Theorem dibintclN 35839

Description: The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dibintcl.h	|- H = ( LHyp ` K )
dibintcl.i	|- I = ( ( DIsoB ` K ) ` W )

Assertion

Ref	Expression
dibintclN	|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| S e. ran I )

Proof of Theorem dibintclN

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dibintcl.h	. . . . . . . 8 |- H = ( LHyp ` K )
2		dibintcl.i	. . . . . . . 8 |- I = ( ( DIsoB ` K ) ` W )
3	1, 2	dibf11N 35833	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )
4	3	adantr 465	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I : dom I -1-1-onto-> ran I )
5		f1ofn 5808	. . . . . 6 |- ( I : dom I -1-1-onto-> ran I -> I Fn dom I )
6	4, 5	syl 16	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I Fn dom I )
7		cnvimass 5348	. . . . 5 |- ( `' I " S ) C_ dom I
8		fnssres 5685	. . . . 5 |- ( ( I Fn dom I /\ ( `' I " S ) C_ dom I ) -> ( I |` ( `' I " S ) ) Fn ( `' I " S ) )
9	6, 7, 8	sylancl 662	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I |` ( `' I " S ) ) Fn ( `' I " S ) )
10		fniinfv 5917	. . . 4 |- ( ( I |` ( `' I " S ) ) Fn ( `' I " S ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| ran ( I |` ( `' I " S ) ) )
11	9, 10	syl 16	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| ran ( I |` ( `' I " S ) ) )
12		df-ima 5005	. . . . 5 |- ( I " ( `' I " S ) ) = ran ( I |` ( `' I " S ) )
13		f1ofo 5814	. . . . . . . 8 |- ( I : dom I -1-1-onto-> ran I -> I : dom I -onto-> ran I )
14	3, 13	syl 16	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> I : dom I -onto-> ran I )
15	14	adantr 465	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I : dom I -onto-> ran I )
16		simprl 755	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> S C_ ran I )
17		foimacnv 5824	. . . . . 6 |- ( ( I : dom I -onto-> ran I /\ S C_ ran I ) -> ( I " ( `' I " S ) ) = S )
18	15, 16, 17	syl2anc 661	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I " ( `' I " S ) ) = S )
19	12, 18	syl5eqr 2515	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ran ( I |` ( `' I " S ) ) = S )
20	19	inteqd 4280	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| ran ( I |` ( `' I " S ) ) = |^| S )
21	11, 20	eqtrd 2501	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| S )
22		simpl 457	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( K e. HL /\ W e. H ) )
23	7	a1i 11	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ dom I )
24		simprr 756	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> S =/= (/) )
25		n0 3787	. . . . . . 7 |- ( S =/= (/) <-> E. y y e. S )
26	24, 25	sylib 196	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> E. y y e. S )
27	16	sselda 3497	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> y e. ran I )
28	3	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> I : dom I -1-1-onto-> ran I )
29	28, 5	syl 16	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> I Fn dom I )
30		fvelrnb 5906	. . . . . . . . 9 |- ( I Fn dom I -> ( y e. ran I <-> E. x e. dom I ( I ` x ) = y ) )
31	29, 30	syl 16	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( y e. ran I <-> E. x e. dom I ( I ` x ) = y ) )
32	27, 31	mpbid 210	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> E. x e. dom I ( I ` x ) = y )
33		f1ofun 5809	. . . . . . . . . . . . . . . 16 |- ( I : dom I -1-1-onto-> ran I -> Fun I )
34	3, 33	syl 16	. . . . . . . . . . . . . . 15 |- ( ( K e. HL /\ W e. H ) -> Fun I )
35	34	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> Fun I )
36		fvimacnv 5987	. . . . . . . . . . . . . 14 |- ( ( Fun I /\ x e. dom I ) -> ( ( I ` x ) e. S <-> x e. ( `' I " S ) ) )
37	35, 36	sylan 471	. . . . . . . . . . . . 13 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ x e. dom I ) -> ( ( I ` x ) e. S <-> x e. ( `' I " S ) ) )
38		ne0i 3784	. . . . . . . . . . . . 13 |- ( x e. ( `' I " S ) -> ( `' I " S ) =/= (/) )
39	37, 38	syl6bi 228	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ x e. dom I ) -> ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) )
40	39	ex 434	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( x e. dom I -> ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) ) )
41		eleq1 2532	. . . . . . . . . . . . 13 |- ( ( I ` x ) = y -> ( ( I ` x ) e. S <-> y e. S ) )
42	41	biimprd 223	. . . . . . . . . . . 12 |- ( ( I ` x ) = y -> ( y e. S -> ( I ` x ) e. S ) )
43	42	imim1d 75	. . . . . . . . . . 11 |- ( ( I ` x ) = y -> ( ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) -> ( y e. S -> ( `' I " S ) =/= (/) ) ) )
44	40, 43	syl9 71	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( I ` x ) = y -> ( x e. dom I -> ( y e. S -> ( `' I " S ) =/= (/) ) ) ) )
45	44	com24 87	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( y e. S -> ( x e. dom I -> ( ( I ` x ) = y -> ( `' I " S ) =/= (/) ) ) ) )
46	45	imp 429	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( x e. dom I -> ( ( I ` x ) = y -> ( `' I " S ) =/= (/) ) ) )
47	46	rexlimdv 2946	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( E. x e. dom I ( I ` x ) = y -> ( `' I " S ) =/= (/) ) )
48	32, 47	mpd 15	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( `' I " S ) =/= (/) )
49	26, 48	exlimddv 1697	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) =/= (/) )
50		eqid 2460	. . . . . 6 |- ( glb ` K ) = ( glb ` K )
51	50, 1, 2	dibglbN 35838	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( `' I " S ) C_ dom I /\ ( `' I " S ) =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) = |^|_ y e. ( `' I " S ) ( I ` y ) )
52	22, 23, 49, 51	syl12anc 1221	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) = |^|_ y e. ( `' I " S ) ( I ` y ) )
53		fvres 5871	. . . . 5 |- ( y e. ( `' I " S ) -> ( ( I |` ( `' I " S ) ) ` y ) = ( I ` y ) )
54	53	iineq2i 4338	. . . 4 |- |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^|_ y e. ( `' I " S ) ( I ` y )
55	52, 54	syl6eqr 2519	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) = |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) )
56		hlclat 34030	. . . . . . 7 |- ( K e. HL -> K e. CLat )
57	56	ad2antrr 725	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> K e. CLat )
58		eqid 2460	. . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
59		eqid 2460	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
60	58, 59, 1, 2	dibdmN 35829	. . . . . . . . 9 |- ( ( K e. HL /\ W e. H ) -> dom I = { x e. ( Base ` K ) | x ( le ` K ) W } )
61		ssrab2 3578	. . . . . . . . 9 |- { x e. ( Base ` K ) | x ( le ` K ) W } C_ ( Base ` K )
62	60, 61	syl6eqss 3547	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> dom I C_ ( Base ` K ) )
63	62	adantr 465	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> dom I C_ ( Base ` K ) )
64	7, 63	syl5ss 3508	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ ( Base ` K ) )
65	58, 50	clatglbcl 15590	. . . . . 6 |- ( ( K e. CLat /\ ( `' I " S ) C_ ( Base ` K ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) )
66	57, 64, 65	syl2anc 661	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) )
67		n0 3787	. . . . . . 7 |- ( ( `' I " S ) =/= (/) <-> E. y y e. ( `' I " S ) )
68	49, 67	sylib 196	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> E. y y e. ( `' I " S ) )
69		hllat 34035	. . . . . . . 8 |- ( K e. HL -> K e. Lat )
70	69	ad3antrrr 729	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> K e. Lat )
71	66	adantr 465	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) )
72	64	sselda 3497	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> y e. ( Base ` K ) )
73	58, 1	lhpbase 34669	. . . . . . . 8 |- ( W e. H -> W e. ( Base ` K ) )
74	73	ad3antlr 730	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> W e. ( Base ` K ) )
75	56	ad3antrrr 729	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> K e. CLat )
76	60	adantr 465	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> dom I = { x e. ( Base ` K ) | x ( le ` K ) W } )
77	7, 76	syl5sseq 3545	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ { x e. ( Base ` K ) | x ( le ` K ) W } )
78	77, 61	syl6ss 3509	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ ( Base ` K ) )
79	78	adantr 465	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( `' I " S ) C_ ( Base ` K ) )
80		simpr 461	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> y e. ( `' I " S ) )
81	58, 59, 50	clatglble 15601	. . . . . . . 8 |- ( ( K e. CLat /\ ( `' I " S ) C_ ( Base ` K ) /\ y e. ( `' I " S ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) y )
82	75, 79, 80, 81	syl3anc 1223	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) y )
83	7	sseli 3493	. . . . . . . . . 10 |- ( y e. ( `' I " S ) -> y e. dom I )
84	83	adantl 466	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> y e. dom I )
85	58, 59, 1, 2	dibeldmN 35830	. . . . . . . . . 10 |- ( ( K e. HL /\ W e. H ) -> ( y e. dom I <-> ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) )
86	85	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( y e. dom I <-> ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) )
87	84, 86	mpbid 210	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( y e. ( Base ` K ) /\ y ( le ` K ) W ) )
88	87	simprd 463	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> y ( le ` K ) W )
89	58, 59, 70, 71, 72, 74, 82, 88	lattrd 15534	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) W )
90	68, 89	exlimddv 1697	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) W )
91	58, 59, 1, 2	dibeldmN 35830	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> ( ( ( glb ` K ) ` ( `' I " S ) ) e. dom I <-> ( ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) /\ ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) W ) ) )
92	91	adantr 465	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( ( glb ` K ) ` ( `' I " S ) ) e. dom I <-> ( ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) /\ ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) W ) ) )
93	66, 90, 92	mpbir2and 915	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. dom I )
94	1, 2	dibclN 35834	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( glb ` K ) ` ( `' I " S ) ) e. dom I ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) e. ran I )
95	93, 94	syldan 470	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) e. ran I )
96	55, 95	eqeltrrd 2549	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) e. ran I )
97	21, 96	eqeltrrd 2549	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| S e. ran I )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   E.wex 1591   e. wcel 1762   =/= wne 2655   E.wrex 2808   {crab 2811   C_ wss 3469   (/)c0 3778   |^|cint 4275   |^|_ciin 4319   class class class wbr 4440   `'ccnv 4991   dom cdm 4992   ran crn 4993   |` cres 4994   "cima 4995   Fun wfun 5573   Fn wfn 5574   -onto->wfo 5577   -1-1-onto->wf1o 5578   ` cfv 5579   Basecbs 14479   lecple 14551   glbcglb 15419   Latclat 15521   CLatccla 15583   HLchlt 34022   LHypclh 34655   DIsoBcdib 35810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-undef 6992  df-map 7412  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830  df-disoa 35701  df-dib 35811

This theorem is referenced by: (None)