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Theorem poldmj1N 30117
Description: DeMorgan's law for polarity of projective sum. (oldmj1 29411 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 30116 . 2  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  .+  T ) )  =  (  ._|_  `  ( S  u.  T ) ) )
5 simp1 955 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  HL )
6 unss 3349 . . . . 5  |-  ( ( S  C_  A  /\  T  C_  A )  <->  ( S  u.  T )  C_  A
)
76biimpi 186 . . . 4  |-  ( ( S  C_  A  /\  T  C_  A )  -> 
( S  u.  T
)  C_  A )
873adant1 973 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  ( S  u.  T )  C_  A )
9 eqid 2283 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
10 eqid 2283 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
11 eqid 2283 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
129, 10, 1, 11, 3polval2N 30095 . . 3  |-  ( ( K  e.  HL  /\  ( S  u.  T
)  C_  A )  ->  (  ._|_  `  ( S  u.  T ) )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  ( S  u.  T
) ) ) ) )
135, 8, 12syl2anc 642 . 2  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  u.  T ) )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  ( S  u.  T ) ) ) ) )
14 hlop 29552 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
15143ad2ant1 976 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  OP )
16 hlclat 29548 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CLat )
17163ad2ant1 976 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  CLat )
18 simp2 956 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  S  C_  A )
19 eqid 2283 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 1atssbase 29480 . . . . . . 7  |-  A  C_  ( Base `  K )
2118, 20syl6ss 3191 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  S  C_  ( Base `  K
) )
2219, 9clatlubcl 14217 . . . . . 6  |-  ( ( K  e.  CLat  /\  S  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  S )  e.  ( Base `  K
) )
2317, 21, 22syl2anc 642 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  S )  e.  ( Base `  K
) )
2419, 10opoccl 29384 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  S )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  S ) )  e.  ( Base `  K
) )
2515, 23, 24syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  S ) )  e.  ( Base `  K
) )
26 simp3 957 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  T  C_  A )
2726, 20syl6ss 3191 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  T  C_  ( Base `  K
) )
2819, 9clatlubcl 14217 . . . . . 6  |-  ( ( K  e.  CLat  /\  T  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  T )  e.  ( Base `  K
) )
2917, 27, 28syl2anc 642 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  T )  e.  ( Base `  K
) )
3019, 10opoccl 29384 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  T )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  T ) )  e.  ( Base `  K
) )
3115, 29, 30syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  T ) )  e.  ( Base `  K
) )
32 eqid 2283 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
3319, 32, 1, 11pmapmeet 29962 . . . 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 1182 . . 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 2283 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
3619, 35, 9lubun 14227 . . . . . . 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 1182 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  ( S  u.  T ) )  =  ( ( ( lub `  K ) `  S
) ( join `  K
) ( ( lub `  K ) `  T
) ) )
3837fveq2d 5529 . . . . 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 29551 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
40393ad2ant1 976 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  OL )
4119, 35, 32, 10oldmj1 29411 . . . . . 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 1182 . . . . 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 2315 . . . 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 5529 . . 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 30095 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
(  ._|_  `  S )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  S ) ) ) )
46453adant3 975 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  S )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  S )
) ) )
479, 10, 1, 11, 3polval2N 30095 . . . . 5  |-  ( ( K  e.  HL  /\  T  C_  A )  -> 
(  ._|_  `  T )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  T ) ) ) )
48473adant2 974 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  T )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  T )
) ) )
4946, 48ineq12d 3371 . . 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 2325 . 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 2319 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    u. cun 3150    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   occoc 13216   lubclub 14076   joincjn 14078   meetcmee 14079   CLatccla 14213   OPcops 29362   OLcol 29364   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   + Pcpadd 29984   _|_ PcpolN 30091
This theorem is referenced by:  pmapj2N  30118  osumcllem3N  30147  pexmidN  30158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-polarityN 30092
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