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Theorem lbspss 17506
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 3887 . . 3  |-  ( C 
C.  B  ->  E. x
( x  e.  B  /\  -.  x  e.  C
) )
213ad2ant3 1014 . 2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  E. x ( x  e.  B  /\  -.  x  e.  C )
)
3 simpl2 995 . . . . . 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 17501 . . . . . 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 755 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  x  e.  B )
97, 8sseldd 3500 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  x  e.  V )
10 simpl1l 1042 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  W  e.  LMod )
117ssdifssd 3637 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( B  \  {
x } )  C_  V )
12 simpl3 996 . . . . . . . . . . 11  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C.  B )
1312pssssd 3596 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C_  B )
1413sseld 3498 . . . . . . . . 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 756 . . . . . . . . . . 11  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  C
)
16 eleq1 2534 . . . . . . . . . . . 12  |-  ( y  =  x  ->  (
y  e.  C  <->  x  e.  C ) )
1716notbid 294 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( -.  y  e.  C  <->  -.  x  e.  C ) )
1815, 17syl5ibrcom 222 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( y  =  x  ->  -.  y  e.  C ) )
1918necon2ad 2675 . . . . . . . . 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 533 . . . . . . . 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 4147 . . . . . . . 8  |-  ( y  e.  ( B  \  { x } )  <-> 
( y  e.  B  /\  y  =/=  x
) )
2220, 21syl6ibr 227 . . . . . . 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 3505 . . . . . 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 17408 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( B  \  { x }
)  C_  V  /\  C  C_  ( B  \  { x } ) )  ->  ( N `  C )  C_  ( N `  ( B  \  { x } ) ) )
2610, 11, 23, 25syl3anc 1223 . . . . 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 1043 . . . . . 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 17505 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  x  e.  B
)  ->  -.  x  e.  ( N `  ( B  \  { x }
) ) )
3210, 27, 3, 8, 31syl211anc 1229 . . . . 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 3502 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  ( N `  C )
)
34 nelne1 2791 . . . 4  |-  ( ( x  e.  V  /\  -.  x  e.  ( N `  C )
)  ->  V  =/=  ( N `  C ) )
359, 33, 34syl2anc 661 . . 3  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  V  =/=  ( N `  C ) )
3635necomd 2733 . 2  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( N `  C
)  =/=  V )
372, 36exlimddv 1697 1  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  ( N `  C )  =/=  V
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2657    \ cdif 3468    C_ wss 3471    C. wpss 3472   {csn 4022   ` cfv 5581   Basecbs 14481  Scalarcsca 14549   0gc0g 14686   1rcur 16938   LModclmod 17290   LSpanclspn 17395  LBasisclbs 17498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-plusg 14559  df-0g 14688  df-mnd 15723  df-grp 15853  df-mgp 16927  df-ur 16939  df-rng 16983  df-lmod 17292  df-lss 17357  df-lsp 17396  df-lbs 17499
This theorem is referenced by:  islbs3  17579
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