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Theorem 2pmaplubN 30042
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 2389 . . . . . . 7  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
51, 2, 3, 42polvalN 30030 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) )  =  ( M `  ( U `
 S ) ) )
65fveq2d 5674 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) ) )  =  ( ( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )
76fveq2d 5674 . . . 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 30024 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  S
)  C_  A )
92, 43polN 30032 . . . . 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 457 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) ) ) )  =  ( ( _|_
P `  K ) `  ( ( _|_ P `  K ) `  S
) ) )
117, 10eqtr3d 2423 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( ( _|_ P `  K
) `  ( ( _|_ P `  K ) `
 S ) ) )
12 hlclat 29475 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
13 eqid 2389 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1413, 2atssbase 29407 . . . . . . 7  |-  A  C_  ( Base `  K )
15 sstr 3301 . . . . . . 7  |-  ( ( S  C_  A  /\  A  C_  ( Base `  K
) )  ->  S  C_  ( Base `  K
) )
1614, 15mpan2 653 . . . . . 6  |-  ( S 
C_  A  ->  S  C_  ( Base `  K
) )
1713, 1clatlubcl 14469 . . . . . 6  |-  ( ( K  e.  CLat  /\  S  C_  ( Base `  K
) )  ->  ( U `  S )  e.  ( Base `  K
) )
1812, 16, 17syl2an 464 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( U `  S
)  e.  ( Base `  K ) )
1913, 2, 3pmapssat 29875 . . . . 5  |-  ( ( K  e.  HL  /\  ( U `  S )  e.  ( Base `  K
) )  ->  ( M `  ( U `  S ) )  C_  A )
2018, 19syldan 457 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( M `  ( U `  S )
)  C_  A )
211, 2, 3, 42polvalN 30030 . . . 4  |-  ( ( K  e.  HL  /\  ( M `  ( U `
 S ) ) 
C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( M `
 ( U `  ( M `  ( U `
 S ) ) ) ) )
2220, 21syldan 457 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( M `
 ( U `  ( M `  ( U `
 S ) ) ) ) )
2311, 22eqtr3d 2423 . 2  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) )  =  ( M `  ( U `
 ( M `  ( U `  S ) ) ) ) )
2423, 5eqtr3d 2423 1  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( M `  ( U `  ( M `  ( U `  S
) ) ) )  =  ( M `  ( U `  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265   ` cfv 5396   Basecbs 13398   lubclub 14328   CLatccla 14465   Atomscatm 29380   HLchlt 29467   pmapcpmap 29613   _|_ PcpolN 30018
This theorem is referenced by:  paddunN  30043
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-psubsp 29619  df-pmap 29620  df-polarityN 30019
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