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Theorem l1cvpat 29852
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 30272 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 2436 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2436 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
6 l1cvpat.a . . . . 5  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29789 . . . 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 202 . 2  |-  ( ph  ->  E. v  e.  ( V  \  { ( 0g `  W ) } ) Q  =  ( ( LSpan `  W
) `  { v } ) )
10 l1cvpat.m . 2  |-  ( ph  ->  -.  Q  C_  U
)
11 eldifi 3469 . . . 4  |-  ( v  e.  ( V  \  { ( 0g `  W ) } )  ->  v  e.  V
)
12 l1cvpat.s . . . . . . . . 9  |-  S  =  ( LSubSp `  W )
13 lveclmod 16178 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
142, 13syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
15143ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  W  e.  LMod )
16 l1cvpat.u . . . . . . . . . 10  |-  ( ph  ->  U  e.  S )
17163ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  U  e.  S )
18 simp2 958 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  v  e.  V )
193, 12, 4, 15, 17, 18lspsnel5 16071 . . . . . . . 8  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( v  e.  U  <->  ( ( LSpan `  W ) `  {
v } )  C_  U ) )
2019notbid 286 . . . . . . 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 2436 . . . . . . . . 9  |-  (LSHyp `  W )  =  (LSHyp `  W )
2323ad2ant1 978 . . . . . . . . 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 29851 . . . . . . . . . . 11  |-  ( ph  ->  ( U  e.  (LSHyp `  W )  <->  ( U  e.  S  /\  U C V ) ) )
2716, 24, 26mpbir2and 889 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (LSHyp `  W ) )
28273ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  U  e.  (LSHyp `  W ) )
293, 4, 21, 22, 23, 28, 18lshpnelb 29782 . . . . . . . 8  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( -.  v  e.  U  <->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) )
3029biimpd 199 . . . . . . 7  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( -.  v  e.  U  ->  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) )  =  V ) )
3120, 30sylbird 227 . . . . . 6  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( -.  ( ( LSpan `  W
) `  { v } )  C_  U  ->  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) )
32 sseq1 3369 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( Q  C_  U  <->  ( ( LSpan `  W ) `  { v } ) 
C_  U ) )
3332notbid 286 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( -.  Q  C_  U 
<->  -.  ( ( LSpan `  W ) `  {
v } )  C_  U ) )
34 oveq2 6089 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  .(+)  Q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
3534eqeq1d 2444 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( U  .(+)  Q )  =  V  <->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) )
3633, 35imbi12d 312 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( -.  Q  C_  U  ->  ( U  .(+) 
Q )  =  V )  <->  ( -.  (
( LSpan `  W ) `  { v } ) 
C_  U  ->  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
37363ad2ant3 980 . . . . . 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 224 . . . . 5  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  ( -.  Q  C_  U  ->  ( U  .(+)  Q )  =  V ) )
39383exp 1152 . . . 4  |-  ( ph  ->  ( v  e.  V  ->  ( Q  =  ( ( LSpan `  W ) `  { v } )  ->  ( -.  Q  C_  U  ->  ( U  .(+) 
Q )  =  V ) ) ) )
4011, 39syl5 30 . . 3  |-  ( ph  ->  ( v  e.  ( V  \  { ( 0g `  W ) } )  ->  ( Q  =  ( ( LSpan `  W ) `  { v } )  ->  ( -.  Q  C_  U  ->  ( U  .(+) 
Q )  =  V ) ) ) )
4140rexlimdv 2829 . 2  |-  ( ph  ->  ( E. v  e.  ( V  \  {
( 0g `  W
) } ) Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( -.  Q  C_  U  ->  ( U  .(+)  Q )  =  V ) ) )
429, 10, 41mp2d 43 1  |-  ( ph  ->  ( U  .(+)  Q )  =  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706    \ cdif 3317    C_ wss 3320   {csn 3814   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   0gc0g 13723   LSSumclsm 15268   LModclmod 15950   LSubSpclss 16008   LSpanclspn 16047   LVecclvec 16174  LSAtomsclsa 29772  LSHypclsh 29773    <oLL clcv 29816
This theorem is referenced by:  l1cvat  29853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lvec 16175  df-lsatoms 29774  df-lshyp 29775  df-lcv 29817
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