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Theorem lspsnss 15986
Description: The span of the singleton of a subspace member is included in the subspace. (spansnss 22914 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
Hypotheses
Ref Expression
lspsnss.s  |-  S  =  ( LSubSp `  W )
lspsnss.n  |-  N  =  ( LSpan `  W )
Assertion
Ref Expression
lspsnss  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)

Proof of Theorem lspsnss
StepHypRef Expression
1 snssi 3878 . 2  |-  ( X  e.  U  ->  { X }  C_  U )
2 lspsnss.s . . 3  |-  S  =  ( LSubSp `  W )
3 lspsnss.n . . 3  |-  N  =  ( LSpan `  W )
42, 3lspssp 15984 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  { X }  C_  U )  -> 
( N `  { X } )  C_  U
)
51, 4syl3an3 1219 1  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3256   {csn 3750   ` cfv 5387   LModclmod 15870   LSubSpclss 15928   LSpanclspn 15967
This theorem is referenced by:  lspsnel3  15987  lspsnel6  15990
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-riota 6478  df-0g 13647  df-mnd 14610  df-grp 14732  df-lmod 15872  df-lss 15929  df-lsp 15968
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