Theorem dibintclN 34729

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 34723	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )
4	3	adantr 467	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I : dom I -1-1-onto-> ran I )
5		f1ofn 5813	. . . . . 6 |- ( I : dom I -1-1-onto-> ran I -> I Fn dom I )
6	4, 5	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I Fn dom I )
7		cnvimass 5187	. . . . 5 |- ( `' I " S ) C_ dom I
8		fnssres 5687	. . . . 5 |- ( ( I Fn dom I /\ ( `' I " S ) C_ dom I ) -> ( I |` ( `' I " S ) ) Fn ( `' I " S ) )
9	6, 7, 8	sylancl 667	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I |` ( `' I " S ) ) Fn ( `' I " S ) )
10		fniinfv 5922	. . . 4 |- ( ( I |` ( `' I " S ) ) Fn ( `' I " S ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| ran ( I |` ( `' I " S ) ) )
11	9, 10	syl 17	. . 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 4846	. . . . 5 |- ( I " ( `' I " S ) ) = ran ( I |` ( `' I " S ) )
13		f1ofo 5819	. . . . . . . 8 |- ( I : dom I -1-1-onto-> ran I -> I : dom I -onto-> ran I )
14	3, 13	syl 17	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> I : dom I -onto-> ran I )
15	14	adantr 467	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I : dom I -onto-> ran I )
16		simprl 763	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> S C_ ran I )
17		foimacnv 5829	. . . . . 6 |- ( ( I : dom I -onto-> ran I /\ S C_ ran I ) -> ( I " ( `' I " S ) ) = S )
18	15, 16, 17	syl2anc 666	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I " ( `' I " S ) ) = S )
19	12, 18	syl5eqr 2498	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ran ( I |` ( `' I " S ) ) = S )
20	19	inteqd 4238	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| ran ( I |` ( `' I " S ) ) = |^| S )
21	11, 20	eqtrd 2484	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| S )
22		simpl 459	. . . . 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 765	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> S =/= (/) )
25		n0 3740	. . . . . . 7 |- ( S =/= (/) <-> E. y y e. S )
26	24, 25	sylib 200	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> E. y y e. S )
27	16	sselda 3431	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> y e. ran I )
28	3	ad2antrr 731	. . . . . . . . . 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 17	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> I Fn dom I )
30		fvelrnb 5910	. . . . . . . . 9 |- ( I Fn dom I -> ( y e. ran I <-> E. x e. dom I ( I ` x ) = y ) )
31	29, 30	syl 17	. . . . . . . 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 214	. . . . . . 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 5814	. . . . . . . . . . . . . . . 16 |- ( I : dom I -1-1-onto-> ran I -> Fun I )
34	3, 33	syl 17	. . . . . . . . . . . . . . 15 |- ( ( K e. HL /\ W e. H ) -> Fun I )
35	34	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> Fun I )
36		fvimacnv 5995	. . . . . . . . . . . . . 14 |- ( ( Fun I /\ x e. dom I ) -> ( ( I ` x ) e. S <-> x e. ( `' I " S ) ) )
37	35, 36	sylan 474	. . . . . . . . . . . . 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 3736	. . . . . . . . . . . . 13 |- ( x e. ( `' I " S ) -> ( `' I " S ) =/= (/) )
39	37, 38	syl6bi 232	. . . . . . . . . . . 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 436	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( x e. dom I -> ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) ) )
41		eleq1 2516	. . . . . . . . . . . . 13 |- ( ( I ` x ) = y -> ( ( I ` x ) e. S <-> y e. S ) )
42	41	biimprd 227	. . . . . . . . . . . 12 |- ( ( I ` x ) = y -> ( y e. S -> ( I ` x ) e. S ) )
43	42	imim1d 78	. . . . . . . . . . 11 |- ( ( I ` x ) = y -> ( ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) -> ( y e. S -> ( `' I " S ) =/= (/) ) ) )
44	40, 43	syl9 73	. . . . . . . . . 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 90	. . . . . . . . 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 431	. . . . . . . 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 2876	. . . . . . 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 1780	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) =/= (/) )
50		eqid 2450	. . . . . 6 |- ( glb ` K ) = ( glb ` K )
51	50, 1, 2	dibglbN 34728	. . . . 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 1265	. . . 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 5877	. . . . 5 |- ( y e. ( `' I " S ) -> ( ( I |` ( `' I " S ) ) ` y ) = ( I ` y ) )
54	53	iineq2i 4297	. . . 4 |- |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^|_ y e. ( `' I " S ) ( I ` y )
55	52, 54	syl6eqr 2502	. . 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 32918	. . . . . . 7 |- ( K e. HL -> K e. CLat )
57	56	ad2antrr 731	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> K e. CLat )
58		eqid 2450	. . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
59		eqid 2450	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
60	58, 59, 1, 2	dibdmN 34719	. . . . . . . . 9 |- ( ( K e. HL /\ W e. H ) -> dom I = { x e. ( Base ` K ) | x ( le ` K ) W } )
61		ssrab2 3513	. . . . . . . . 9 |- { x e. ( Base ` K ) | x ( le ` K ) W } C_ ( Base ` K )
62	60, 61	syl6eqss 3481	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> dom I C_ ( Base ` K ) )
63	62	adantr 467	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> dom I C_ ( Base ` K ) )
64	7, 63	syl5ss 3442	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ ( Base ` K ) )
65	58, 50	clatglbcl 16353	. . . . . 6 |- ( ( K e. CLat /\ ( `' I " S ) C_ ( Base ` K ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) )
66	57, 64, 65	syl2anc 666	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) )
67		n0 3740	. . . . . . 7 |- ( ( `' I " S ) =/= (/) <-> E. y y e. ( `' I " S ) )
68	49, 67	sylib 200	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> E. y y e. ( `' I " S ) )
69		hllat 32923	. . . . . . . 8 |- ( K e. HL -> K e. Lat )
70	69	ad3antrrr 735	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> K e. Lat )
71	66	adantr 467	. . . . . . 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 3431	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> y e. ( Base ` K ) )
73	58, 1	lhpbase 33557	. . . . . . . 8 |- ( W e. H -> W e. ( Base ` K ) )
74	73	ad3antlr 736	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> W e. ( Base ` K ) )
75	56	ad3antrrr 735	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> K e. CLat )
76	60	adantr 467	. . . . . . . . . . 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 3479	. . . . . . . . . 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 3443	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ ( Base ` K ) )
79	78	adantr 467	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( `' I " S ) C_ ( Base ` K ) )
80		simpr 463	. . . . . . . 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 16364	. . . . . . . 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 1267	. . . . . . 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 3427	. . . . . . . . . 10 |- ( y e. ( `' I " S ) -> y e. dom I )
84	83	adantl 468	. . . . . . . . 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 34720	. . . . . . . . . 10 |- ( ( K e. HL /\ W e. H ) -> ( y e. dom I <-> ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) )
86	85	ad2antrr 731	. . . . . . . . 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 214	. . . . . . . 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 465	. . . . . . 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 16297	. . . . . 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 1780	. . . . 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 34720	. . . . . 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 467	. . . . 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 932	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. dom I )
94	1, 2	dibclN 34724	. . . 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 473	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) e. ran I )
96	55, 95	eqeltrrd 2529	. 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 2529	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| S e. ran I )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   E.wex 1662   e. wcel 1886   =/= wne 2621   E.wrex 2737   {crab 2740   C_ wss 3403   (/)c0 3730   |^|cint 4233   |^|_ciin 4278   class class class wbr 4401   `'ccnv 4832   dom cdm 4833   ran crn 4834   |` cres 4835   "cima 4836   Fun wfun 5575   Fn wfn 5576   -onto->wfo 5579   -1-1-onto->wf1o 5580   ` cfv 5581   Basecbs 15114   lecple 15190   glbcglb 16181   Latclat 16284   CLatccla 16346   HLchlt 32910   LHypclh 33543   DIsoBcdib 34700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-riotaBAD 32519

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-undef 7017  df-map 7471  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-oposet 32736  df-ol 32738  df-oml 32739  df-covers 32826  df-ats 32827  df-atl 32858  df-cvlat 32882  df-hlat 32911  df-llines 33057  df-lplanes 33058  df-lvols 33059  df-lines 33060  df-psubsp 33062  df-pmap 33063  df-padd 33355  df-lhyp 33547  df-laut 33548  df-ldil 33663  df-ltrn 33664  df-trl 33719  df-disoa 34591  df-dib 34701

This theorem is referenced by: (None)