) )
27263ad2ant3 1011 . . . . . . . . 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 1009 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  A )
302, 3, 11, 13, 29lsatnem0 32530 . . . . . . . 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 32525 . . . . . . 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 16970 . . . . . . . 8  |-  ( W  e.  LMod  ->  W  e. 
Abel )
3512, 34syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  W  e.  Abel )
361lsssssubg 17016 . . . . . . . . 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 3352 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  e.  (SubGrp `  W ) )
3937, 17sseldd 3352 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  Q  e.  (SubGrp `  W ) )
4018lsmcom 16331 . . . . . . 7  |-  ( ( W  e.  Abel  /\  x  e.  (SubGrp `  W )  /\  Q  e.  (SubGrp `  W ) )  -> 
( x  .(+)  Q )  =  ( Q  .(+)  x ) )
4135, 38, 39, 40syl3anc 1218 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( x  .(+) 
Q )  =  ( Q  .(+)  x )
)
4233, 41breqtrd 4311 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x (  <oLL  `  W ) ( Q 
.(+)  x ) )
43 simp3 990 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  C_  U
)
44 lsatcvat.l . . . . . . 7  |-  ( ph  ->  U  C.  ( Q  .(+) 
R ) )
45443ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C.  ( Q  .(+)  R ) )
4618lsmub1 16146 . . . . . . . 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 1009 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  A )
5044pssssd 3448 . . . . . . . . . 10  |-  ( ph  ->  U  C_  ( Q  .(+) 
R ) )
51503ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C_  ( Q  .(+)  R ) )
5243, 51sstrd 3361 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  x  C_  ( Q  .(+)  R ) )
5318, 3, 11, 13, 49, 29, 52, 28lsatexch1 32531 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  C_  ( Q  .(+)  x ) )
541, 3, 6, 48lsatlssel 32482 . . . . . . . . . 10  |-  ( ph  ->  R  e.  S )
55543ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  S )
5637, 55sseldd 3352 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  R  e.  (SubGrp `  W ) )
5737, 20sseldd 3352 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
x )  e.  (SubGrp `  W ) )
5818lsmlub 16153 . . . . . . . 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 1218 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( ( Q  C_  ( Q  .(+)  x )  /\  R  C_  ( Q  .(+)  x ) )  <->  ( Q  .(+)  R )  C_  ( Q  .(+) 
x ) ) )
6047, 53, 59mpbi2and 912 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  ( Q  .(+) 
R )  C_  ( Q  .(+)  x ) )
6145, 60psssstrd 3460 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  x  C_  U
)  ->  U  C.  ( Q  .(+)  x ) )
621, 10, 11, 14, 20, 21, 42, 43, 61lcvnbtwn3 32513 . . . 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 2838 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711    i^i cin 3322    C_ wss 3323    C. wpss 3324   {csn 3872   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   0gc0g 14370  SubGrpcsubg 15666   LSSumclsm 16124   Abelcabel 16269   LModclmod 16926   LSubSpclss 16990   LVecclvec 17160  LSAtomsclsa 32459    <oLL clcv 32503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-cntz 15826  df-oppg 15852  df-lsm 16126  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-drng 16812  df-lmod 16928  df-lss 16991  df-lsp 17030  df-lvec 17161  df-lsatoms 32461  df-lcv 32504
This theorem is referenced by:  lsatcvat  32535
  Copyright terms: Public domain W3C validator