MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lbsind2 Unicode version

Theorem lbsind2 15834
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 simp2 956 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  B  e.  J )
2 simp3 957 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  E  e.  B )
3 simp1l 979 . . . 4  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  W  e.  LMod )
4 lbsind2.f . . . . 5  |-  F  =  (Scalar `  W )
54lmodrng 15635 . . . 4  |-  ( W  e.  LMod  ->  F  e. 
Ring )
6 eqid 2283 . . . . 5  |-  ( Base `  F )  =  (
Base `  F )
7 lbsind2.o . . . . 5  |-  .1.  =  ( 1r `  F )
86, 7rngidcl 15361 . . . 4  |-  ( F  e.  Ring  ->  .1.  e.  ( Base `  F )
)
93, 5, 83syl 18 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  .1.  e.  ( Base `  F )
)
10 simp1r 980 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  .1.  =/=  .0.  )
11 eqid 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
12 lbsind2.j . . . 4  |-  J  =  (LBasis `  W )
13 lbsind2.n . . . 4  |-  N  =  ( LSpan `  W )
14 eqid 2283 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
15 lbsind2.z . . . 4  |-  .0.  =  ( 0g `  F )
1611, 12, 13, 4, 14, 6, 15lbsind 15833 . . 3  |-  ( ( ( B  e.  J  /\  E  e.  B
)  /\  (  .1.  e.  ( Base `  F
)  /\  .1.  =/=  .0.  ) )  ->  -.  (  .1.  ( .s `  W ) E )  e.  ( N `  ( B  \  { E } ) ) )
171, 2, 9, 10, 16syl22anc 1183 . 2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  -.  (  .1.  ( .s `  W
) E )  e.  ( N `  ( B  \  { E }
) ) )
1811, 12lbsel 15831 . . . . 5  |-  ( ( B  e.  J  /\  E  e.  B )  ->  E  e.  ( Base `  W ) )
191, 2, 18syl2anc 642 . . . 4  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  E  e.  ( Base `  W )
)
2011, 4, 14, 7lmodvs1 15658 . . . 4  |-  ( ( W  e.  LMod  /\  E  e.  ( Base `  W
) )  ->  (  .1.  ( .s `  W
) E )  =  E )
213, 19, 20syl2anc 642 . . 3  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  (  .1.  ( .s `  W ) E )  =  E )
2221eleq1d 2349 . 2  |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B
)  ->  ( (  .1.  ( .s `  W
) E )  e.  ( N `  ( B  \  { E }
) )  <->  E  e.  ( N `  ( B 
\  { E }
) ) ) )
2317, 22mtbid 291 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Ringcrg 15337   1rcur 15339   LModclmod 15627   LSpanclspn 15728  LBasisclbs 15827
This theorem is referenced by:  lbspss  15835  islbs2  15907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lbs 15828
  Copyright terms: Public domain W3C validator