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Theorem lbsind2 16143
Description: A basis is linearly independent; that is, every element is not in the span of the remainder of the basis. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 12-Jan-2015.)
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 )
Assertion
Ref Expression
lbsind2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  -.  E  e.  ( N `  ( B  \  { E }
) ) )

Proof of Theorem lbsind2
StepHypRef Expression
1 simp1l 981 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  W  e.  LMod )
2 simp2 958 . . . 4  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  B  e.  J )
3 simp3 959 . . . 4  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  E  e.  B )
4 eqid 2435 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
5 lbsind2.j . . . . 5  |-  J  =  (LBasis `  W )
64, 5lbsel 16140 . . . 4  |-  ( ( B  e.  J  /\  E  e.  B )  ->  E  e.  ( Base `  W ) )
72, 3, 6syl2anc 643 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  E  e.  ( Base `  W )
)
8 lbsind2.f . . . 4  |-  F  =  (Scalar `  W )
9 eqid 2435 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
10 lbsind2.o . . . 4  |-  .1.  =  ( 1r `  F )
114, 8, 9, 10lmodvs1 15968 . . 3  |-  ( ( W  e.  LMod  /\  E  e.  ( Base `  W
) )  ->  (  .1.  ( .s `  W
) E )  =  E )
121, 7, 11syl2anc 643 . 2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  (  .1.  ( .s `  W ) E )  =  E )
138lmodrng 15948 . . . 4  |-  ( W  e.  LMod  ->  F  e. 
Ring )
14 eqid 2435 . . . . 5  |-  ( Base `  F )  =  (
Base `  F )
1514, 10rngidcl 15674 . . . 4  |-  ( F  e.  Ring  ->  .1.  e.  ( Base `  F )
)
161, 13, 153syl 19 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  .1.  e.  ( Base `  F )
)
17 simp1r 982 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  .1.  =/=  .0.  )
18 lbsind2.n . . . 4  |-  N  =  ( LSpan `  W )
19 lbsind2.z . . . 4  |-  .0.  =  ( 0g `  F )
204, 5, 18, 8, 9, 14, 19lbsind 16142 . . 3  |-  ( ( ( B  e.  J  /\  E  e.  B
)  /\  (  .1.  e.  ( Base `  F
)  /\  .1.  =/=  .0.  ) )  ->  -.  (  .1.  ( .s `  W ) E )  e.  ( N `  ( B  \  { E } ) ) )
212, 3, 16, 17, 20syl22anc 1185 . 2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  -.  (  .1.  ( .s `  W
) E )  e.  ( N `  ( B  \  { E }
) ) )
2212, 21eqneltrrd 2529 1  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  -.  E  e.  ( N `  ( B  \  { E }
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13459  Scalarcsca 13522   .scvsca 13523   0gc0g 13713   Ringcrg 15650   1rcur 15652   LModclmod 15940   LSpanclspn 16037  LBasisclbs 16136
This theorem is referenced by:  lbspss  16144  islbs2  16216
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
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-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 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-plusg 13532  df-0g 13717  df-mnd 14680  df-mgp 15639  df-rng 15653  df-ur 15655  df-lmod 15942  df-lbs 16137
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