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Theorem lbspss 18046
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 3836 . . 3  |-  ( C 
C.  B  ->  E. x
( x  e.  B  /\  -.  x  e.  C
) )
213ad2ant3 1020 . 2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  E. x ( x  e.  B  /\  -.  x  e.  C )
)
3 simpl2 1001 . . . . . 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 18041 . . . . . 6  |-  ( B  e.  J  ->  B  C_  V )
73, 6syl 17 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  B  C_  V )
8 simprl 756 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  x  e.  B )
97, 8sseldd 3442 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  x  e.  V )
10 simpl1l 1048 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  W  e.  LMod )
117ssdifssd 3580 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( B  \  {
x } )  C_  V )
12 simpl3 1002 . . . . . . . . . . 11  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C.  B )
1312pssssd 3539 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  C  C_  B )
1413sseld 3440 . . . . . . . . 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 758 . . . . . . . . . . 11  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  C
)
16 eleq1 2474 . . . . . . . . . . . 12  |-  ( y  =  x  ->  (
y  e.  C  <->  x  e.  C ) )
1716notbid 292 . . . . . . . . . . 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 2616 . . . . . . . . 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 531 . . . . . . . 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 4096 . . . . . . . 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 3447 . . . . . 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 17948 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( B  \  { x }
)  C_  V  /\  C  C_  ( B  \  { x } ) )  ->  ( N `  C )  C_  ( N `  ( B  \  { x } ) ) )
2610, 11, 23, 25syl3anc 1230 . . . . 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 1049 . . . . . 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 18045 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  x  e.  B
)  ->  -.  x  e.  ( N `  ( B  \  { x }
) ) )
3210, 27, 3, 8, 31syl211anc 1236 . . . . 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 3444 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  -.  x  e.  ( N `  C )
)
34 nelne1 2732 . . . 4  |-  ( ( x  e.  V  /\  -.  x  e.  ( N `  C )
)  ->  V  =/=  ( N `  C ) )
359, 33, 34syl2anc 659 . . 3  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  ->  V  =/=  ( N `  C ) )
3635necomd 2674 . 2  |-  ( ( ( ( W  e. 
LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  /\  ( x  e.  B  /\  -.  x  e.  C ) )  -> 
( N `  C
)  =/=  V )
372, 36exlimddv 1747 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 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598    \ cdif 3410    C_ wss 3413    C. wpss 3414   {csn 3971   ` cfv 5568   Basecbs 14839  Scalarcsca 14910   0gc0g 15052   1rcur 17471   LModclmod 17830   LSpanclspn 17935  LBasisclbs 18038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-plusg 14920  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-mgp 17460  df-ur 17472  df-ring 17518  df-lmod 17832  df-lss 17897  df-lsp 17936  df-lbs 18039
This theorem is referenced by:  islbs3  18119
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