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Theorem l1cvpat 33860
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 34280 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 2467 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2467 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
6 l1cvpat.a . . . . 5  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 33797 . . . 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 3626 . . . 4  |-  ( v  e.  ( V  \  { ( 0g `  W ) } )  ->  v  e.  V
)
12 l1cvpat.s . . . . . . . . 9  |-  S  =  ( LSubSp `  W )
13 lveclmod 17547 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
142, 13syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
15143ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  W  e.  LMod )
16 l1cvpat.u . . . . . . . . . 10  |-  ( ph  ->  U  e.  S )
17163ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  U  e.  S )
18 simp2 997 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  v  e.  V )
193, 12, 4, 15, 17, 18lspsnel5 17436 . . . . . . . 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 2467 . . . . . . . . 9  |-  (LSHyp `  W )  =  (LSHyp `  W )
2323ad2ant1 1017 . . . . . . . . 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 33859 . . . . . . . . . . 11  |-  ( ph  ->  ( U  e.  (LSHyp `  W )  <->  ( U  e.  S  /\  U C V ) ) )
2716, 24, 26mpbir2and 920 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (LSHyp `  W ) )
28273ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  V  /\  Q  =  ( ( LSpan `  W ) `  { v } ) )  ->  U  e.  (LSHyp `  W ) )
293, 4, 21, 22, 23, 28, 18lshpnelb 33790 . . . . . . . 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 3525 . . . . . . . . 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 6291 . . . . . . . . 9  |-  ( Q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  .(+)  Q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
3534eqeq1d 2469 . . . . . . . 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 1019 . . . . . 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 1195 . . . 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 2953 . 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 973    = wceq 1379    e. wcel 1767   E.wrex 2815    \ cdif 3473    C_ wss 3476   {csn 4027   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   Basecbs 14489   0gc0g 14694   LSSumclsm 16457   LModclmod 17307   LSubSpclss 17373   LSpanclspn 17412   LVecclvec 17543  LSAtomsclsa 33780  LSHypclsh 33781    <oLL clcv 33824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-3 10594  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-0g 14696  df-mnd 15731  df-submnd 15784  df-grp 15864  df-minusg 15865  df-sbg 15866  df-subg 16000  df-cntz 16157  df-lsm 16459  df-cmn 16603  df-abl 16604  df-mgp 16941  df-ur 16953  df-rng 16997  df-oppr 17068  df-dvdsr 17086  df-unit 17087  df-invr 17117  df-drng 17193  df-lmod 17309  df-lss 17374  df-lsp 17413  df-lvec 17544  df-lsatoms 33782  df-lshyp 33783  df-lcv 33825
This theorem is referenced by:  l1cvat  33861
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