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Theorem 2pmaplubN 30737
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 2296 . . . . . . 7  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
51, 2, 3, 42polvalN 30725 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) )  =  ( M `  ( U `
 S ) ) )
65fveq2d 5545 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) ) )  =  ( ( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )
76fveq2d 5545 . . . 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 30719 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  S
)  C_  A )
92, 43polN 30727 . . . . 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 456 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) ) ) )  =  ( ( _|_
P `  K ) `  ( ( _|_ P `  K ) `  S
) ) )
117, 10eqtr3d 2330 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( ( _|_ P `  K
) `  ( ( _|_ P `  K ) `
 S ) ) )
12 hlclat 30170 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
13 eqid 2296 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1413, 2atssbase 30102 . . . . . . 7  |-  A  C_  ( Base `  K )
15 sstr 3200 . . . . . . 7  |-  ( ( S  C_  A  /\  A  C_  ( Base `  K
) )  ->  S  C_  ( Base `  K
) )
1614, 15mpan2 652 . . . . . 6  |-  ( S 
C_  A  ->  S  C_  ( Base `  K
) )
1713, 1clatlubcl 14233 . . . . . 6  |-  ( ( K  e.  CLat  /\  S  C_  ( Base `  K
) )  ->  ( U `  S )  e.  ( Base `  K
) )
1812, 16, 17syl2an 463 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( U `  S
)  e.  ( Base `  K ) )
1913, 2, 3pmapssat 30570 . . . . 5  |-  ( ( K  e.  HL  /\  ( U `  S )  e.  ( Base `  K
) )  ->  ( M `  ( U `  S ) )  C_  A )
2018, 19syldan 456 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( M `  ( U `  S )
)  C_  A )
211, 2, 3, 42polvalN 30725 . . . 4  |-  ( ( K  e.  HL  /\  ( M `  ( U `
 S ) ) 
C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( M `
 ( U `  ( M `  ( U `
 S ) ) ) ) )
2220, 21syldan 456 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( M `
 ( U `  ( M `  ( U `
 S ) ) ) ) )
2311, 22eqtr3d 2330 . 2  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  S
) )  =  ( M `  ( U `
 ( M `  ( U `  S ) ) ) ) )
2423, 5eqtr3d 2330 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 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271   Basecbs 13164   lubclub 14092   CLatccla 14229   Atomscatm 30075   HLchlt 30162   pmapcpmap 30308   _|_ PcpolN 30713
This theorem is referenced by:  paddunN  30738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-polarityN 30714
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