Theorem poldmj1N 35214

Description: De Morgan's law for polarity of projective sum. (oldmj1 34508 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

Step	Hyp	Ref	Expression
1		paddun.a	. . 3 |- A = ( Atoms ` K )
2		paddun.p	. . 3 |- .+ = ( +P ` K )
3		paddun.o	. . 3 |- ._|_ = ( _|_P ` K )
4	1, 2, 3	paddunN 35213	. 2 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ._|_ ` ( S .+ T ) ) = ( ._|_ ` ( S u. T ) ) )
5		simp1 1061	. . 3 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> K e. HL )
6		unss 3787	. . . . 5 |- ( ( S C_ A /\ T C_ A ) <-> ( S u. T ) C_ A )
7	6	biimpi 206	. . . 4 |- ( ( S C_ A /\ T C_ A ) -> ( S u. T ) C_ A )
8	7	3adant1 1079	. . 3 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( S u. T ) C_ A )
9		eqid 2622	. . . 4 |- ( lub ` K ) = ( lub ` K )
10		eqid 2622	. . . 4 |- ( oc ` K ) = ( oc ` K )
11		eqid 2622	. . . 4 |- ( pmap ` K ) = ( pmap ` K )
12	9, 10, 1, 11, 3	polval2N 35192	. . 3 |- ( ( K e. HL /\ ( S u. T ) C_ A ) -> ( ._|_ ` ( S u. T ) ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( S u. T ) ) ) ) )
13	5, 8, 12	syl2anc 693	. 2 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ._|_ ` ( S u. T ) ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( S u. T ) ) ) ) )
14		hlop 34649	. . . . . 6 |- ( K e. HL -> K e. OP )
15	14	3ad2ant1 1082	. . . . 5 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> K e. OP )
16		hlclat 34645	. . . . . . 7 |- ( K e. HL -> K e. CLat )
17	16	3ad2ant1 1082	. . . . . 6 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> K e. CLat )
18		simp2 1062	. . . . . . 7 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> S C_ A )
19		eqid 2622	. . . . . . . 8 |- ( Base ` K ) = ( Base ` K )
20	19, 1	atssbase 34577	. . . . . . 7 |- A C_ ( Base ` K )
21	18, 20	syl6ss 3615	. . . . . 6 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> S C_ ( Base ` K ) )
22	19, 9	clatlubcl 17112	. . . . . 6 |- ( ( K e. CLat /\ S C_ ( Base ` K ) ) -> ( ( lub ` K ) ` S ) e. ( Base ` K ) )
23	17, 21, 22	syl2anc 693	. . . . 5 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ( lub ` K ) ` S ) e. ( Base ` K ) )
24	19, 10	opoccl 34481	. . . . 5 |- ( ( K e. OP /\ ( ( lub ` K ) ` S ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` S ) ) e. ( Base ` K ) )
25	15, 23, 24	syl2anc 693	. . . 4 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` S ) ) e. ( Base ` K ) )
26		simp3 1063	. . . . . . 7 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> T C_ A )
27	26, 20	syl6ss 3615	. . . . . 6 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> T C_ ( Base ` K ) )
28	19, 9	clatlubcl 17112	. . . . . 6 |- ( ( K e. CLat /\ T C_ ( Base ` K ) ) -> ( ( lub ` K ) ` T ) e. ( Base ` K ) )
29	17, 27, 28	syl2anc 693	. . . . 5 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ( lub ` K ) ` T ) e. ( Base ` K ) )
30	19, 10	opoccl 34481	. . . . 5 |- ( ( K e. OP /\ ( ( lub ` K ) ` T ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` T ) ) e. ( Base ` K ) )
31	15, 29, 30	syl2anc 693	. . . 4 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` T ) ) e. ( Base ` K ) )
32		eqid 2622	. . . . 5 |- ( meet ` K ) = ( meet ` K )
33	19, 32, 1, 11	pmapmeet 35059	. . . 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 ) ) ) ) )
34	5, 25, 31, 33	syl3anc 1326	. . 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 2622	. . . . . . . 8 |- ( join ` K ) = ( join ` K )
36	19, 35, 9	lubun 17123	. . . . . . 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 ) ) )
37	17, 21, 27, 36	syl3anc 1326	. . . . . 6 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ( lub ` K ) ` ( S u. T ) ) = ( ( ( lub ` K ) ` S ) ( join ` K ) ( ( lub ` K ) ` T ) ) )
38	37	fveq2d 6195	. . . . 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 34648	. . . . . . 7 |- ( K e. HL -> K e. OL )
40	39	3ad2ant1 1082	. . . . . 6 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> K e. OL )
41	19, 35, 32, 10	oldmj1 34508	. . . . . 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 ) ) ) )
42	40, 23, 29, 41	syl3anc 1326	. . . . 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 ) ) ) )
43	38, 42	eqtrd 2656	. . . 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 ) ) ) )
44	43	fveq2d 6195	. . 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 ) ) ) ) )
45	9, 10, 1, 11, 3	polval2N 35192	. . . . 5 |- ( ( K e. HL /\ S C_ A ) -> ( ._|_ ` S ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` S ) ) ) )
46	45	3adant3 1081	. . . 4 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ._|_ ` S ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` S ) ) ) )
47	9, 10, 1, 11, 3	polval2N 35192	. . . . 5 |- ( ( K e. HL /\ T C_ A ) -> ( ._|_ ` T ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` T ) ) ) )
48	47	3adant2 1080	. . . 4 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ._|_ ` T ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` T ) ) ) )
49	46, 48	ineq12d 3815	. . 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 ) ) ) ) )
50	34, 44, 49	3eqtr4d 2666	. 2 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( S u. T ) ) ) ) = ( ( ._|_ ` S ) i^i ( ._|_ ` T ) ) )
51	4, 13, 50	3eqtrd 2660	1 |- ( ( K e. HL /\ S C_ A /\ T C_ A ) -> ( ._|_ ` ( S .+ T ) ) = ( ( ._|_ ` S ) i^i ( ._|_ ` T ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   occoc 15949   lubclub 16942   joincjn 16944   meetcmee 16945   CLatccla 17107   OPcops 34459   OLcol 34461   Atomscatm 34550   HLchlt 34637   pmapcpmap 34783   +Pcpadd 35081   _|_PcpolN 35188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-polarityN 35189

This theorem is referenced by:  pmapj2N  35215  osumcllem3N  35244  pexmidN  35255