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Theorem lspss 16061
Description: Span preserves subset ordering. (spanss 22851 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 963 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  /\  t  e.  ( LSubSp `  W ) )  ->  T  C_  U )
2 sstr2 3356 . . . . 5  |-  ( T 
C_  U  ->  ( U  C_  t  ->  T  C_  t ) )
31, 2syl 16 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  /\  t  e.  ( LSubSp `  W ) )  -> 
( U  C_  t  ->  T  C_  t )
)
43ss2rabdv 3425 . . 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 4072 . . 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 16 . 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 958 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  W  e.  LMod )
8 simp3 960 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  T  C_  U )
9 simp2 959 . . . 4  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  U  C_  V )
108, 9sstrd 3359 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  T  C_  V )
11 lspss.v . . . 4  |-  V  =  ( Base `  W
)
12 eqid 2437 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
13 lspss.n . . . 4  |-  N  =  ( LSpan `  W )
1411, 12, 13lspval 16052 . . 3  |-  ( ( W  e.  LMod  /\  T  C_  V )  ->  ( N `  T )  =  |^| { t  e.  ( LSubSp `  W )  |  T  C_  t } )
157, 10, 14syl2anc 644 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  T )  =  |^| { t  e.  ( LSubSp `  W )  |  T  C_  t } )
1611, 12, 13lspval 16052 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  |^| { t  e.  ( LSubSp `  W )  |  U  C_  t } )
17163adant3 978 . 2  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  U )  =  |^| { t  e.  ( LSubSp `  W )  |  U  C_  t } )
186, 15, 173sstr4d 3392 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2710    C_ wss 3321   |^|cint 4051   ` cfv 5455   Basecbs 13470   LModclmod 15951   LSubSpclss 16009   LSpanclspn 16048
This theorem is referenced by:  lspun  16064  lspssp  16065  lspprid1  16074  lbspss  16155  lspsolvlem  16215  lspsolv  16216  lsppratlem3  16222  lbsextlem2  16232  lbsextlem3  16233  lbsextlem4  16234  lindfrn  27269  f1lindf  27270  lssats  29811  lpssat  29812  lssatle  29814  lssat  29815  dvhdimlem  32243  dvh3dim3N  32248  mapdindp2  32520  lspindp5  32569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-riota 6550  df-0g 13728  df-mnd 14691  df-grp 14813  df-lmod 15953  df-lss 16010  df-lsp 16049
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