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Theorem lspid 15755
Description: The span of a subspace is itself. (spanid 21942 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 2296 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 lspid.s . . . 4  |-  S  =  ( LSubSp `  W )
31, 2lssss 15710 . . 3  |-  ( U  e.  S  ->  U  C_  ( Base `  W
) )
4 lspid.n . . . 4  |-  N  =  ( LSpan `  W )
51, 2, 4lspval 15748 . . 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 3898 . . 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 2328 1  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   |^|cint 3878   ` cfv 5271   Basecbs 13164   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744
This theorem is referenced by:  lspidm  15759  lspssp  15761  lspsn0  15781  lspsolvlem  15911  lbsextlem3  15929  filnm  27295  islshpsm  29792  lshpnel2N  29797  lssats  29824  lkrlsp3  29916  dochspocN  32192  dochsatshp  32263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-lmod 15645  df-lss 15706  df-lsp 15745
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