Theorem pnonsingN 35219

Description: The intersection of a set of atoms and its polarity is empty. Definition of nonsingular in [Holland95] p. 214. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
2polat.a	|- A = ( Atoms ` K )
2polat.p	|- P = ( _|_P ` K )

Assertion

Ref	Expression
pnonsingN	|- ( ( K e. HL /\ X C_ A ) -> ( X i^i ( P ` X ) ) = (/) )

Proof of Theorem pnonsingN

Step	Hyp	Ref	Expression
1		2polat.a	. . . . 5 |- A = ( Atoms ` K )
2		2polat.p	. . . . 5 |- P = ( _|_P ` K )
3	1, 2	2polssN 35201	. . . 4 |- ( ( K e. HL /\ X C_ A ) -> X C_ ( P ` ( P ` X ) ) )
4		ssrin 3838	. . . 4 |- ( X C_ ( P ` ( P ` X ) ) -> ( X i^i ( P ` X ) ) C_ ( ( P ` ( P ` X ) ) i^i ( P ` X ) ) )
5	3, 4	syl 17	. . 3 |- ( ( K e. HL /\ X C_ A ) -> ( X i^i ( P ` X ) ) C_ ( ( P ` ( P ` X ) ) i^i ( P ` X ) ) )
6		eqid 2622	. . . . . 6 |- ( lub ` K ) = ( lub ` K )
7		eqid 2622	. . . . . 6 |- ( pmap ` K ) = ( pmap ` K )
8	6, 1, 7, 2	2polvalN 35200	. . . . 5 |- ( ( K e. HL /\ X C_ A ) -> ( P ` ( P ` X ) ) = ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) )
9		eqid 2622	. . . . . 6 |- ( oc ` K ) = ( oc ` K )
10	6, 9, 1, 7, 2	polval2N 35192	. . . . 5 |- ( ( K e. HL /\ X C_ A ) -> ( P ` X ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) )
11	8, 10	ineq12d 3815	. . . 4 |- ( ( K e. HL /\ X C_ A ) -> ( ( P ` ( P ` X ) ) i^i ( P ` X ) ) = ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) i^i ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) ) )
12		hlop 34649	. . . . . . . 8 |- ( K e. HL -> K e. OP )
13	12	adantr 481	. . . . . . 7 |- ( ( K e. HL /\ X C_ A ) -> K e. OP )
14		hlclat 34645	. . . . . . . 8 |- ( K e. HL -> K e. CLat )
15		eqid 2622	. . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
16	15, 1	atssbase 34577	. . . . . . . . 9 |- A C_ ( Base ` K )
17		sstr 3611	. . . . . . . . 9 |- ( ( X C_ A /\ A C_ ( Base ` K ) ) -> X C_ ( Base ` K ) )
18	16, 17	mpan2 707	. . . . . . . 8 |- ( X C_ A -> X C_ ( Base ` K ) )
19	15, 6	clatlubcl 17112	. . . . . . . 8 |- ( ( K e. CLat /\ X C_ ( Base ` K ) ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
20	14, 18, 19	syl2an 494	. . . . . . 7 |- ( ( K e. HL /\ X C_ A ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
21		eqid 2622	. . . . . . . 8 |- ( meet ` K ) = ( meet ` K )
22		eqid 2622	. . . . . . . 8 |- ( 0. ` K ) = ( 0. ` K )
23	15, 9, 21, 22	opnoncon 34495	. . . . . . 7 |- ( ( K e. OP /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) ) -> ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) = ( 0. ` K ) )
24	13, 20, 23	syl2anc 693	. . . . . 6 |- ( ( K e. HL /\ X C_ A ) -> ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) = ( 0. ` K ) )
25	24	fveq2d 6195	. . . . 5 |- ( ( K e. HL /\ X C_ A ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) ) = ( ( pmap ` K ) ` ( 0. ` K ) ) )
26		simpl 473	. . . . . 6 |- ( ( K e. HL /\ X C_ A ) -> K e. HL )
27	15, 9	opoccl 34481	. . . . . . 7 |- ( ( K e. OP /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) )
28	13, 20, 27	syl2anc 693	. . . . . 6 |- ( ( K e. HL /\ X C_ A ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) )
29	15, 21, 1, 7	pmapmeet 35059	. . . . . 6 |- ( ( K e. HL /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) /\ ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) ) = ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) i^i ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) ) )
30	26, 20, 28, 29	syl3anc 1326	. . . . 5 |- ( ( K e. HL /\ X C_ A ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) ) = ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) i^i ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) ) )
31		hlatl 34647	. . . . . . 7 |- ( K e. HL -> K e. AtLat )
32	31	adantr 481	. . . . . 6 |- ( ( K e. HL /\ X C_ A ) -> K e. AtLat )
33	22, 7	pmap0 35051	. . . . . 6 |- ( K e. AtLat -> ( ( pmap ` K ) ` ( 0. ` K ) ) = (/) )
34	32, 33	syl 17	. . . . 5 |- ( ( K e. HL /\ X C_ A ) -> ( ( pmap ` K ) ` ( 0. ` K ) ) = (/) )
35	25, 30, 34	3eqtr3d 2664	. . . 4 |- ( ( K e. HL /\ X C_ A ) -> ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) i^i ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) ) = (/) )
36	11, 35	eqtrd 2656	. . 3 |- ( ( K e. HL /\ X C_ A ) -> ( ( P ` ( P ` X ) ) i^i ( P ` X ) ) = (/) )
37	5, 36	sseqtrd 3641	. 2 |- ( ( K e. HL /\ X C_ A ) -> ( X i^i ( P ` X ) ) C_ (/) )
38		ss0b 3973	. 2 |- ( ( X i^i ( P ` X ) ) C_ (/) <-> ( X i^i ( P ` X ) ) = (/) )
39	37, 38	sylib 208	1 |- ( ( K e. HL /\ X C_ A ) -> ( X i^i ( P ` X ) ) = (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Basecbs 15857   occoc 15949   lubclub 16942   meetcmee 16945   0.cp0 17037   CLatccla 17107   OPcops 34459   Atomscatm 34550   AtLatcal 34551   HLchlt 34637   pmapcpmap 34783   _|_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-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-pmap 34790  df-polarityN 35189

This theorem is referenced by:  osumcllem4N  35245  pexmidN  35255  pexmidlem1N  35256