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Theorem poldmj1N 33878
Description: De Morgan's law for polarity of projective sum. (oldmj1 33172 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 33877 . 2  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  ( S  .+  T ) )  =  (  ._|_  `  ( S  u.  T ) ) )
5 simp1 988 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  HL )
6 unss 3628 . . . . 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 1006 . . 3  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  ( S  u.  T )  C_  A )
9 eqid 2451 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
10 eqid 2451 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
11 eqid 2451 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
129, 10, 1, 11, 3polval2N 33856 . . 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 33313 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
15143ad2ant1 1009 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  OP )
16 hlclat 33309 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CLat )
17163ad2ant1 1009 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  CLat )
18 simp2 989 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  S  C_  A )
19 eqid 2451 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 1atssbase 33241 . . . . . . 7  |-  A  C_  ( Base `  K )
2118, 20syl6ss 3466 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  S  C_  ( Base `  K
) )
2219, 9clatlubcl 15384 . . . . . 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 33145 . . . . 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 990 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  T  C_  A )
2726, 20syl6ss 3466 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  T  C_  ( Base `  K
) )
2819, 9clatlubcl 15384 . . . . . 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 33145 . . . . 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 2451 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
3319, 32, 1, 11pmapmeet 33723 . . . 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 1219 . . 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 2451 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
3619, 35, 9lubun 15395 . . . . . . 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 1219 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (
( lub `  K
) `  ( S  u.  T ) )  =  ( ( ( lub `  K ) `  S
) ( join `  K
) ( ( lub `  K ) `  T
) ) )
3837fveq2d 5793 . . . . 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 33312 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
40393ad2ant1 1009 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  OL )
4119, 35, 32, 10oldmj1 33172 . . . . . 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 1219 . . . . 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 2492 . . . 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 5793 . . 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 33856 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A )  -> 
(  ._|_  `  S )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  S ) ) ) )
46453adant3 1008 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  S )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  S )
) ) )
479, 10, 1, 11, 3polval2N 33856 . . . . 5  |-  ( ( K  e.  HL  /\  T  C_  A )  -> 
(  ._|_  `  T )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  T ) ) ) )
48473adant2 1007 . . . 4  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  (  ._|_  `  T )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  T )
) ) )
4946, 48ineq12d 3651 . . 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 2502 . 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 2496 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 965    = wceq 1370    e. wcel 1758    u. cun 3424    i^i cin 3425    C_ wss 3426   ` cfv 5516  (class class class)co 6190   Basecbs 14276   occoc 14348   lubclub 15214   joincjn 15216   meetcmee 15217   CLatccla 15379   OPcops 33123   OLcol 33125   Atomscatm 33214   HLchlt 33301   pmapcpmap 33447   +Pcpadd 33745   _|_PcpolN 33852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-riotaBAD 32910
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-undef 6892  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-p1 15312  df-lat 15318  df-clat 15380  df-oposet 33127  df-ol 33129  df-oml 33130  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302  df-psubsp 33453  df-pmap 33454  df-padd 33746  df-polarityN 33853
This theorem is referenced by:  pmapj2N  33879  osumcllem3N  33908  pexmidN  33919
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