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Theorem pnonsingN 29289
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 29271 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  ( P `  ( P `  X ) ) )
4 ssrin 3369 . . . 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 2258 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
7 eqid 2258 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
86, 1, 7, 22polvalN 29270 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  ( P `  X )
)  =  ( (
pmap `  K ) `  ( ( lub `  K
) `  X )
) )
9 eqid 2258 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
106, 9, 1, 7, 2polval2N 29262 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( (
pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )
118, 10ineq12d 3346 . . . 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 28719 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1312adantr 453 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
14 hlclat 28715 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CLat )
15 eqid 2258 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
1615, 1atssbase 28647 . . . . . . . . 9  |-  A  C_  ( Base `  K )
17 sstr 3162 . . . . . . . . 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 14179 . . . . . . . 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 2258 . . . . . . . 8  |-  ( meet `  K )  =  (
meet `  K )
22 eqid 2258 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
2315, 9, 21, 22opnoncon 28565 . . . . . . 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 5462 . . . . 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 28551 . . . . . . 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 29129 . . . . . 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 28717 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
3231adantr 453 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  AtLat )
3322, 7pmap0 29121 . . . . . 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 2298 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( ( pmap `  K ) `  (
( lub `  K
) `  X )
)  i^i  ( ( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  X )
) ) )  =  (/) )
3611, 35eqtrd 2290 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( P `  ( P `  X ) )  i^i  ( P `
 X ) )  =  (/) )
375, 36sseqtrd 3189 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  i^i  ( P `  X )
)  C_  (/) )
38 ss0b 3459 . 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 1619    e. wcel 1621    i^i cin 3126    C_ wss 3127   (/)c0 3430   ` cfv 4673  (class class class)co 5792   Basecbs 13110   occoc 13178   lubclub 14038   meetcmee 14041   0.cp0 14105   CLatccla 14175   OPcops 28529   Atomscatm 28620   AtLatcal 28621   HLchlt 28707   pmapcpmap 28853   _|_
PcpolN 29258
This theorem is referenced by:  osumcllem4N  29315  pexmidN  29325  pexmidlem1N  29326
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-pmap 28860  df-polarityN 29259
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