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Theorem lspsnel3 17205
Description: A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 25154 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 17204 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
71, 2, 3, 6syl3anc 1219 . 2  |-  ( ph  ->  ( N `  { X } )  C_  U
)
8 lspsnel3.y . 2  |-  ( ph  ->  Y  e.  ( N `
 { X }
) )
97, 8sseldd 3468 1  |-  ( ph  ->  Y  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    C_ wss 3439   {csn 3988   ` cfv 5529   LModclmod 17081   LSubSpclss 17146   LSpanclspn 17185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-0g 14503  df-mnd 15538  df-grp 15668  df-lmod 17083  df-lss 17147  df-lsp 17186
This theorem is referenced by:  lspsnel4  17338
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