Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmapidclN Unicode version

Theorem pmapidclN 30131
Description: Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapidcl.u  |-  U  =  ( lub `  K
)
pmapidcl.m  |-  M  =  ( pmap `  K
)
pmapidcl.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
pmapidclN  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( M `  ( U `  X )
)  =  X )

Proof of Theorem pmapidclN
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 pmapidcl.c . . . 4  |-  C  =  ( PSubCl `  K )
31, 2psubclssatN 30130 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Atoms `  K ) )
4 pmapidcl.u . . . 4  |-  U  =  ( lub `  K
)
5 pmapidcl.m . . . 4  |-  M  =  ( pmap `  K
)
6 eqid 2283 . . . 4  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
74, 1, 5, 62polvalN 30103 . . 3  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( M `  ( U `
 X ) ) )
83, 7syldan 456 . 2  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( M `  ( U `
 X ) ) )
96, 2psubcli2N 30128 . 2  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )
108, 9eqtr3d 2317 1  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( M `  ( U `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255   lubclub 14076   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   _|_ PcpolN 30091   PSubClcpscN 30123
This theorem is referenced by:  psubclinN  30137  paddatclN  30138
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  ax-un 4512
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-nel 2449  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-iun 3907  df-iin 3908  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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-pmap 29693  df-polarityN 30092  df-psubclN 30124
  Copyright terms: Public domain W3C validator