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Theorem poldmj1N 30739
Description: De Morgan's law for polarity of projective sum. (oldmj1 30033 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 30738 . 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 3362 . . . . 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 2296 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
10 eqid 2296 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
11 eqid 2296 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
129, 10, 1, 11, 3polval2N 30717 . . 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 30174 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
15143ad2ant1 976 . . . . 5  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  K  e.  OP )
16 hlclat 30170 . . . . . . 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 2296 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 1atssbase 30102 . . . . . . 7  |-  A  C_  ( Base `  K )
2118, 20syl6ss 3204 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  S  C_  ( Base `  K
) )
2219, 9clatlubcl 14233 . . . . . 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 30006 . . . . 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 3204 . . . . . 6  |-  ( ( K  e.  HL  /\  S  C_  A  /\  T  C_  A )  ->  T  C_  ( Base `  K
) )
2819, 9clatlubcl 14233 . . . . . 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 30006 . . . . 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 2296 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
3319, 32, 1, 11pmapmeet 30584 . . . 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 2296 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
3619, 35, 9lubun 14243 . . . . . . 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 5545 . . . . 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 30173 . . . . . . 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 30033 . . . . . 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 2328 . . . 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 5545 . . 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 30717 . . . . 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 30717 . . . . 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 3384 . . 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 2338 . 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 2332 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 1632    e. wcel 1696    u. cun 3163    i^i cin 3164    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   occoc 13232   lubclub 14092   joincjn 14094   meetcmee 14095   CLatccla 14229   OPcops 29984   OLcol 29986   Atomscatm 30075   HLchlt 30162   pmapcpmap 30308   + Pcpadd 30606   _|_ PcpolN 30713
This theorem is referenced by:  pmapj2N  30740  osumcllem3N  30769  pexmidN  30780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-polarityN 30714
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