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Theorem islsati 29806
Description: A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.)
Hypotheses
Ref Expression
islsati.v  |-  V  =  ( Base `  W
)
islsati.n  |-  N  =  ( LSpan `  W )
islsati.a  |-  A  =  (LSAtoms `  W )
Assertion
Ref Expression
islsati  |-  ( ( W  e.  X  /\  U  e.  A )  ->  E. v  e.  V  U  =  ( N `  { v } ) )
Distinct variable groups:    v, N    v, U    v, V    v, W    v, X
Allowed substitution hint:    A( v)

Proof of Theorem islsati
StepHypRef Expression
1 difss 3316 . 2  |-  ( V 
\  { ( 0g
`  W ) } )  C_  V
2 islsati.v . . . 4  |-  V  =  ( Base `  W
)
3 islsati.n . . . 4  |-  N  =  ( LSpan `  W )
4 eqid 2296 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 islsati.a . . . 4  |-  A  =  (LSAtoms `  W )
62, 3, 4, 5islsat 29803 . . 3  |-  ( W  e.  X  ->  ( U  e.  A  <->  E. v  e.  ( V  \  {
( 0g `  W
) } ) U  =  ( N `  { v } ) ) )
76biimpa 470 . 2  |-  ( ( W  e.  X  /\  U  e.  A )  ->  E. v  e.  ( V  \  { ( 0g `  W ) } ) U  =  ( N `  {
v } ) )
8 ssrexv 3251 . 2  |-  ( ( V  \  { ( 0g `  W ) } )  C_  V  ->  ( E. v  e.  ( V  \  {
( 0g `  W
) } ) U  =  ( N `  { v } )  ->  E. v  e.  V  U  =  ( N `  { v } ) ) )
91, 7, 8mpsyl 59 1  |-  ( ( W  e.  X  /\  U  e.  A )  ->  E. v  e.  V  U  =  ( N `  { v } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    \ cdif 3162    C_ wss 3165   {csn 3653   ` cfv 5271   Basecbs 13164   0gc0g 13416   LSpanclspn 15744  LSAtomsclsa 29786
This theorem is referenced by:  lsmsatcv  29822  dihjat2  32243  dvh4dimlem  32255  lcfl8  32314  mapdval2N  32442  mapdspex  32480  hdmaprnlem16N  32677
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-fv 5279  df-lsatoms 29788
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