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Theorem 2pmaplubN 32907
Description: Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspmaplub.u  |-  U  =  ( lub `  K
)
sspmaplub.a  |-  A  =  ( Atoms `  K )
sspmaplub.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
2pmaplubN  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( M `  ( U `  ( M `  ( U `  S
) ) ) )  =  ( M `  ( U `  S ) ) )

Proof of Theorem 2pmaplubN
StepHypRef Expression
1 sspmaplub.u . . . . . . 7  |-  U  =  ( lub `  K
)
2 sspmaplub.a . . . . . . 7  |-  A  =  ( Atoms `  K )
3 sspmaplub.m . . . . . . 7  |-  M  =  ( pmap `  K
)
4 eqid 2400 . . . . . . 7  |-  ( _|_P `  K )  =  ( _|_P `  K )
51, 2, 3, 42polvalN 32895 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  S
) )  =  ( M `  ( U `
 S ) ) )
65fveq2d 5807 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  S
) ) )  =  ( ( _|_P `  K ) `  ( M `  ( U `  S ) ) ) )
76fveq2d 5807 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  S
) ) ) )  =  ( ( _|_P `  K ) `
 ( ( _|_P `  K ) `
 ( M `  ( U `  S ) ) ) ) )
82, 4polssatN 32889 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  S
)  C_  A )
92, 43polN 32897 . . . . 5  |-  ( ( K  e.  HL  /\  ( ( _|_P `  K ) `  S
)  C_  A )  ->  ( ( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  S
) ) ) )  =  ( ( _|_P `  K ) `
 ( ( _|_P `  K ) `
 S ) ) )
108, 9syldan 468 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  S
) ) ) )  =  ( ( _|_P `  K ) `
 ( ( _|_P `  K ) `
 S ) ) )
117, 10eqtr3d 2443 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( ( _|_P `  K
) `  ( ( _|_P `  K ) `
 S ) ) )
12 hlclat 32340 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
13 eqid 2400 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1413, 2atssbase 32272 . . . . . . 7  |-  A  C_  ( Base `  K )
15 sstr 3447 . . . . . . 7  |-  ( ( S  C_  A  /\  A  C_  ( Base `  K
) )  ->  S  C_  ( Base `  K
) )
1614, 15mpan2 669 . . . . . 6  |-  ( S 
C_  A  ->  S  C_  ( Base `  K
) )
1713, 1clatlubcl 15956 . . . . . 6  |-  ( ( K  e.  CLat  /\  S  C_  ( Base `  K
) )  ->  ( U `  S )  e.  ( Base `  K
) )
1812, 16, 17syl2an 475 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( U `  S
)  e.  ( Base `  K ) )
1913, 2, 3pmapssat 32740 . . . . 5  |-  ( ( K  e.  HL  /\  ( U `  S )  e.  ( Base `  K
) )  ->  ( M `  ( U `  S ) )  C_  A )
2018, 19syldan 468 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( M `  ( U `  S )
)  C_  A )
211, 2, 3, 42polvalN 32895 . . . 4  |-  ( ( K  e.  HL  /\  ( M `  ( U `
 S ) ) 
C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( M `
 ( U `  ( M `  ( U `
 S ) ) ) ) )
2220, 21syldan 468 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( M `
 ( U `  ( M `  ( U `
 S ) ) ) ) )
2311, 22eqtr3d 2443 . 2  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  S
) )  =  ( M `  ( U `
 ( M `  ( U `  S ) ) ) ) )
2423, 5eqtr3d 2443 1  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( M `  ( U `  ( M `  ( U `  S
) ) ) )  =  ( M `  ( U `  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    C_ wss 3411   ` cfv 5523   Basecbs 14731   lubclub 15785   CLatccla 15951   Atomscatm 32245   HLchlt 32332   pmapcpmap 32478   _|_PcpolN 32883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-riotaBAD 31941
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-undef 6957  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-p1 15884  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-psubsp 32484  df-pmap 32485  df-polarityN 32884
This theorem is referenced by:  paddunN  32908
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