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Theorem pnonsingN 30744
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 30726 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  ( P `  ( P `  X ) ) )
4 ssrin 3407 . . . 4  |-  ( X 
C_  ( P `  ( P `  X ) )  ->  ( X  i^i  ( P `  X
) )  C_  (
( P `  ( P `  X )
)  i^i  ( P `  X ) ) )
53, 4syl 15 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  C_  ( ( P `  ( P `  X ) )  i^i  ( P `  X
) ) )
6 eqid 2296 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
7 eqid 2296 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
86, 1, 7, 22polvalN 30725 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  ( P `  X )
)  =  ( (
pmap `  K ) `  ( ( lub `  K
) `  X )
) )
9 eqid 2296 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
106, 9, 1, 7, 2polval2N 30717 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( (
pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )
118, 10ineq12d 3384 . . . 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 30174 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1312adantr 451 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
14 hlclat 30170 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CLat )
15 eqid 2296 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
1615, 1atssbase 30102 . . . . . . . . 9  |-  A  C_  ( Base `  K )
17 sstr 3200 . . . . . . . . 9  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1816, 17mpan2 652 . . . . . . . 8  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1915, 6clatlubcl 14233 . . . . . . . 8  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
2014, 18, 19syl2an 463 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
21 eqid 2296 . . . . . . . 8  |-  ( meet `  K )  =  (
meet `  K )
22 eqid 2296 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
2315, 9, 21, 22opnoncon 30020 . . . . . . 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 642 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( ( lub `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  ( ( lub `  K
) `  X )
) )  =  ( 0. `  K ) )
2524fveq2d 5545 . . . . 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 443 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  HL )
2715, 9opoccl 30006 . . . . . . 7  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  X ) )  e.  ( Base `  K
) )
2813, 20, 27syl2anc 642 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  (
( lub `  K
) `  X )
)  e.  ( Base `  K ) )
2915, 21, 1, 7pmapmeet 30584 . . . . . 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 1182 . . . . 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 30172 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
3231adantr 451 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  AtLat )
3322, 7pmap0 30576 . . . . . 6  |-  ( K  e.  AtLat  ->  ( ( pmap `  K ) `  ( 0. `  K ) )  =  (/) )
3432, 33syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( 0. `  K ) )  =  (/) )
3525, 30, 343eqtr3d 2336 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )  =  (/) )
3611, 35eqtrd 2328 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( P `  ( P `  X ) )  i^i  ( P `
 X ) )  =  (/) )
375, 36sseqtrd 3227 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  C_  (/) )
38 ss0b 3497 . 2  |-  ( ( X  i^i  ( P `
 X ) ) 
C_  (/)  <->  ( X  i^i  ( P `  X ) )  =  (/) )
3937, 38sylib 188 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   (/)c0 3468   ` cfv 5271  (class class class)co 5874   Basecbs 13164   occoc 13232   lubclub 14092   meetcmee 14095   0.cp0 14159   CLatccla 14229   OPcops 29984   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   pmapcpmap 30308   _|_
PcpolN 30713
This theorem is referenced by:  osumcllem4N  30770  pexmidN  30780  pexmidlem1N  30781
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-pmap 30315  df-polarityN 30714
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