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Theorem 2pmaplubN 30650
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 2435 . . . . . . 7  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
51, 2, 3, 42polvalN 30638 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) )  =  ( M `  ( U `
 S ) ) )
65fveq2d 5724 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) ) )  =  ( ( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )
76fveq2d 5724 . . . 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 30632 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  S
)  C_  A )
92, 43polN 30640 . . . . 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 2469 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( ( _|_ P `  K
) `  ( ( _|_ P `  K ) `
 S ) ) )
12 hlclat 30083 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
13 eqid 2435 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1413, 2atssbase 30015 . . . . . . 7  |-  A  C_  ( Base `  K )
15 sstr 3348 . . . . . . 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 14532 . . . . . 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 30483 . . . . 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 30638 . . . 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 2469 . 2  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) )  =  ( M `  ( U `
 ( M `  ( U `  S ) ) ) ) )
2423, 5eqtr3d 2469 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 1652    e. wcel 1725    C_ wss 3312   ` cfv 5446   Basecbs 13461   lubclub 14391   CLatccla 14528   Atomscatm 29988   HLchlt 30075   pmapcpmap 30221   _|_ PcpolN 30626
This theorem is referenced by:  paddunN  30651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-psubsp 30227  df-pmap 30228  df-polarityN 30627
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