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Theorem pmaplubN 30722
Description: The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmaplub.b  |-  B  =  ( Base `  K
)
pmaplub.u  |-  U  =  ( lub `  K
)
pmaplub.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmaplubN  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( U `  ( M `  X )
)  =  X )

Proof of Theorem pmaplubN
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 pmaplub.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2437 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2437 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 pmaplub.m . . . 4  |-  M  =  ( pmap `  K
)
51, 2, 3, 4pmapval 30555 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  =  { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
)
65fveq2d 5733 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( U `  ( M `  X )
)  =  ( U `
 { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
) )
7 hlomcmat 30163 . . 3  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
8 pmaplub.u . . . 4  |-  U  =  ( lub `  K
)
91, 2, 8, 3atlatmstc 30118 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B )  ->  ( U `  { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
)  =  X )
107, 9sylan 459 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( U `  {
p  e.  ( Atoms `  K )  |  p ( le `  K
) X } )  =  X )
116, 10eqtrd 2469 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( U `  ( M `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2710   class class class wbr 4213   ` cfv 5455   Basecbs 13470   lecple 13537   lubclub 14400   CLatccla 14537   OMLcoml 29974   Atomscatm 30062   AtLatcal 30063   HLchlt 30149   pmapcpmap 30295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-pmap 30302
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