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Theorem pmapsubclN 30841
Description: A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapsubcl.b  |-  B  =  ( Base `  K
)
pmapsubcl.m  |-  M  =  ( pmap `  K
)
pmapsubcl.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
pmapsubclN  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  e.  C )

Proof of Theorem pmapsubclN
StepHypRef Expression
1 pmapsubcl.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2442 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 pmapsubcl.m . . 3  |-  M  =  ( pmap `  K
)
41, 2, 3pmapssat 30654 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  ( Atoms `  K ) )
5 eqid 2442 . . 3  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
61, 3, 52polpmapN 30808 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  X )
) )  =  ( M `  X ) )
7 pmapsubcl.c . . . 4  |-  C  =  ( PSubCl `  K )
82, 5, 7ispsubclN 30832 . . 3  |-  ( K  e.  HL  ->  (
( M `  X
)  e.  C  <->  ( ( M `  X )  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  X )
) )  =  ( M `  X ) ) ) )
98adantr 453 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  C  <->  ( ( M `  X
)  C_  ( Atoms `  K )  /\  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  X )
) )  =  ( M `  X ) ) ) )
104, 6, 9mpbir2and 890 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    C_ wss 3306   ` cfv 5483   Basecbs 13500   Atomscatm 30159   HLchlt 30246   pmapcpmap 30392   _|_
PcpolN 30797   PSubClcpscN 30829
This theorem is referenced by:  psubclinN  30843  paddatclN  30844  linepsubclN  30846  polsubclN  30847  pmapojoinN  30863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-p1 14500  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-pmap 30399  df-polarityN 30798  df-psubclN 30830
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