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Theorem lbspss 15851
Description: No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.)
Hypotheses
Ref Expression
lbsind2.j  |-  J  =  (LBasis `  W )
lbsind2.n  |-  N  =  ( LSpan `  W )
lbsind2.f  |-  F  =  (Scalar `  W )
lbsind2.o  |-  .1.  =  ( 1r `  F )
lbsind2.z  |-  .0.  =  ( 0g `  F )
lbspss.v  |-  V  =  ( Base `  W
)
Assertion
Ref Expression
lbspss  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  ( N `  C )  =/=  V
)

Proof of Theorem lbspss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pssnel 3532 . . 3  |-  ( C 
C.  B  ->  E. x
( x  e.  B  /\  -.  x  e.  C
) )
213ad2ant3 978 . 2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  E. x ( x  e.  B  /\  -.  x  e.  C )
)
3 simpl2 959 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  B  e.  J )
4 lbspss.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
5 lbsind2.j . . . . . . . . 9  |-  J  =  (LBasis `  W )
64, 5lbsss 15846 . . . . . . . 8  |-  ( B  e.  J  ->  B  C_  V )
73, 6syl 15 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  B  C_  V )
8 simprl 732 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  x  e.  B )
97, 8sseldd 3194 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  x  e.  V )
10 simpl1l 1006 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  W  e.  LMod )
11 simpl1r 1007 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  .1.  =/=  .0.  )
12 lbsind2.n . . . . . . . . 9  |-  N  =  ( LSpan `  W )
13 lbsind2.f . . . . . . . . 9  |-  F  =  (Scalar `  W )
14 lbsind2.o . . . . . . . . 9  |-  .1.  =  ( 1r `  F )
15 lbsind2.z . . . . . . . . 9  |-  .0.  =  ( 0g `  F )
165, 12, 13, 14, 15lbsind2 15850 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  x  e.  B
)  ->  -.  x  e.  ( N `  ( B  \  { x }
) ) )
1710, 11, 3, 8, 16syl211anc 1188 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  ( N `  ( B  \  { x } ) ) )
18 difss 3316 . . . . . . . . . 10  |-  ( B 
\  { x }
)  C_  B
1918, 7syl5ss 3203 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( B  \  {
x } )  C_  V )
20 simpl3 960 . . . . . . . . . . . . . 14  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C.  B )
2120pssssd 3286 . . . . . . . . . . . . 13  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C_  B )
2221sseld 3192 . . . . . . . . . . . 12  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  e.  C  ->  y  e.  B ) )
23 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  C
)
24 eleq1 2356 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
y  e.  C  <->  x  e.  C ) )
2524notbid 285 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( -.  y  e.  C  <->  -.  x  e.  C ) )
2623, 25syl5ibrcom 213 . . . . . . . . . . . . 13  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  =  x  ->  -.  y  e.  C ) )
2726necon2ad 2507 . . . . . . . . . . . 12  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  e.  C  ->  y  =/=  x ) )
2822, 27jcad 519 . . . . . . . . . . 11  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  e.  C  ->  ( y  e.  B  /\  y  =/=  x
) ) )
29 eldifsn 3762 . . . . . . . . . . 11  |-  ( y  e.  ( B  \  { x } )  <-> 
( y  e.  B  /\  y  =/=  x
) )
3028, 29syl6ibr 218 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  e.  C  ->  y  e.  ( B 
\  { x }
) ) )
3130ssrdv 3198 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C_  ( B  \  { x } ) )
324, 12lspss 15757 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  ( B  \  { x }
)  C_  V  /\  C  C_  ( B  \  { x } ) )  ->  ( N `  C )  C_  ( N `  ( B  \  { x } ) ) )
3310, 19, 31, 32syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( N `  C
)  C_  ( N `  ( B  \  {
x } ) ) )
3433sseld 3192 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( x  e.  ( N `  C )  ->  x  e.  ( N `  ( B 
\  { x }
) ) ) )
3517, 34mtod 168 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  ( N `  C )
)
36 nelne1 2548 . . . . . 6  |-  ( ( x  e.  V  /\  -.  x  e.  ( N `  C )
)  ->  V  =/=  ( N `  C ) )
379, 35, 36syl2anc 642 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  V  =/=  ( N `  C ) )
3837necomd 2542 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( N `  C
)  =/=  V )
3938ex 423 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  ( ( x  e.  B  /\  -.  x  e.  C )  ->  ( N `  C
)  =/=  V ) )
4039exlimdv 1626 . 2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  ( E. x
( x  e.  B  /\  -.  x  e.  C
)  ->  ( N `  C )  =/=  V
) )
412, 40mpd 14 1  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  ( N `  C )  =/=  V
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165    C. wpss 3166   {csn 3653   ` cfv 5271   Basecbs 13164  Scalarcsca 13227   0gc0g 13416   1rcur 15355   LModclmod 15643   LSpanclspn 15744  LBasisclbs 15843
This theorem is referenced by:  islbs3  15924
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-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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lbs 15844
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