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Theorem lbspss 16074
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 3629 . . 3  |-  ( C 
C.  B  ->  E. x
( x  e.  B  /\  -.  x  e.  C
) )
213ad2ant3 980 . 2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  E. x ( x  e.  B  /\  -.  x  e.  C )
)
3 simpl2 961 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  B  e.  J )
4 lbspss.v . . . . . . 7  |-  V  =  ( Base `  W
)
5 lbsind2.j . . . . . . 7  |-  J  =  (LBasis `  W )
64, 5lbsss 16069 . . . . . 6  |-  ( B  e.  J  ->  B  C_  V )
73, 6syl 16 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  B  C_  V )
8 simprl 733 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  x  e.  B )
97, 8sseldd 3285 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  x  e.  V )
10 simpl1l 1008 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  W  e.  LMod )
117ssdifssd 3421 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( B  \  {
x } )  C_  V )
12 simpl3 962 . . . . . . . . . . 11  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C.  B )
1312pssssd 3380 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C_  B )
1413sseld 3283 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  e.  C  ->  y  e.  B ) )
15 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  C
)
16 eleq1 2440 . . . . . . . . . . . 12  |-  ( y  =  x  ->  (
y  e.  C  <->  x  e.  C ) )
1716notbid 286 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( -.  y  e.  C  <->  -.  x  e.  C ) )
1815, 17syl5ibrcom 214 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  =  x  ->  -.  y  e.  C ) )
1918necon2ad 2591 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  e.  C  ->  y  =/=  x ) )
2014, 19jcad 520 . . . . . . . 8  |-  ( ( ( ( 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
) ) )
21 eldifsn 3863 . . . . . . . 8  |-  ( y  e.  ( B  \  { x } )  <-> 
( y  e.  B  /\  y  =/=  x
) )
2220, 21syl6ibr 219 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  e.  C  ->  y  e.  ( B 
\  { x }
) ) )
2322ssrdv 3290 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C_  ( B  \  { x } ) )
24 lbsind2.n . . . . . . 7  |-  N  =  ( LSpan `  W )
254, 24lspss 15980 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( B  \  { x }
)  C_  V  /\  C  C_  ( B  \  { x } ) )  ->  ( N `  C )  C_  ( N `  ( B  \  { x } ) ) )
2610, 11, 23, 25syl3anc 1184 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( N `  C
)  C_  ( N `  ( B  \  {
x } ) ) )
27 simpl1r 1009 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  .1.  =/=  .0.  )
28 lbsind2.f . . . . . . 7  |-  F  =  (Scalar `  W )
29 lbsind2.o . . . . . . 7  |-  .1.  =  ( 1r `  F )
30 lbsind2.z . . . . . . 7  |-  .0.  =  ( 0g `  F )
315, 24, 28, 29, 30lbsind2 16073 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  x  e.  B
)  ->  -.  x  e.  ( N `  ( B  \  { x }
) ) )
3210, 27, 3, 8, 31syl211anc 1190 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  ( N `  ( B  \  { x } ) ) )
3326, 32ssneldd 3287 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  ( N `  C )
)
34 nelne1 2632 . . . 4  |-  ( ( x  e.  V  /\  -.  x  e.  ( N `  C )
)  ->  V  =/=  ( N `  C ) )
359, 33, 34syl2anc 643 . . 3  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  V  =/=  ( N `  C ) )
3635necomd 2626 . 2  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( N `  C
)  =/=  V )
372, 36exlimddv 1645 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 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2543    \ cdif 3253    C_ wss 3256    C. wpss 3257   {csn 3750   ` cfv 5387   Basecbs 13389  Scalarcsca 13452   0gc0g 13643   1rcur 15582   LModclmod 15870   LSpanclspn 15967  LBasisclbs 16066
This theorem is referenced by:  islbs3  16147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-plusg 13462  df-0g 13647  df-mnd 14610  df-grp 14732  df-mgp 15569  df-rng 15583  df-ur 15585  df-lmod 15872  df-lss 15929  df-lsp 15968  df-lbs 16067
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