Theorem pol1N 35196

Description: The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
pol1N	|- ( K e. HL -> ( ._|_ ` A ) = (/) )

Proof of Theorem pol1N

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3624	. . 3 |- A C_ A
2		eqid 2622	. . . 4 |- ( lub ` K ) = ( lub ` K )
3		eqid 2622	. . . 4 |- ( oc ` K ) = ( oc ` K )
4		polssat.a	. . . 4 |- A = ( Atoms ` K )
5		eqid 2622	. . . 4 |- ( pmap ` K ) = ( pmap ` K )
6		polssat.p	. . . 4 |- ._|_ = ( _|_P ` K )
7	2, 3, 4, 5, 6	polval2N 35192	. . 3 |- ( ( K e. HL /\ A C_ A ) -> ( ._|_ ` A ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) ) )
8	1, 7	mpan2 707	. 2 |- ( K e. HL -> ( ._|_ ` A ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) ) )
9		hlop 34649	. . . . . . . . . 10 |- ( K e. HL -> K e. OP )
10		eqid 2622	. . . . . . . . . . 11 |- ( Base ` K ) = ( Base ` K )
11	10, 4	atbase 34576	. . . . . . . . . 10 |- ( p e. A -> p e. ( Base ` K ) )
12		eqid 2622	. . . . . . . . . . 11 |- ( le ` K ) = ( le ` K )
13		eqid 2622	. . . . . . . . . . 11 |- ( 1. ` K ) = ( 1. ` K )
14	10, 12, 13	ople1 34478	. . . . . . . . . 10 |- ( ( K e. OP /\ p e. ( Base ` K ) ) -> p ( le ` K ) ( 1. ` K ) )
15	9, 11, 14	syl2an 494	. . . . . . . . 9 |- ( ( K e. HL /\ p e. A ) -> p ( le ` K ) ( 1. ` K ) )
16	15	ralrimiva 2966	. . . . . . . 8 |- ( K e. HL -> A. p e. A p ( le ` K ) ( 1. ` K ) )
17		rabid2 3118	. . . . . . . 8 |- ( A = { p e. A | p ( le ` K ) ( 1. ` K ) } <-> A. p e. A p ( le ` K ) ( 1. ` K ) )
18	16, 17	sylibr 224	. . . . . . 7 |- ( K e. HL -> A = { p e. A | p ( le ` K ) ( 1. ` K ) } )
19	18	fveq2d 6195	. . . . . 6 |- ( K e. HL -> ( ( lub ` K ) ` A ) = ( ( lub ` K ) ` { p e. A | p ( le ` K ) ( 1. ` K ) } ) )
20		hlomcmat 34651	. . . . . . 7 |- ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. AtLat ) )
21	10, 13	op1cl 34472	. . . . . . . 8 |- ( K e. OP -> ( 1. ` K ) e. ( Base ` K ) )
22	9, 21	syl 17	. . . . . . 7 |- ( K e. HL -> ( 1. ` K ) e. ( Base ` K ) )
23	10, 12, 2, 4	atlatmstc 34606	. . . . . . 7 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ ( 1. ` K ) e. ( Base ` K ) ) -> ( ( lub ` K ) ` { p e. A | p ( le ` K ) ( 1. ` K ) } ) = ( 1. ` K ) )
24	20, 22, 23	syl2anc 693	. . . . . 6 |- ( K e. HL -> ( ( lub ` K ) ` { p e. A | p ( le ` K ) ( 1. ` K ) } ) = ( 1. ` K ) )
25	19, 24	eqtr2d 2657	. . . . 5 |- ( K e. HL -> ( 1. ` K ) = ( ( lub ` K ) ` A ) )
26	25	fveq2d 6195	. . . 4 |- ( K e. HL -> ( ( oc ` K ) ` ( 1. ` K ) ) = ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) )
27		eqid 2622	. . . . . 6 |- ( 0. ` K ) = ( 0. ` K )
28	27, 13, 3	opoc1 34489	. . . . 5 |- ( K e. OP -> ( ( oc ` K ) ` ( 1. ` K ) ) = ( 0. ` K ) )
29	9, 28	syl 17	. . . 4 |- ( K e. HL -> ( ( oc ` K ) ` ( 1. ` K ) ) = ( 0. ` K ) )
30	26, 29	eqtr3d 2658	. . 3 |- ( K e. HL -> ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) = ( 0. ` K ) )
31	30	fveq2d 6195	. 2 |- ( K e. HL -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) ) = ( ( pmap ` K ) ` ( 0. ` K ) ) )
32		hlatl 34647	. . 3 |- ( K e. HL -> K e. AtLat )
33	27, 5	pmap0 35051	. . 3 |- ( K e. AtLat -> ( ( pmap ` K ) ` ( 0. ` K ) ) = (/) )
34	32, 33	syl 17	. 2 |- ( K e. HL -> ( ( pmap ` K ) ` ( 0. ` K ) ) = (/) )
35	8, 31, 34	3eqtrd 2660	1 |- ( K e. HL -> ( ._|_ ` A ) = (/) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   occoc 15949   lubclub 16942   0.cp0 17037   1.cp1 17038   CLatccla 17107   OPcops 34459   OMLcoml 34462   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:  2pol0N  35197  1psubclN  35230