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Theorem lbspss 16146
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 3685 . . 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 16141 . . . . . 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 3341 . . . 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 3477 . . . . . 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 3436 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C_  B )
1413sseld 3339 . . . . . . . . 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 2495 . . . . . . . . . . . 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 2646 . . . . . . . . 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 3919 . . . . . . . 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 3346 . . . . . 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 16052 . . . . . 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 16145 . . . . . 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 3343 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  ( N `  C )
)
34 nelne1 2687 . . . 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 2681 . 2  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( N `  C
)  =/=  V )
372, 36exlimddv 1648 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    C_ wss 3312    C. wpss 3313   {csn 3806   ` cfv 5446   Basecbs 13461  Scalarcsca 13524   0gc0g 13715   1rcur 15654   LModclmod 15942   LSpanclspn 16039  LBasisclbs 16138
This theorem is referenced by:  islbs3  16219
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-0g 13719  df-mnd 14682  df-grp 14804  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lbs 16139
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