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Theorem pnonsingN 29389
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 29371 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  ( P `  ( P `  X ) ) )
4 ssrin 3395 . . . 4  |-  ( X 
C_  ( P `  ( P `  X ) )  ->  ( X  i^i  ( P `  X
) )  C_  (
( P `  ( P `  X )
)  i^i  ( P `  X ) ) )
53, 4syl 17 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  C_  ( ( P `  ( P `  X ) )  i^i  ( P `  X
) ) )
6 eqid 2284 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
7 eqid 2284 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
86, 1, 7, 22polvalN 29370 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  ( P `  X )
)  =  ( (
pmap `  K ) `  ( ( lub `  K
) `  X )
) )
9 eqid 2284 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
106, 9, 1, 7, 2polval2N 29362 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( (
pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )
118, 10ineq12d 3372 . . . 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 28819 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1312adantr 453 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
14 hlclat 28815 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CLat )
15 eqid 2284 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
1615, 1atssbase 28747 . . . . . . . . 9  |-  A  C_  ( Base `  K )
17 sstr 3188 . . . . . . . . 9  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1816, 17mpan2 655 . . . . . . . 8  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1915, 6clatlubcl 14211 . . . . . . . 8  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
2014, 18, 19syl2an 465 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
21 eqid 2284 . . . . . . . 8  |-  ( meet `  K )  =  (
meet `  K )
22 eqid 2284 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
2315, 9, 21, 22opnoncon 28665 . . . . . . 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 645 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( ( lub `  K ) `  X
) ( meet `  K
) ( ( oc
`  K ) `  ( ( lub `  K
) `  X )
) )  =  ( 0. `  K ) )
2524fveq2d 5489 . . . . 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 445 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  HL )
2715, 9opoccl 28651 . . . . . . 7  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  X ) )  e.  ( Base `  K
) )
2813, 20, 27syl2anc 645 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  (
( lub `  K
) `  X )
)  e.  ( Base `  K ) )
2915, 21, 1, 7pmapmeet 29229 . . . . . 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 1187 . . . . 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 28817 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
3231adantr 453 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  AtLat )
3322, 7pmap0 29221 . . . . . 6  |-  ( K  e.  AtLat  ->  ( ( pmap `  K ) `  ( 0. `  K ) )  =  (/) )
3432, 33syl 17 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( 0. `  K ) )  =  (/) )
3525, 30, 343eqtr3d 2324 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )  =  (/) )
3611, 35eqtrd 2316 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( P `  ( P `  X ) )  i^i  ( P `
 X ) )  =  (/) )
375, 36sseqtrd 3215 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  C_  (/) )
38 ss0b 3485 . 2  |-  ( ( X  i^i  ( P `
 X ) ) 
C_  (/)  <->  ( X  i^i  ( P `  X ) )  =  (/) )
3937, 38sylib 190 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1628    e. wcel 1688    i^i cin 3152    C_ wss 3153   (/)c0 3456   ` cfv 5221  (class class class)co 5819   Basecbs 13142   occoc 13210   lubclub 14070   meetcmee 14073   0.cp0 14137   CLatccla 14207   OPcops 28629   Atomscatm 28720   AtLatcal 28721   HLchlt 28807   pmapcpmap 28953   _|_
PcpolN 29358
This theorem is referenced by:  osumcllem4N  29415  pexmidN  29425  pexmidlem1N  29426
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-pmap 28960  df-polarityN 29359
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