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Theorem pmapidclN 29949
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 2316 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 pmapidcl.c . . . 4  |-  C  =  ( PSubCl `  K )
31, 2psubclssatN 29948 . . 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 2316 . . . 4  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
74, 1, 5, 62polvalN 29921 . . 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 29946 . 2  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )
108, 9eqtr3d 2350 1  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( M `  ( U `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701    C_ wss 3186   ` cfv 5292   lubclub 14125   Atomscatm 29271   HLchlt 29358   pmapcpmap 29504   _|_ PcpolN 29909   PSubClcpscN 29941
This theorem is referenced by:  psubclinN  29955  paddatclN  29956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-pmap 29511  df-polarityN 29910  df-psubclN 29942
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