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Theorem lspsnel5a 15769
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
Hypotheses
Ref Expression
lspsnel5a.s  |-  S  =  ( LSubSp `  W )
lspsnel5a.n  |-  N  =  ( LSpan `  W )
lspsnel5a.w  |-  ( ph  ->  W  e.  LMod )
lspsnel5a.a  |-  ( ph  ->  U  e.  S )
lspsnel5a.x  |-  ( ph  ->  X  e.  U )
Assertion
Ref Expression
lspsnel5a  |-  ( ph  ->  ( N `  { X } )  C_  U
)

Proof of Theorem lspsnel5a
StepHypRef Expression
1 lspsnel5a.x . 2  |-  ( ph  ->  X  e.  U )
2 eqid 2296 . . 3  |-  ( Base `  W )  =  (
Base `  W )
3 lspsnel5a.s . . 3  |-  S  =  ( LSubSp `  W )
4 lspsnel5a.n . . 3  |-  N  =  ( LSpan `  W )
5 lspsnel5a.w . . 3  |-  ( ph  ->  W  e.  LMod )
6 lspsnel5a.a . . 3  |-  ( ph  ->  U  e.  S )
72, 3lssel 15711 . . . 4  |-  ( ( U  e.  S  /\  X  e.  U )  ->  X  e.  ( Base `  W ) )
86, 1, 7syl2anc 642 . . 3  |-  ( ph  ->  X  e.  ( Base `  W ) )
92, 3, 4, 5, 6, 8lspsnel5 15768 . 2  |-  ( ph  ->  ( X  e.  U  <->  ( N `  { X } )  C_  U
) )
101, 9mpbid 201 1  |-  ( ph  ->  ( N `  { X } )  C_  U
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    C_ wss 3165   {csn 3653   ` cfv 5271   Basecbs 13164   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744
This theorem is referenced by:  lssats2  15773  lspsn  15775  lspsnvsi  15777  lsmelval2  15854  lspprabs  15864  lspvadd  15865  lspabs3  15890  lsmcv  15910  lspsnat  15914  lsppratlem6  15921  issubassa2  16100  lshpnel  29795  lsatel  29817  lsmsat  29820  lssatomic  29823  lssats  29824  lsat0cv  29845  dia2dimlem10  31885  dochsatshpb  32264  lclkrlem2f  32324  lcfrlem25  32379  lcfrlem35  32389  mapdval2N  32442  mapdrvallem2  32457  mapdpglem8  32491  mapdpglem13  32496  mapdindp0  32531  mapdh6aN  32547  mapdh8e  32596  mapdh9a  32602  hdmap1l6a  32622  hdmapval0  32648  hdmapval3lemN  32652  hdmap10lem  32654  hdmap11lem1  32656  hdmap11lem2  32657  hdmaprnlem4N  32668  hdmaprnlem3eN  32673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-lmod 15645  df-lss 15706  df-lsp 15745
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