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Theorem lspid 17406
Description: The span of a subspace is itself. (spanid 25929 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 2462 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 lspid.s . . . 4  |-  S  =  ( LSubSp `  W )
31, 2lssss 17361 . . 3  |-  ( U  e.  S  ->  U  C_  ( Base `  W
) )
4 lspid.n . . . 4  |-  N  =  ( LSpan `  W )
51, 2, 4lspval 17399 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  ( Base `  W
) )  ->  ( N `  U )  =  |^| { t  e.  S  |  U  C_  t } )
63, 5sylan2 474 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  |^| { t  e.  S  |  U  C_  t } )
7 intmin 4297 . . 3  |-  ( U  e.  S  ->  |^| { t  e.  S  |  U  C_  t }  =  U )
87adantl 466 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  |^| { t  e.  S  |  U  C_  t }  =  U )
96, 8eqtrd 2503 1  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {crab 2813    C_ wss 3471   |^|cint 4277   ` cfv 5581   Basecbs 14481   LModclmod 17290   LSubSpclss 17356   LSpanclspn 17395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-0g 14688  df-mnd 15723  df-grp 15853  df-lmod 17292  df-lss 17357  df-lsp 17396
This theorem is referenced by:  lspidm  17410  lspssp  17412  lspsn0  17432  lspsolvlem  17566  lbsextlem3  17584  filnm  30631  islshpsm  33654  lshpnel2N  33659  lssats  33686  lkrlsp3  33778  dochspocN  36054  dochsatshp  36125
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