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Theorem islsati 29184
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 3303 . 2  |-  ( V 
\  { ( 0g
`  W ) } )  C_  V
2 islsati.v . . . 4  |-  V  =  ( Base `  W
)
3 islsati.n . . . 4  |-  N  =  ( LSpan `  W )
4 eqid 2283 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 islsati.a . . . 4  |-  A  =  (LSAtoms `  W )
62, 3, 4, 5islsat 29181 . . 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 3238 . 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 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149    C_ wss 3152   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   LSpanclspn 15728  LSAtomsclsa 29164
This theorem is referenced by:  lsmsatcv  29200  dihjat2  31621  dvh4dimlem  31633  lcfl8  31692  mapdval2N  31820  mapdspex  31858  hdmaprnlem16N  32055
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
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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-fv 5263  df-lsatoms 29166
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