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Theorem pnonsingN 35397
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 35379 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  ( P `  ( P `  X ) ) )
4 ssrin 3708 . . . 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 2443 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
7 eqid 2443 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
86, 1, 7, 22polvalN 35378 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  ( P `  X )
)  =  ( (
pmap `  K ) `  ( ( lub `  K
) `  X )
) )
9 eqid 2443 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
106, 9, 1, 7, 2polval2N 35370 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( (
pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )
118, 10ineq12d 3686 . . . 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 34827 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1312adantr 465 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
14 hlclat 34823 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CLat )
15 eqid 2443 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
1615, 1atssbase 34755 . . . . . . . . 9  |-  A  C_  ( Base `  K )
17 sstr 3497 . . . . . . . . 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 15720 . . . . . . . 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 2443 . . . . . . . 8  |-  ( meet `  K )  =  (
meet `  K )
22 eqid 2443 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
2315, 9, 21, 22opnoncon 34673 . . . . . . 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 5860 . . . . 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 34659 . . . . . . 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 35237 . . . . . 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 1229 . . . . 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 34825 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
3231adantr 465 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  AtLat )
3322, 7pmap0 35229 . . . . . 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 2492 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )  =  (/) )
3611, 35eqtrd 2484 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( P `  ( P `  X ) )  i^i  ( P `
 X ) )  =  (/) )
375, 36sseqtrd 3525 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  C_  (/) )
38 ss0b 3801 . 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 1383    e. wcel 1804    i^i cin 3460    C_ wss 3461   (/)c0 3770   ` cfv 5578  (class class class)co 6281   Basecbs 14613   occoc 14686   lubclub 15549   meetcmee 15552   0.cp0 15645   CLatccla 15715   OPcops 34637   Atomscatm 34728   AtLatcal 34729   HLchlt 34815   pmapcpmap 34961   _|_PcpolN 35366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-riotaBAD 34424
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-undef 7004  df-preset 15535  df-poset 15553  df-plt 15566  df-lub 15582  df-glb 15583  df-join 15584  df-meet 15585  df-p0 15647  df-p1 15648  df-lat 15654  df-clat 15716  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-pmap 34968  df-polarityN 35367
This theorem is referenced by:  osumcllem4N  35423  pexmidN  35433  pexmidlem1N  35434
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