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Theorem poldmj1N 35795
Description: De Morgan's law for polarity of projective sum. (oldmj1 35089 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
paddun.a  |-  A  =  ( Atoms `  K )
paddun.p  |-  .+  =  ( +P `  K
)
paddun.o  |-  ._|_  =  ( _|_P `  K
)
Assertion
Ref Expression
poldmj1N  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  .+  T ) )  =  ( (  ._|_  `  S
)  i^i  (  ._|_  `  T ) ) )

Proof of Theorem poldmj1N
StepHypRef Expression
1 paddun.a . . 3  |-  A  =  ( Atoms `  K )
2 paddun.p . . 3  |-  .+  =  ( +P `  K
)
3 paddun.o . . 3  |-  ._|_  =  ( _|_P `  K
)
41, 2, 3paddunN 35794 . 2  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  .+  T ) )  =  (  ._|_  `  ( S  u.  T ) ) )
5 simp1 996 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  HL )
6 unss 3674 . . . . 5  |-  ( ( S  C_  A  /\  T  C_  A )  <->  ( S  u.  T )  C_  A
)
76biimpi 194 . . . 4  |-  ( ( S  C_  A  /\  T  C_  A )  -> 
( S  u.  T
)  C_  A )
873adant1 1014 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  ( S  u.  T )  C_  A )
9 eqid 2457 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
10 eqid 2457 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
11 eqid 2457 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
129, 10, 1, 11, 3polval2N 35773 . . 3  |-  ( ( K  e.  HL  /\  ( S  u.  T
)  C_  A )  ->  (  ._|_  `  ( S  u.  T ) )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  ( S  u.  T
) ) ) ) )
135, 8, 12syl2anc 661 . 2  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  u.  T ) )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  ( S  u.  T ) ) ) ) )
14 hlop 35230 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
15143ad2ant1 1017 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  OP )
16 hlclat 35226 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CLat )
17163ad2ant1 1017 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  CLat )
18 simp2 997 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  S  C_  A )
19 eqid 2457 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 1atssbase 35158 . . . . . . 7  |-  A  C_  ( Base `  K )
2118, 20syl6ss 3511 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  S  C_  ( Base `  K
) )
2219, 9clatlubcl 15869 . . . . . 6  |-  ( ( K  e.  CLat  /\  S  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  S )  e.  ( Base `  K
) )
2317, 21, 22syl2anc 661 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  S )  e.  ( Base `  K
) )
2419, 10opoccl 35062 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  S )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  S ) )  e.  ( Base `  K
) )
2515, 23, 24syl2anc 661 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  S ) )  e.  ( Base `  K
) )
26 simp3 998 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  T  C_  A )
2726, 20syl6ss 3511 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  T  C_  ( Base `  K
) )
2819, 9clatlubcl 15869 . . . . . 6  |-  ( ( K  e.  CLat  /\  T  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  T )  e.  ( Base `  K
) )
2917, 27, 28syl2anc 661 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  T )  e.  ( Base `  K
) )
3019, 10opoccl 35062 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  T )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  T ) )  e.  ( Base `  K
) )
3115, 29, 30syl2anc 661 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  T ) )  e.  ( Base `  K
) )
32 eqid 2457 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
3319, 32, 1, 11pmapmeet 35640 . . . 4  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  (
( lub `  K
) `  S )
)  e.  ( Base `  K )  /\  (
( oc `  K
) `  ( ( lub `  K ) `  T ) )  e.  ( Base `  K
) )  ->  (
( pmap `  K ) `  ( ( ( oc
`  K ) `  ( ( lub `  K
) `  S )
) ( meet `  K
) ( ( oc
`  K ) `  ( ( lub `  K
) `  T )
) ) )  =  ( ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  S ) ) )  i^i  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  T ) ) ) ) )
345, 25, 31, 33syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( pmap `  K ) `  ( ( ( oc
`  K ) `  ( ( lub `  K
) `  S )
) ( meet `  K
) ( ( oc
`  K ) `  ( ( lub `  K
) `  T )
) ) )  =  ( ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  S ) ) )  i^i  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  T ) ) ) ) )
35 eqid 2457 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
3619, 35, 9lubun 15880 . . . . . . 7  |-  ( ( K  e.  CLat  /\  S  C_  ( Base `  K
)  /\  T  C_  ( Base `  K ) )  ->  ( ( lub `  K ) `  ( S  u.  T )
)  =  ( ( ( lub `  K
) `  S )
( join `  K )
( ( lub `  K
) `  T )
) )
3717, 21, 27, 36syl3anc 1228 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  ( S  u.  T ) )  =  ( ( ( lub `  K ) `  S
) ( join `  K
) ( ( lub `  K ) `  T
) ) )
3837fveq2d 5876 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  ( S  u.  T
) ) )  =  ( ( oc `  K ) `  (
( ( lub `  K
) `  S )
( join `  K )
( ( lub `  K
) `  T )
) ) )
39 hlol 35229 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
40393ad2ant1 1017 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  OL )
4119, 35, 32, 10oldmj1 35089 . . . . . 6  |-  ( ( K  e.  OL  /\  ( ( lub `  K
) `  S )  e.  ( Base `  K
)  /\  ( ( lub `  K ) `  T )  e.  (
Base `  K )
)  ->  ( ( oc `  K ) `  ( ( ( lub `  K ) `  S
) ( join `  K
) ( ( lub `  K ) `  T
) ) )  =  ( ( ( oc
`  K ) `  ( ( lub `  K
) `  S )
) ( meet `  K
) ( ( oc
`  K ) `  ( ( lub `  K
) `  T )
) ) )
4240, 23, 29, 41syl3anc 1228 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( oc `  K
) `  ( (
( lub `  K
) `  S )
( join `  K )
( ( lub `  K
) `  T )
) )  =  ( ( ( oc `  K ) `  (
( lub `  K
) `  S )
) ( meet `  K
) ( ( oc
`  K ) `  ( ( lub `  K
) `  T )
) ) )
4338, 42eqtrd 2498 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  ( S  u.  T
) ) )  =  ( ( ( oc
`  K ) `  ( ( lub `  K
) `  S )
) ( meet `  K
) ( ( oc
`  K ) `  ( ( lub `  K
) `  T )
) ) )
4443fveq2d 5876 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  ( S  u.  T ) ) ) )  =  ( (
pmap `  K ) `  ( ( ( oc
`  K ) `  ( ( lub `  K
) `  S )
) ( meet `  K
) ( ( oc
`  K ) `  ( ( lub `  K
) `  T )
) ) ) )
459, 10, 1, 11, 3polval2N 35773 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
(  ._|_  `  S )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  S ) ) ) )
46453adant3 1016 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  S )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  S )
) ) )
479, 10, 1, 11, 3polval2N 35773 . . . . 5  |-  ( ( K  e.  HL  /\  T  C_  A )  -> 
(  ._|_  `  T )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  T ) ) ) )
48473adant2 1015 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  T )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  T )
) ) )
4946, 48ineq12d 3697 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
(  ._|_  `  S )  i^i  (  ._|_  `  T
) )  =  ( ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  S )
) )  i^i  (
( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  T )
) ) ) )
5034, 44, 493eqtr4d 2508 . 2  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  ( S  u.  T ) ) ) )  =  ( ( 
._|_  `  S )  i^i  (  ._|_  `  T ) ) )
514, 13, 503eqtrd 2502 1  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  .+  T ) )  =  ( (  ._|_  `  S
)  i^i  (  ._|_  `  T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    u. cun 3469    i^i cin 3470    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14644   occoc 14720   lubclub 15698   joincjn 15700   meetcmee 15701   CLatccla 15864   OPcops 35040   OLcol 35042   Atomscatm 35131   HLchlt 35218   pmapcpmap 35364   +Pcpadd 35662   _|_PcpolN 35769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-polarityN 35770
This theorem is referenced by:  pmapj2N  35796  osumcllem3N  35825  pexmidN  35836
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