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Theorem poldmj1N 30787
Description: De Morgan's law for polarity of projective sum. (oldmj1 30081 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 30786 . 2  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  .+  T ) )  =  (  ._|_  `  ( S  u.  T ) ) )
5 simp1 958 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  HL )
6 unss 3523 . . . . 5  |-  ( ( S  C_  A  /\  T  C_  A )  <->  ( S  u.  T )  C_  A
)
76biimpi 188 . . . 4  |-  ( ( S  C_  A  /\  T  C_  A )  -> 
( S  u.  T
)  C_  A )
873adant1 976 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  ( S  u.  T )  C_  A )
9 eqid 2438 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
10 eqid 2438 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
11 eqid 2438 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
129, 10, 1, 11, 3polval2N 30765 . . 3  |-  ( ( K  e.  HL  /\  ( S  u.  T
)  C_  A )  ->  (  ._|_  `  ( S  u.  T ) )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  ( S  u.  T
) ) ) ) )
135, 8, 12syl2anc 644 . 2  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  u.  T ) )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  ( S  u.  T ) ) ) ) )
14 hlop 30222 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
15143ad2ant1 979 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  OP )
16 hlclat 30218 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CLat )
17163ad2ant1 979 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  CLat )
18 simp2 959 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  S  C_  A )
19 eqid 2438 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 1atssbase 30150 . . . . . . 7  |-  A  C_  ( Base `  K )
2118, 20syl6ss 3362 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  S  C_  ( Base `  K
) )
2219, 9clatlubcl 14542 . . . . . 6  |-  ( ( K  e.  CLat  /\  S  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  S )  e.  ( Base `  K
) )
2317, 21, 22syl2anc 644 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  S )  e.  ( Base `  K
) )
2419, 10opoccl 30054 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  S )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  S ) )  e.  ( Base `  K
) )
2515, 23, 24syl2anc 644 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  S ) )  e.  ( Base `  K
) )
26 simp3 960 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  T  C_  A )
2726, 20syl6ss 3362 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  T  C_  ( Base `  K
) )
2819, 9clatlubcl 14542 . . . . . 6  |-  ( ( K  e.  CLat  /\  T  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  T )  e.  ( Base `  K
) )
2917, 27, 28syl2anc 644 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  T )  e.  ( Base `  K
) )
3019, 10opoccl 30054 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  T )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  T ) )  e.  ( Base `  K
) )
3115, 29, 30syl2anc 644 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  T ) )  e.  ( Base `  K
) )
32 eqid 2438 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
3319, 32, 1, 11pmapmeet 30632 . . . 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 1185 . . 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 2438 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
3619, 35, 9lubun 14552 . . . . . . 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 1185 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  ( S  u.  T ) )  =  ( ( ( lub `  K ) `  S
) ( join `  K
) ( ( lub `  K ) `  T
) ) )
3837fveq2d 5734 . . . . 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 30221 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
40393ad2ant1 979 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  OL )
4119, 35, 32, 10oldmj1 30081 . . . . . 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 1185 . . . . 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 2470 . . . 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 5734 . . 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 30765 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
(  ._|_  `  S )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  S ) ) ) )
46453adant3 978 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  S )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  S )
) ) )
479, 10, 1, 11, 3polval2N 30765 . . . . 5  |-  ( ( K  e.  HL  /\  T  C_  A )  -> 
(  ._|_  `  T )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  T ) ) ) )
48473adant2 977 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  T )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  T )
) ) )
4946, 48ineq12d 3545 . . 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 2480 . 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 2474 1  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  .+  T ) )  =  ( (  ._|_  `  S
)  i^i  (  ._|_  `  T ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    u. cun 3320    i^i cin 3321    C_ wss 3322   ` cfv 5456  (class class class)co 6083   Basecbs 13471   occoc 13539   lubclub 14401   joincjn 14403   meetcmee 14404   CLatccla 14538   OPcops 30032   OLcol 30034   Atomscatm 30123   HLchlt 30210   pmapcpmap 30356   + Pcpadd 30654   _|_ PcpolN 30761
This theorem is referenced by:  pmapj2N  30788  osumcllem3N  30817  pexmidN  30828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-polarityN 30762
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