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Theorem l1cvpat 32699
Description: A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 33119 analog.) (Contributed by NM, 11-Jan-2015.)
Hypotheses
Ref Expression
l1cvpat.v  |-  V  =  ( Base `  W
)
l1cvpat.s  |-  S  =  ( LSubSp `  W )
l1cvpat.p  |-  .(+)  =  (
LSSum `  W )
l1cvpat.a  |-  A  =  (LSAtoms `  W )
l1cvpat.c  |-  C  =  (  <oLL  `  W )
l1cvpat.w  |-  ( ph  ->  W  e.  LVec )
l1cvpat.u  |-  ( ph  ->  U  e.  S )
l1cvpat.q  |-  ( ph  ->  Q  e.  A )
l1cvpat.l  |-  ( ph  ->  U C V )
l1cvpat.m  |-  ( ph  ->  -.  Q  C_  U
)
Assertion
Ref Expression
l1cvpat  |-  ( ph  ->  ( U  .(+)  Q )  =  V )

Proof of Theorem l1cvpat
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 l1cvpat.q . . 3  |-  ( ph  ->  Q  e.  A )
2 l1cvpat.w . . . 4  |-  ( ph  ->  W  e.  LVec )
3 l1cvpat.v . . . . 5  |-  V  =  ( Base `  W
)
4 eqid 2443 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2443 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
6 l1cvpat.a . . . . 5  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 32636 . . . 4  |-  ( W  e.  LVec  ->  ( Q  e.  A  <->  E. v  e.  ( V  \  {
( 0g `  W
) } ) Q  =  ( ( LSpan `  W ) `  {
v } ) ) )
82, 7syl 16 . . 3  |-  ( ph  ->  ( Q  e.  A  <->  E. v  e.  ( V 
\  { ( 0g
`  W ) } ) Q  =  ( ( LSpan `  W ) `  { v } ) ) )
91, 8mpbid 210 . 2  |-  ( ph  ->  E. v  e.  ( V  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W
) `  { v } ) )
10 l1cvpat.m . 2  |-  ( ph  ->  -.  Q  C_  U
)
11 eldifi 3478 . . . 4  |-  ( v  e.  ( V  \  { ( 0g `  W ) } )  ->  v  e.  V
)
12 l1cvpat.s . . . . . . . . 9  |-  S  =  ( LSubSp `  W )
13 lveclmod 17187 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
142, 13syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
15143ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  W  e.  LMod )
16 l1cvpat.u . . . . . . . . . 10  |-  ( ph  ->  U  e.  S )
17163ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  U  e.  S )
18 simp2 989 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  v  e.  V )
193, 12, 4, 15, 17, 18lspsnel5 17076 . . . . . . . 8  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( v  e.  U  <->  ( ( LSpan `  W ) `  {
v } )  C_  U ) )
2019notbid 294 . . . . . . 7  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( -.  v  e.  U  <->  -.  (
( LSpan `  W ) `  { v } ) 
C_  U ) )
21 l1cvpat.p . . . . . . . . 9  |-  .(+)  =  (
LSSum `  W )
22 eqid 2443 . . . . . . . . 9  |-  (LSHyp `  W )  =  (LSHyp `  W )
2323ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  W  e.  LVec )
24 l1cvpat.l . . . . . . . . . . 11  |-  ( ph  ->  U C V )
25 l1cvpat.c . . . . . . . . . . . 12  |-  C  =  (  <oLL  `  W )
263, 12, 22, 25, 2islshpcv 32698 . . . . . . . . . . 11  |-  ( ph  ->  ( U  e.  (LSHyp `  W )  <->  ( U  e.  S  /\  U C V ) ) )
2716, 24, 26mpbir2and 913 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (LSHyp `  W ) )
28273ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  U  e.  (LSHyp `  W ) )
293, 4, 21, 22, 23, 28, 18lshpnelb 32629 . . . . . . . 8  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( -.  v  e.  U  <->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) )
3029biimpd 207 . . . . . . 7  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( -.  v  e.  U  ->  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) )  =  V ) )
3120, 30sylbird 235 . . . . . 6  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( -.  ( ( LSpan `  W
) `  { v } )  C_  U  ->  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) )
32 sseq1 3377 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( Q  C_  U  <->  ( ( LSpan `  W ) `  { v } ) 
C_  U ) )
3332notbid 294 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( -.  Q  C_  U 
<->  -.  ( ( LSpan `  W ) `  {
v } )  C_  U ) )
34 oveq2 6099 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  .(+)  Q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
3534eqeq1d 2451 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( U  .(+)  Q )  =  V  <->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) )
3633, 35imbi12d 320 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( -.  Q  C_  U  ->  ( U  .(+) 
Q )  =  V )  <->  ( -.  (
( LSpan `  W ) `  { v } ) 
C_  U  ->  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
37363ad2ant3 1011 . . . . . 6  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( ( -.  Q  C_  U  -> 
( U  .(+)  Q )  =  V )  <->  ( -.  ( ( LSpan `  W
) `  { v } )  C_  U  ->  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) ) )
3831, 37mpbird 232 . . . . 5  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( -.  Q  C_  U  ->  ( U  .(+)  Q )  =  V ) )
39383exp 1186 . . . 4  |-  ( ph  ->  ( v  e.  V  ->  ( Q  =  ( ( LSpan `  W ) `  { v } )  ->  ( -.  Q  C_  U  ->  ( U  .(+) 
Q )  =  V ) ) ) )
4011, 39syl5 32 . . 3  |-  ( ph  ->  ( v  e.  ( V  \  { ( 0g `  W ) } )  ->  ( Q  =  ( ( LSpan `  W ) `  { v } )  ->  ( -.  Q  C_  U  ->  ( U  .(+) 
Q )  =  V ) ) ) )
4140rexlimdv 2840 . 2  |-  ( ph  ->  ( E. v  e.  ( V  \  {
( 0g `  W
) } ) Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( -.  Q  C_  U  ->  ( U  .(+)  Q )  =  V ) ) )
429, 10, 41mp2d 45 1  |-  ( ph  ->  ( U  .(+)  Q )  =  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2716    \ cdif 3325    C_ wss 3328   {csn 3877   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   0gc0g 14378   LSSumclsm 16133   LModclmod 16948   LSubSpclss 17013   LSpanclspn 17052   LVecclvec 17183  LSAtomsclsa 32619  LSHypclsh 32620    <oLL clcv 32663
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-0g 14380  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-cntz 15835  df-lsm 16135  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-drng 16834  df-lmod 16950  df-lss 17014  df-lsp 17053  df-lvec 17184  df-lsatoms 32621  df-lshyp 32622  df-lcv 32664
This theorem is referenced by:  l1cvat  32700
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