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Theorem 2pmaplubN 33928
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 2454 . . . . . . 7  |-  ( _|_P `  K )  =  ( _|_P `  K )
51, 2, 3, 42polvalN 33916 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  S
) )  =  ( M `  ( U `
 S ) ) )
65fveq2d 5806 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  S
) ) )  =  ( ( _|_P `  K ) `  ( M `  ( U `  S ) ) ) )
76fveq2d 5806 . . . 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 33910 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  S
)  C_  A )
92, 43polN 33918 . . . . 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 470 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  (
( _|_P `  K ) `  S
) ) ) )  =  ( ( _|_P `  K ) `
 ( ( _|_P `  K ) `
 S ) ) )
117, 10eqtr3d 2497 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( ( _|_P `  K
) `  ( ( _|_P `  K ) `
 S ) ) )
12 hlclat 33361 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
13 eqid 2454 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1413, 2atssbase 33293 . . . . . . 7  |-  A  C_  ( Base `  K )
15 sstr 3475 . . . . . . 7  |-  ( ( S  C_  A  /\  A  C_  ( Base `  K
) )  ->  S  C_  ( Base `  K
) )
1614, 15mpan2 671 . . . . . 6  |-  ( S 
C_  A  ->  S  C_  ( Base `  K
) )
1713, 1clatlubcl 15404 . . . . . 6  |-  ( ( K  e.  CLat  /\  S  C_  ( Base `  K
) )  ->  ( U `  S )  e.  ( Base `  K
) )
1812, 16, 17syl2an 477 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( U `  S
)  e.  ( Base `  K ) )
1913, 2, 3pmapssat 33761 . . . . 5  |-  ( ( K  e.  HL  /\  ( U `  S )  e.  ( Base `  K
) )  ->  ( M `  ( U `  S ) )  C_  A )
2018, 19syldan 470 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( M `  ( U `  S )
)  C_  A )
211, 2, 3, 42polvalN 33916 . . . 4  |-  ( ( K  e.  HL  /\  ( M `  ( U `
 S ) ) 
C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( M `
 ( U `  ( M `  ( U `
 S ) ) ) ) )
2220, 21syldan 470 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  ( M `  ( U `  S ) ) ) )  =  ( M `
 ( U `  ( M `  ( U `
 S ) ) ) ) )
2311, 22eqtr3d 2497 . 2  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  S
) )  =  ( M `  ( U `
 ( M `  ( U `  S ) ) ) ) )
2423, 5eqtr3d 2497 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 369    = wceq 1370    e. wcel 1758    C_ wss 3439   ` cfv 5529   Basecbs 14295   lubclub 15234   CLatccla 15399   Atomscatm 33266   HLchlt 33353   pmapcpmap 33499   _|_PcpolN 33904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-riotaBAD 32962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-undef 6905  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-psubsp 33505  df-pmap 33506  df-polarityN 33905
This theorem is referenced by:  paddunN  33929
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