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Theorem islsati 29729
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 3466 . 2  |-  ( V 
\  { ( 0g
`  W ) } )  C_  V
2 islsati.v . . . 4  |-  V  =  ( Base `  W
)
3 islsati.n . . . 4  |-  N  =  ( LSpan `  W )
4 eqid 2435 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 islsati.a . . . 4  |-  A  =  (LSAtoms `  W )
62, 3, 4, 5islsat 29726 . . 3  |-  ( W  e.  X  ->  ( U  e.  A  <->  E. v  e.  ( V  \  {
( 0g `  W
) } ) U  =  ( N `  { v } ) ) )
76biimpa 471 . 2  |-  ( ( W  e.  X  /\  U  e.  A )  ->  E. v  e.  ( V  \  { ( 0g `  W ) } ) U  =  ( N `  {
v } ) )
8 ssrexv 3400 . 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 61 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 359    = wceq 1652    e. wcel 1725   E.wrex 2698    \ cdif 3309    C_ wss 3312   {csn 3806   ` cfv 5446   Basecbs 13461   0gc0g 13715   LSpanclspn 16039  LSAtomsclsa 29709
This theorem is referenced by:  lsmsatcv  29745  dihjat2  32166  dvh4dimlem  32178  lcfl8  32237  mapdval2N  32365  mapdspex  32403  hdmaprnlem16N  32600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-lsatoms 29711
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