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Theorem lcv1 29853
Description: Covering property of a subspace plus an atom. (chcv1 22951 analog.) (Contributed by NM, 10-Jan-2015.)
Hypotheses
Ref Expression
lcv1.s  |-  S  =  ( LSubSp `  W )
lcv1.p  |-  .(+)  =  (
LSSum `  W )
lcv1.a  |-  A  =  (LSAtoms `  W )
lcv1.c  |-  C  =  (  <oLL  `  W )
lcv1.w  |-  ( ph  ->  W  e.  LVec )
lcv1.u  |-  ( ph  ->  U  e.  S )
lcv1.q  |-  ( ph  ->  Q  e.  A )
Assertion
Ref Expression
lcv1  |-  ( ph  ->  ( -.  Q  C_  U 
<->  U C ( U 
.(+)  Q ) ) )

Proof of Theorem lcv1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lcv1.q . . . . 5  |-  ( ph  ->  Q  e.  A )
2 lcv1.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
3 eqid 2296 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2296 . . . . . . 7  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2296 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
6 lcv1.a . . . . . . 7  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29803 . . . . . 6  |-  ( W  e.  LVec  ->  ( Q  e.  A  <->  E. x  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W ) `  { x } ) ) )
82, 7syl 15 . . . . 5  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( (
Base `  W )  \  { ( 0g `  W ) } ) Q  =  ( (
LSpan `  W ) `  { x } ) ) )
91, 8mpbid 201 . . . 4  |-  ( ph  ->  E. x  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W ) `  { x } ) )
109adantr 451 . . 3  |-  ( (
ph  /\  -.  Q  C_  U )  ->  E. x  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W ) `  { x } ) )
11 lcv1.s . . . . . 6  |-  S  =  ( LSubSp `  W )
12 lcv1.p . . . . . 6  |-  .(+)  =  (
LSSum `  W )
13 lcv1.c . . . . . 6  |-  C  =  (  <oLL  `  W )
142adantr 451 . . . . . . 7  |-  ( (
ph  /\  -.  Q  C_  U )  ->  W  e.  LVec )
15143ad2ant1 976 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  W  e.  LVec )
16 lcv1.u . . . . . . . 8  |-  ( ph  ->  U  e.  S )
1716adantr 451 . . . . . . 7  |-  ( (
ph  /\  -.  Q  C_  U )  ->  U  e.  S )
18173ad2ant1 976 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  U  e.  S )
19 eldifi 3311 . . . . . . 7  |-  ( x  e.  ( ( Base `  W )  \  {
( 0g `  W
) } )  ->  x  e.  ( Base `  W ) )
20193ad2ant2 977 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  x  e.  ( Base `  W )
)
21 simp1r 980 . . . . . . 7  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  -.  Q  C_  U )
22 simp3 957 . . . . . . . 8  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  Q  =  ( ( LSpan `  W
) `  { x } ) )
2322sseq1d 3218 . . . . . . 7  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  ( Q  C_  U  <->  ( ( LSpan `  W ) `  {
x } )  C_  U ) )
2421, 23mtbid 291 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  -.  (
( LSpan `  W ) `  { x } ) 
C_  U )
253, 11, 4, 12, 13, 15, 18, 20, 24lsmcv2 29841 . . . . 5  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  U C
( U  .(+)  ( (
LSpan `  W ) `  { x } ) ) )
2622oveq2d 5890 . . . . 5  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  ( U  .(+) 
Q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { x } ) ) )
2725, 26breqtrrd 4065 . . . 4  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  U C
( U  .(+)  Q ) )
2827rexlimdv3a 2682 . . 3  |-  ( (
ph  /\  -.  Q  C_  U )  ->  ( E. x  e.  (
( Base `  W )  \  { ( 0g `  W ) } ) Q  =  ( (
LSpan `  W ) `  { x } )  ->  U C ( U  .(+)  Q )
) )
2910, 28mpd 14 . 2  |-  ( (
ph  /\  -.  Q  C_  U )  ->  U C ( U  .(+)  Q ) )
302adantr 451 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  W  e.  LVec )
3116adantr 451 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  U  e.  S )
32 lveclmod 15875 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
332, 32syl 15 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
3411, 6, 33, 1lsatlssel 29809 . . . . . 6  |-  ( ph  ->  Q  e.  S )
3511, 12lsmcl 15852 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  Q  e.  S )  ->  ( U  .(+)  Q )  e.  S )
3633, 16, 34, 35syl3anc 1182 . . . . 5  |-  ( ph  ->  ( U  .(+)  Q )  e.  S )
3736adantr 451 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  ( U  .(+) 
Q )  e.  S
)
38 simpr 447 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  U C
( U  .(+)  Q ) )
3911, 13, 30, 31, 37, 38lcvpss 29836 . . 3  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  U  C.  ( U  .(+)  Q ) )
4011lsssssubg 15731 . . . . . . 7  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
4133, 40syl 15 . . . . . 6  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
4241, 16sseldd 3194 . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
4341, 34sseldd 3194 . . . . 5  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
4412, 42, 43lssnle 14999 . . . 4  |-  ( ph  ->  ( -.  Q  C_  U 
<->  U  C.  ( U 
.(+)  Q ) ) )
4544adantr 451 . . 3  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  ( -.  Q  C_  U  <->  U  C.  ( U  .(+)  Q ) ) )
4639, 45mpbird 223 . 2  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  -.  Q  C_  U )
4729, 46impbida 805 1  |-  ( ph  ->  ( -.  Q  C_  U 
<->  U C ( U 
.(+)  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    \ cdif 3162    C_ wss 3165    C. wpss 3166   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   0gc0g 13416  SubGrpcsubg 14631   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871  LSAtomsclsa 29786    <oLL clcv 29830
This theorem is referenced by:  lcv2  29854  lsatnle  29856  lsatcvat3  29864
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-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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  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-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  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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