Theorem ispsubcl2N 35233

Description: Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pmapsubcl.b	|- B = ( Base ` K )
pmapsubcl.m	|- M = ( pmap ` K )
pmapsubcl.c	|- C = ( PSubCl ` K )

Assertion

Ref	Expression
ispsubcl2N	|- ( K e. HL -> ( X e. C <-> E. y e. B X = ( M ` y ) ) )

Distinct variable groups:   y, B   y, K   y, M   y, X

Allowed substitution hint:   C( y)

Proof of Theorem ispsubcl2N

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Atoms ` K ) = ( Atoms ` K )
2		eqid 2622	. . 3 |- ( _|_P ` K ) = ( _|_P ` K )
3		pmapsubcl.c	. . 3 |- C = ( PSubCl ` K )
4	1, 2, 3	ispsubclN 35223	. 2 |- ( K e. HL -> ( X e. C <-> ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) ) )
5		hlop 34649	. . . . . . . . 9 |- ( K e. HL -> K e. OP )
6	5	adantr 481	. . . . . . . 8 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) ) -> K e. OP )
7		hlclat 34645	. . . . . . . . . 10 |- ( K e. HL -> K e. CLat )
8	7	adantr 481	. . . . . . . . 9 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) ) -> K e. CLat )
9	1, 2	polssatN 35194	. . . . . . . . . 10 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) ) -> ( ( _|_P ` K ) ` X ) C_ ( Atoms ` K ) )
10		pmapsubcl.b	. . . . . . . . . . 11 |- B = ( Base ` K )
11	10, 1	atssbase 34577	. . . . . . . . . 10 |- ( Atoms ` K ) C_ B
12	9, 11	syl6ss 3615	. . . . . . . . 9 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) ) -> ( ( _|_P ` K ) ` X ) C_ B )
13		eqid 2622	. . . . . . . . . 10 |- ( lub ` K ) = ( lub ` K )
14	10, 13	clatlubcl 17112	. . . . . . . . 9 |- ( ( K e. CLat /\ ( ( _|_P ` K ) ` X ) C_ B ) -> ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) e. B )
15	8, 12, 14	syl2anc 693	. . . . . . . 8 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) ) -> ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) e. B )
16		eqid 2622	. . . . . . . . 9 |- ( oc ` K ) = ( oc ` K )
17	10, 16	opoccl 34481	. . . . . . . 8 |- ( ( K e. OP /\ ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) e. B ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) e. B )
18	6, 15, 17	syl2anc 693	. . . . . . 7 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) e. B )
19	18	ex 450	. . . . . 6 |- ( K e. HL -> ( X C_ ( Atoms ` K ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) e. B ) )
20	19	adantrd 484	. . . . 5 |- ( K e. HL -> ( ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) e. B ) )
21		pmapsubcl.m	. . . . . . . . . 10 |- M = ( pmap ` K )
22	13, 16, 1, 21, 2	polval2N 35192	. . . . . . . . 9 |- ( ( K e. HL /\ ( ( _|_P ` K ) ` X ) C_ ( Atoms ` K ) ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) )
23	9, 22	syldan 487	. . . . . . . 8 |- ( ( K e. HL /\ X C_ ( Atoms ` K ) ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) )
24	23	ex 450	. . . . . . 7 |- ( K e. HL -> ( X C_ ( Atoms ` K ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) ) )
25		eqeq1 2626	. . . . . . . 8 |- ( ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X -> ( ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) <-> X = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) ) )
26	25	biimpcd 239	. . . . . . 7 |- ( ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) -> ( ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X -> X = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) ) )
27	24, 26	syl6 35	. . . . . 6 |- ( K e. HL -> ( X C_ ( Atoms ` K ) -> ( ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X -> X = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) ) ) )
28	27	impd 447	. . . . 5 |- ( K e. HL -> ( ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) -> X = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) ) )
29	20, 28	jcad 555	. . . 4 |- ( K e. HL -> ( ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) -> ( ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) e. B /\ X = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) ) ) )
30		fveq2 6191	. . . . . 6 |- ( y = ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) -> ( M ` y ) = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) )
31	30	eqeq2d 2632	. . . . 5 |- ( y = ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) -> ( X = ( M ` y ) <-> X = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) ) )
32	31	rspcev 3309	. . . 4 |- ( ( ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) e. B /\ X = ( M ` ( ( oc ` K ) ` ( ( lub ` K ) ` ( ( _|_P ` K ) ` X ) ) ) ) ) -> E. y e. B X = ( M ` y ) )
33	29, 32	syl6 35	. . 3 |- ( K e. HL -> ( ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) -> E. y e. B X = ( M ` y ) ) )
34	10, 1, 21	pmapssat 35045	. . . . 5 |- ( ( K e. HL /\ y e. B ) -> ( M ` y ) C_ ( Atoms ` K ) )
35	10, 21, 2	2polpmapN 35199	. . . . 5 |- ( ( K e. HL /\ y e. B ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` ( M ` y ) ) ) = ( M ` y ) )
36		sseq1 3626	. . . . . . 7 |- ( X = ( M ` y ) -> ( X C_ ( Atoms ` K ) <-> ( M ` y ) C_ ( Atoms ` K ) ) )
37		fveq2 6191	. . . . . . . . 9 |- ( X = ( M ` y ) -> ( ( _|_P ` K ) ` X ) = ( ( _|_P ` K ) ` ( M ` y ) ) )
38	37	fveq2d 6195	. . . . . . . 8 |- ( X = ( M ` y ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` ( M ` y ) ) ) )
39		id 22	. . . . . . . 8 |- ( X = ( M ` y ) -> X = ( M ` y ) )
40	38, 39	eqeq12d 2637	. . . . . . 7 |- ( X = ( M ` y ) -> ( ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X <-> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` ( M ` y ) ) ) = ( M ` y ) ) )
41	36, 40	anbi12d 747	. . . . . 6 |- ( X = ( M ` y ) -> ( ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) <-> ( ( M ` y ) C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` ( M ` y ) ) ) = ( M ` y ) ) ) )
42	41	biimprcd 240	. . . . 5 |- ( ( ( M ` y ) C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` ( M ` y ) ) ) = ( M ` y ) ) -> ( X = ( M ` y ) -> ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) ) )
43	34, 35, 42	syl2anc 693	. . . 4 |- ( ( K e. HL /\ y e. B ) -> ( X = ( M ` y ) -> ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) ) )
44	43	rexlimdva 3031	. . 3 |- ( K e. HL -> ( E. y e. B X = ( M ` y ) -> ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) ) )
45	33, 44	impbid 202	. 2 |- ( K e. HL -> ( ( X C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X ) <-> E. y e. B X = ( M ` y ) ) )
46	4, 45	bitrd 268	1 |- ( K e. HL -> ( X e. C <-> E. y e. B X = ( M ` y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   ` cfv 5888   Basecbs 15857   occoc 15949   lubclub 16942   CLatccla 17107   OPcops 34459   Atomscatm 34550   HLchlt 34637   pmapcpmap 34783   _|_PcpolN 35188   PSubClcpscN 35220

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-psubsp 34789  df-pmap 34790  df-polarityN 35189  df-psubclN 35221

This theorem is referenced by: (None)