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Theorem lspsnel5 15752
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)
Hypotheses
Ref Expression
lspsnel5.v  |-  V  =  ( Base `  W
)
lspsnel5.s  |-  S  =  ( LSubSp `  W )
lspsnel5.n  |-  N  =  ( LSpan `  W )
lspsnel5.w  |-  ( ph  ->  W  e.  LMod )
lspsnel5.a  |-  ( ph  ->  U  e.  S )
lspsnel5.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
lspsnel5  |-  ( ph  ->  ( X  e.  U  <->  ( N `  { X } )  C_  U
) )

Proof of Theorem lspsnel5
StepHypRef Expression
1 lspsnel5.v . . 3  |-  V  =  ( Base `  W
)
2 lspsnel5.s . . 3  |-  S  =  ( LSubSp `  W )
3 lspsnel5.n . . 3  |-  N  =  ( LSpan `  W )
4 lspsnel5.w . . 3  |-  ( ph  ->  W  e.  LMod )
5 lspsnel5.a . . 3  |-  ( ph  ->  U  e.  S )
61, 2, 3, 4, 5lspsnel6 15751 . 2  |-  ( ph  ->  ( X  e.  U  <->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) ) )
7 lspsnel5.x . . 3  |-  ( ph  ->  X  e.  V )
87biantrurd 494 . 2  |-  ( ph  ->  ( ( N `  { X } )  C_  U 
<->  ( X  e.  V  /\  ( N `  { X } )  C_  U
) ) )
96, 8bitr4d 247 1  |-  ( ph  ->  ( X  e.  U  <->  ( N `  { X } )  C_  U
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   ` cfv 5255   Basecbs 13148   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728
This theorem is referenced by:  lspsnel5a  15753  lspprid1  15754  lspsnss2  15762  lsmelpr  15844  lspsncmp  15869  lspsnne1  15870  lspsnne2  15871  lspsneq  15875  lspindpi  15885  islbs2  15907  lsatelbN  29196  lsmsat  29198  lsatfixedN  29199  l1cvpat  29244  dia2dimlem5  31258  dochsncom  31572  dihjat1lem  31618  dvh4dimlem  31633  lclkrlem2a  31697  lcfrlem6  31737  lcfrlem20  31752  lcfrlem26  31758  lcfrlem36  31768  mapdval2N  31820  mapdrvallem2  31835  mapdindp  31861  mapdh6aN  31925  lspindp5  31960  mapdh8ab  31967  mapdh8e  31974  hdmap1l6a  32000  hdmaprnlem3eN  32051  hdmapoc  32124
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629  df-lss 15690  df-lsp 15729
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