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Theorem lcv1 29901
Description: Covering property of a subspace plus an atom. (chcv1 23860 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 2438 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2438 . . . . . . 7  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2438 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
6 lcv1.a . . . . . . 7  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29851 . . . . . 6  |-  ( W  e.  LVec  ->  ( Q  e.  A  <->  E. x  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W ) `  { x } ) ) )
82, 7syl 16 . . . . 5  |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( (
Base `  W )  \  { ( 0g `  W ) } ) Q  =  ( (
LSpan `  W ) `  { x } ) ) )
91, 8mpbid 203 . . . 4  |-  ( ph  ->  E. x  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W ) `  { x } ) )
109adantr 453 . . 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 453 . . . . . . 7  |-  ( (
ph  /\  -.  Q  C_  U )  ->  W  e.  LVec )
15143ad2ant1 979 . . . . . 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 453 . . . . . . 7  |-  ( (
ph  /\  -.  Q  C_  U )  ->  U  e.  S )
18173ad2ant1 979 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  U  e.  S )
19 eldifi 3471 . . . . . . 7  |-  ( x  e.  ( ( Base `  W )  \  {
( 0g `  W
) } )  ->  x  e.  ( Base `  W ) )
20193ad2ant2 980 . . . . . 6  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  x  e.  ( Base `  W )
)
21 simp1r 983 . . . . . . 7  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  -.  Q  C_  U )
22 simp3 960 . . . . . . . 8  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  Q  =  ( ( LSpan `  W
) `  { x } ) )
2322sseq1d 3377 . . . . . . 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 293 . . . . . 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 29889 . . . . 5  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  U C
( U  .(+)  ( (
LSpan `  W ) `  { x } ) ) )
2622oveq2d 6099 . . . . 5  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  ( U  .(+) 
Q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { x } ) ) )
2725, 26breqtrrd 4240 . . . 4  |-  ( ( ( ph  /\  -.  Q  C_  U )  /\  x  e.  ( ( Base `  W )  \  { ( 0g `  W ) } )  /\  Q  =  ( ( LSpan `  W ) `  { x } ) )  ->  U C
( U  .(+)  Q ) )
2827rexlimdv3a 2834 . . 3  |-  ( (
ph  /\  -.  Q  C_  U )  ->  ( E. x  e.  (
( Base `  W )  \  { ( 0g `  W ) } ) Q  =  ( (
LSpan `  W ) `  { x } )  ->  U C ( U  .(+)  Q )
) )
2910, 28mpd 15 . 2  |-  ( (
ph  /\  -.  Q  C_  U )  ->  U C ( U  .(+)  Q ) )
302adantr 453 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  W  e.  LVec )
3116adantr 453 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  U  e.  S )
32 lveclmod 16180 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
332, 32syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
3411, 6, 33, 1lsatlssel 29857 . . . . . 6  |-  ( ph  ->  Q  e.  S )
3511, 12lsmcl 16157 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  Q  e.  S )  ->  ( U  .(+)  Q )  e.  S )
3633, 16, 34, 35syl3anc 1185 . . . . 5  |-  ( ph  ->  ( U  .(+)  Q )  e.  S )
3736adantr 453 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  ( U  .(+) 
Q )  e.  S
)
38 simpr 449 . . . 4  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  U C
( U  .(+)  Q ) )
3911, 13, 30, 31, 37, 38lcvpss 29884 . . 3  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  U  C.  ( U  .(+)  Q ) )
4011lsssssubg 16036 . . . . . . 7  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
4133, 40syl 16 . . . . . 6  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
4241, 16sseldd 3351 . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
4341, 34sseldd 3351 . . . . 5  |-  ( ph  ->  Q  e.  (SubGrp `  W ) )
4412, 42, 43lssnle 15308 . . . 4  |-  ( ph  ->  ( -.  Q  C_  U 
<->  U  C.  ( U 
.(+)  Q ) ) )
4544adantr 453 . . 3  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  ( -.  Q  C_  U  <->  U  C.  ( U  .(+)  Q ) ) )
4639, 45mpbird 225 . 2  |-  ( (
ph  /\  U C
( U  .(+)  Q ) )  ->  -.  Q  C_  U )
4729, 46impbida 807 1  |-  ( ph  ->  ( -.  Q  C_  U 
<->  U C ( U 
.(+)  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708    \ cdif 3319    C_ wss 3322    C. wpss 3323   {csn 3816   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   0gc0g 13725  SubGrpcsubg 14940   LSSumclsm 15270   LModclmod 15952   LSubSpclss 16010   LSpanclspn 16049   LVecclvec 16176  LSAtomsclsa 29834    <oLL clcv 29878
This theorem is referenced by:  lcv2  29902  lsatnle  29904  lsatcvat3  29912
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-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-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  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-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  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 29836  df-lcv 29879
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