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Theorem lspsnel3 17847
Description: A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 26785 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 17846 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
71, 2, 3, 6syl3anc 1228 . 2  |-  ( ph  ->  ( N `  { X } )  C_  U
)
8 lspsnel3.y . 2  |-  ( ph  ->  Y  e.  ( N `
 { X }
) )
97, 8sseldd 3440 1  |-  ( ph  ->  Y  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840    C_ wss 3411   {csn 3969   ` cfv 5523   LModclmod 17722   LSubSpclss 17788   LSpanclspn 17827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-grp 16271  df-lmod 17724  df-lss 17789  df-lsp 17828
This theorem is referenced by:  lspsnel4  17980
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