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Theorem lspid 15739
Description: The span of a subspace is itself. (spanid 21926 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lspid.s  |-  S  =  ( LSubSp `  W )
lspid.n  |-  N  =  ( LSpan `  W )
Assertion
Ref Expression
lspid  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )

Proof of Theorem lspid
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 lspid.s . . . 4  |-  S  =  ( LSubSp `  W )
31, 2lssss 15694 . . 3  |-  ( U  e.  S  ->  U  C_  ( Base `  W
) )
4 lspid.n . . . 4  |-  N  =  ( LSpan `  W )
51, 2, 4lspval 15732 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  ( Base `  W
) )  ->  ( N `  U )  =  |^| { t  e.  S  |  U  C_  t } )
63, 5sylan2 460 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  |^| { t  e.  S  |  U  C_  t } )
7 intmin 3882 . . 3  |-  ( U  e.  S  ->  |^| { t  e.  S  |  U  C_  t }  =  U )
87adantl 452 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  |^| { t  e.  S  |  U  C_  t }  =  U )
96, 8eqtrd 2315 1  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = 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:  lspidm  15743  lspssp  15745  lspsn0  15765  lspsolvlem  15895  lbsextlem3  15913  filnm  27192  islshpsm  29170  lshpnel2N  29175  lssats  29202  lkrlsp3  29294  dochspocN  31570  dochsatshp  31641
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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