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Theorem lspss 18207
Description: Span preserves subset ordering. (spanss 27001 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lspss.v  |-  V  =  ( Base `  W
)
lspss.n  |-  N  =  ( LSpan `  W )
Assertion
Ref Expression
lspss  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  T )  C_  ( N `  U
) )

Proof of Theorem lspss
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 simpl3 1013 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  /\  t  e.  ( LSubSp `  W ) )  ->  T  C_  U )
2 sstr2 3439 . . . . 5  |-  ( T 
C_  U  ->  ( U  C_  t  ->  T  C_  t ) )
31, 2syl 17 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  /\  t  e.  ( LSubSp `  W ) )  -> 
( U  C_  t  ->  T  C_  t )
)
43ss2rabdv 3510 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  { t  e.  ( LSubSp `  W
)  |  U  C_  t }  C_  { t  e.  ( LSubSp `  W
)  |  T  C_  t } )
5 intss 4255 . . 3  |-  ( { t  e.  ( LSubSp `  W )  |  U  C_  t }  C_  { t  e.  ( LSubSp `  W
)  |  T  C_  t }  ->  |^| { t  e.  ( LSubSp `  W
)  |  T  C_  t }  C_  |^| { t  e.  ( LSubSp `  W
)  |  U  C_  t } )
64, 5syl 17 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  |^| { t  e.  ( LSubSp `  W
)  |  T  C_  t }  C_  |^| { t  e.  ( LSubSp `  W
)  |  U  C_  t } )
7 simp1 1008 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  W  e.  LMod )
8 simp3 1010 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  T  C_  U )
9 simp2 1009 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  U  C_  V )
108, 9sstrd 3442 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  T  C_  V )
11 lspss.v . . . 4  |-  V  =  ( Base `  W
)
12 eqid 2451 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
13 lspss.n . . . 4  |-  N  =  ( LSpan `  W )
1411, 12, 13lspval 18198 . . 3  |-  ( ( W  e.  LMod  /\  T  C_  V )  ->  ( N `  T )  =  |^| { t  e.  ( LSubSp `  W )  |  T  C_  t } )
157, 10, 14syl2anc 667 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  T )  =  |^| { t  e.  ( LSubSp `  W )  |  T  C_  t } )
1611, 12, 13lspval 18198 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  |^| { t  e.  ( LSubSp `  W )  |  U  C_  t } )
17163adant3 1028 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  U )  =  |^| { t  e.  ( LSubSp `  W )  |  U  C_  t } )
186, 15, 173sstr4d 3475 1  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  T )  C_  ( N `  U
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   {crab 2741    C_ wss 3404   |^|cint 4234   ` cfv 5582   Basecbs 15121   LModclmod 18091   LSubSpclss 18155   LSpanclspn 18194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-lmod 18093  df-lss 18156  df-lsp 18195
This theorem is referenced by:  lspun  18210  lspssp  18211  lspprid1  18220  lbspss  18305  lspsolvlem  18365  lspsolv  18366  lsppratlem3  18372  lbsextlem2  18382  lbsextlem3  18383  lbsextlem4  18384  lindfrn  19379  f1lindf  19380  lssats  32578  lpssat  32579  lssatle  32581  lssat  32582  dvhdimlem  35012  dvh3dim3N  35017  mapdindp2  35289  lspindp5  35338
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