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Theorem lspsnel3 15994
Description: A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 22922 analog.) (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lspsnss.s  |-  S  =  ( LSubSp `  W )
lspsnss.n  |-  N  =  ( LSpan `  W )
lspsnel3.w  |-  ( ph  ->  W  e.  LMod )
lspsnel3.u  |-  ( ph  ->  U  e.  S )
lspsnel3.x  |-  ( ph  ->  X  e.  U )
lspsnel3.y  |-  ( ph  ->  Y  e.  ( N `
 { X }
) )
Assertion
Ref Expression
lspsnel3  |-  ( ph  ->  Y  e.  U )

Proof of Theorem lspsnel3
StepHypRef Expression
1 lspsnel3.w . . 3  |-  ( ph  ->  W  e.  LMod )
2 lspsnel3.u . . 3  |-  ( ph  ->  U  e.  S )
3 lspsnel3.x . . 3  |-  ( ph  ->  X  e.  U )
4 lspsnss.s . . . 4  |-  S  =  ( LSubSp `  W )
5 lspsnss.n . . . 4  |-  N  =  ( LSpan `  W )
64, 5lspsnss 15993 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
71, 2, 3, 6syl3anc 1184 . 2  |-  ( ph  ->  ( N `  { X } )  C_  U
)
8 lspsnel3.y . 2  |-  ( ph  ->  Y  e.  ( N `
 { X }
) )
97, 8sseldd 3292 1  |-  ( ph  ->  Y  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    C_ wss 3263   {csn 3757   ` cfv 5394   LModclmod 15877   LSubSpclss 15935   LSpanclspn 15974
This theorem is referenced by:  lspsnel4  16123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-riota 6485  df-0g 13654  df-mnd 14617  df-grp 14739  df-lmod 15879  df-lss 15936  df-lsp 15975
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