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Theorem lspss 15741
Description: Span preserves subset ordering. (spanss 21927 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 960 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  /\  t  e.  ( LSubSp `  W ) )  ->  T  C_  U )
2 sstr2 3186 . . . . 5  |-  ( T 
C_  U  ->  ( U  C_  t  ->  T  C_  t ) )
31, 2syl 15 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  /\  t  e.  ( LSubSp `  W ) )  -> 
( U  C_  t  ->  T  C_  t )
)
43ss2rabdv 3254 . . 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 3883 . . 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 15 . 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 955 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  W  e.  LMod )
8 simp3 957 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  T  C_  U )
9 simp2 956 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  U  C_  V )
108, 9sstrd 3189 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  T  C_  V )
11 lspss.v . . . 4  |-  V  =  ( Base `  W
)
12 eqid 2283 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
13 lspss.n . . . 4  |-  N  =  ( LSpan `  W )
1411, 12, 13lspval 15732 . . 3  |-  ( ( W  e.  LMod  /\  T  C_  V )  ->  ( N `  T )  =  |^| { t  e.  ( LSubSp `  W )  |  T  C_  t } )
157, 10, 14syl2anc 642 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  T )  =  |^| { t  e.  ( LSubSp `  W )  |  T  C_  t } )
1611, 12, 13lspval 15732 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  |^| { t  e.  ( LSubSp `  W )  |  U  C_  t } )
17163adant3 975 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  U )  =  |^| { t  e.  ( LSubSp `  W )  |  U  C_  t } )
186, 15, 173sstr4d 3221 1  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  T )  C_  ( N `  U
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   |^|cint 3862   ` cfv 5255   Basecbs 13148   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728
This theorem is referenced by:  lspun  15744  lspssp  15745  lspprid1  15754  lbspss  15835  lspsolvlem  15895  lspsolv  15896  lsppratlem3  15902  lbsextlem2  15912  lbsextlem3  15913  lbsextlem4  15914  lindfrn  27291  f1lindf  27292  lssats  29202  lpssat  29203  lssatle  29205  lssat  29206  dvhdimlem  31634  dvh3dim3N  31639  mapdindp2  31911  lspindp5  31960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629  df-lss 15690  df-lsp 15729
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