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Theorem pnonsingN 34604
Description: The intersection of a set of atoms and its polarity is empty. Definition of nonsingular in [Holland95] p. 214. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polat.a  |-  A  =  ( Atoms `  K )
2polat.p  |-  P  =  ( _|_P `  K )
Assertion
Ref Expression
pnonsingN  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  =  (/) )

Proof of Theorem pnonsingN
StepHypRef Expression
1 2polat.a . . . . 5  |-  A  =  ( Atoms `  K )
2 2polat.p . . . . 5  |-  P  =  ( _|_P `  K )
31, 22polssN 34586 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  ( P `  ( P `  X ) ) )
4 ssrin 3716 . . . 4  |-  ( X 
C_  ( P `  ( P `  X ) )  ->  ( X  i^i  ( P `  X
) )  C_  (
( P `  ( P `  X )
)  i^i  ( P `  X ) ) )
53, 4syl 16 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  C_  ( ( P `  ( P `  X ) )  i^i  ( P `  X
) ) )
6 eqid 2460 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
7 eqid 2460 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
86, 1, 7, 22polvalN 34585 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  ( P `  X )
)  =  ( (
pmap `  K ) `  ( ( lub `  K
) `  X )
) )
9 eqid 2460 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
106, 9, 1, 7, 2polval2N 34577 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( (
pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )
118, 10ineq12d 3694 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( P `  ( P `  X ) )  i^i  ( P `
 X ) )  =  ( ( (
pmap `  K ) `  ( ( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) ) )
12 hlop 34034 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1312adantr 465 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
14 hlclat 34030 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CLat )
15 eqid 2460 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
1615, 1atssbase 33962 . . . . . . . . 9  |-  A  C_  ( Base `  K )
17 sstr 3505 . . . . . . . . 9  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1816, 17mpan2 671 . . . . . . . 8  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1915, 6clatlubcl 15588 . . . . . . . 8  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
2014, 18, 19syl2an 477 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
21 eqid 2460 . . . . . . . 8  |-  ( meet `  K )  =  (
meet `  K )
22 eqid 2460 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
2315, 9, 21, 22opnoncon 33880 . . . . . . 7  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )  ->  (
( ( lub `  K
) `  X )
( meet `  K )
( ( oc `  K ) `  (
( lub `  K
) `  X )
) )  =  ( 0. `  K ) )
2413, 20, 23syl2anc 661 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( ( lub `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  ( ( lub `  K
) `  X )
) )  =  ( 0. `  K ) )
2524fveq2d 5861 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( meet `  K )
( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )  =  ( ( pmap `  K
) `  ( 0. `  K ) ) )
26 simpl 457 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  HL )
2715, 9opoccl 33866 . . . . . . 7  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  X ) )  e.  ( Base `  K
) )
2813, 20, 27syl2anc 661 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  (
( lub `  K
) `  X )
)  e.  ( Base `  K ) )
2915, 21, 1, 7pmapmeet 34444 . . . . . 6  |-  ( ( K  e.  HL  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
)  /\  ( ( oc `  K ) `  ( ( lub `  K
) `  X )
)  e.  ( Base `  K ) )  -> 
( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( meet `  K )
( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )  =  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) ) )
3026, 20, 28, 29syl3anc 1223 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( (
( lub `  K
) `  X )
( meet `  K )
( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )  =  ( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) ) )
31 hlatl 34032 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
3231adantr 465 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  AtLat )
3322, 7pmap0 34436 . . . . . 6  |-  ( K  e.  AtLat  ->  ( ( pmap `  K ) `  ( 0. `  K ) )  =  (/) )
3432, 33syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( 0. `  K ) )  =  (/) )
3525, 30, 343eqtr3d 2509 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )  =  (/) )
3611, 35eqtrd 2501 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( P `  ( P `  X ) )  i^i  ( P `
 X ) )  =  (/) )
375, 36sseqtrd 3533 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  C_  (/) )
38 ss0b 3808 . 2  |-  ( ( X  i^i  ( P `
 X ) ) 
C_  (/)  <->  ( X  i^i  ( P `  X ) )  =  (/) )
3937, 38sylib 196 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    i^i cin 3468    C_ wss 3469   (/)c0 3778   ` cfv 5579  (class class class)co 6275   Basecbs 14479   occoc 14552   lubclub 15418   meetcmee 15421   0.cp0 15513   CLatccla 15583   OPcops 33844   Atomscatm 33935   AtLatcal 33936   HLchlt 34022   pmapcpmap 34168   _|_PcpolN 34573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-undef 6992  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-pmap 34175  df-polarityN 34574
This theorem is referenced by:  osumcllem4N  34630  pexmidN  34640  pexmidlem1N  34641
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