Theorem dih1 36575

Description: The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)

Hypotheses

Ref	Expression
dih1.m	|- .1. = ( 1. ` K )
dih1.h	|- H = ( LHyp ` K )
dih1.i	|- I = ( ( DIsoH ` K ) ` W )
dih1.u	|- U = ( ( DVecH ` K ) ` W )
dih1.v	|- V = ( Base ` U )

Assertion

Ref	Expression
dih1	|- ( ( K e. HL /\ W e. H ) -> ( I ` .1. ) = V )

Proof of Theorem dih1

Dummy variables f g s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dih1.h	. . 3 |- H = ( LHyp ` K )
2		dih1.i	. . 3 |- I = ( ( DIsoH ` K ) ` W )
3	1, 2	dihvalrel 36568	. 2 |- ( ( K e. HL /\ W e. H ) -> Rel ( I ` .1. ) )
4		relxp 5227	. . 3 |- Rel ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) )
5		eqid 2622	. . . . 5 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
6		eqid 2622	. . . . 5 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
7		dih1.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
8		dih1.v	. . . . 5 |- V = ( Base ` U )
9	1, 5, 6, 7, 8	dvhvbase 36376	. . . 4 |- ( ( K e. HL /\ W e. H ) -> V = ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
10	9	releqd 5203	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( Rel V <-> Rel ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) ) )
11	4, 10	mpbiri 248	. 2 |- ( ( K e. HL /\ W e. H ) -> Rel V )
12		id 22	. 2 |- ( ( K e. HL /\ W e. H ) -> ( K e. HL /\ W e. H ) )
13		hlop 34649	. . . . . . . 8 |- ( K e. HL -> K e. OP )
14	13	ad2antrr 762	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> K e. OP )
15		simpl 473	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> ( K e. HL /\ W e. H ) )
16		simprl 794	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> f e. ( ( LTrn ` K ) ` W ) )
17		simprr 796	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> s e. ( ( TEndo ` K ) ` W ) )
18		eqid 2622	. . . . . . . . . . . . . 14 |- ( le ` K ) = ( le ` K )
19		eqid 2622	. . . . . . . . . . . . . 14 |- ( oc ` K ) = ( oc ` K )
20		eqid 2622	. . . . . . . . . . . . . 14 |- ( Atoms ` K ) = ( Atoms ` K )
21	18, 19, 20, 1	lhpocnel 35304	. . . . . . . . . . . . 13 |- ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) e. ( Atoms ` K ) /\ -. ( ( oc ` K ) ` W ) ( le ` K ) W ) )
22	21	adantr 481	. . . . . . . . . . . 12 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> ( ( ( oc ` K ) ` W ) e. ( Atoms ` K ) /\ -. ( ( oc ` K ) ` W ) ( le ` K ) W ) )
23		eqid 2622	. . . . . . . . . . . . 13 |- ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) = ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) )
24	18, 20, 1, 5, 23	ltrniotacl 35867	. . . . . . . . . . . 12 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( ( oc ` K ) ` W ) e. ( Atoms ` K ) /\ -. ( ( oc ` K ) ` W ) ( le ` K ) W ) /\ ( ( ( oc ` K ) ` W ) e. ( Atoms ` K ) /\ -. ( ( oc ` K ) ` W ) ( le ` K ) W ) ) -> ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) e. ( ( LTrn ` K ) ` W ) )
25	15, 22, 22, 24	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) e. ( ( LTrn ` K ) ` W ) )
26	1, 5, 6	tendocl 36055	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ s e. ( ( TEndo ` K ) ` W ) /\ ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) e. ( ( LTrn ` K ) ` W ) ) -> ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) e. ( ( LTrn ` K ) ` W ) )
27	15, 17, 25, 26	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) e. ( ( LTrn ` K ) ` W ) )
28	1, 5	ltrncnv 35432	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) e. ( ( LTrn ` K ) ` W ) ) -> `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) e. ( ( LTrn ` K ) ` W ) )
29	27, 28	syldan 487	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) e. ( ( LTrn ` K ) ` W ) )
30	1, 5	ltrnco 36007	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) /\ `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) e. ( ( LTrn ` K ) ` W ) ) -> ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) e. ( ( LTrn ` K ) ` W ) )
31	15, 16, 29, 30	syl3anc 1326	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) e. ( ( LTrn ` K ) ` W ) )
32		eqid 2622	. . . . . . . . 9 |- ( Base ` K ) = ( Base ` K )
33		eqid 2622	. . . . . . . . 9 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
34	32, 1, 5, 33	trlcl 35451	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( trL ` K ) ` W ) ` ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) ) e. ( Base ` K ) )
35	31, 34	syldan 487	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> ( ( ( trL ` K ) ` W ) ` ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) ) e. ( Base ` K ) )
36		dih1.m	. . . . . . . 8 |- .1. = ( 1. ` K )
37	32, 18, 36	ople1 34478	. . . . . . 7 |- ( ( K e. OP /\ ( ( ( trL ` K ) ` W ) ` ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) ) e. ( Base ` K ) ) -> ( ( ( trL ` K ) ` W ) ` ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) ) ( le ` K ) .1. )
38	14, 35, 37	syl2anc 693	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) -> ( ( ( trL ` K ) ` W ) ` ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) ) ( le ` K ) .1. )
39	38	ex 450	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) -> ( ( ( trL ` K ) ` W ) ` ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) ) ( le ` K ) .1. ) )
40	39	pm4.71d 666	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) <-> ( ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) /\ ( ( ( trL ` K ) ` W ) ` ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) ) ( le ` K ) .1. ) ) )
41	9	eleq2d 2687	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( <. f , s >. e. V <-> <. f , s >. e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) ) )
42		opelxp 5146	. . . . 5 |- ( <. f , s >. e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) <-> ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) )
43	41, 42	syl6bb 276	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( <. f , s >. e. V <-> ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) )
44	13	adantr 481	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> K e. OP )
45	32, 36	op1cl 34472	. . . . . 6 |- ( K e. OP -> .1. e. ( Base ` K ) )
46	44, 45	syl 17	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> .1. e. ( Base ` K ) )
47		hlpos 34652	. . . . . . 7 |- ( K e. HL -> K e. Poset )
48	47	adantr 481	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> K e. Poset )
49	32, 1	lhpbase 35284	. . . . . . 7 |- ( W e. H -> W e. ( Base ` K ) )
50	49	adantl 482	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> W e. ( Base ` K ) )
51		eqid 2622	. . . . . . 7 |- ( <o ` K ) = ( <o ` K )
52	36, 51, 1	lhp1cvr 35285	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> W ( <o ` K ) .1. )
53	32, 18, 51	cvrnle 34567	. . . . . 6 |- ( ( ( K e. Poset /\ W e. ( Base ` K ) /\ .1. e. ( Base ` K ) ) /\ W ( <o ` K ) .1. ) -> -. .1. ( le ` K ) W )
54	48, 50, 46, 52, 53	syl31anc 1329	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> -. .1. ( le ` K ) W )
55		hlol 34648	. . . . . . . 8 |- ( K e. HL -> K e. OL )
56		eqid 2622	. . . . . . . . 9 |- ( meet ` K ) = ( meet ` K )
57	32, 56, 36	olm12 34515	. . . . . . . 8 |- ( ( K e. OL /\ W e. ( Base ` K ) ) -> ( .1. ( meet ` K ) W ) = W )
58	55, 49, 57	syl2an 494	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> ( .1. ( meet ` K ) W ) = W )
59	58	oveq2d 6666	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) ( join ` K ) ( .1. ( meet ` K ) W ) ) = ( ( ( oc ` K ) ` W ) ( join ` K ) W ) )
60		hllat 34650	. . . . . . . 8 |- ( K e. HL -> K e. Lat )
61	60	adantr 481	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> K e. Lat )
62	32, 19	opoccl 34481	. . . . . . . 8 |- ( ( K e. OP /\ W e. ( Base ` K ) ) -> ( ( oc ` K ) ` W ) e. ( Base ` K ) )
63	13, 49, 62	syl2an 494	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> ( ( oc ` K ) ` W ) e. ( Base ` K ) )
64		eqid 2622	. . . . . . . 8 |- ( join ` K ) = ( join ` K )
65	32, 64	latjcom 17059	. . . . . . 7 |- ( ( K e. Lat /\ ( ( oc ` K ) ` W ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( ( oc ` K ) ` W ) ( join ` K ) W ) = ( W ( join ` K ) ( ( oc ` K ) ` W ) ) )
66	61, 63, 50, 65	syl3anc 1326	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) ( join ` K ) W ) = ( W ( join ` K ) ( ( oc ` K ) ` W ) ) )
67	32, 19, 64, 36	opexmid 34494	. . . . . . 7 |- ( ( K e. OP /\ W e. ( Base ` K ) ) -> ( W ( join ` K ) ( ( oc ` K ) ` W ) ) = .1. )
68	13, 49, 67	syl2an 494	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> ( W ( join ` K ) ( ( oc ` K ) ` W ) ) = .1. )
69	59, 66, 68	3eqtrd 2660	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) ( join ` K ) ( .1. ( meet ` K ) W ) ) = .1. )
70		eqid 2622	. . . . . 6 |- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
71		vex 3203	. . . . . 6 |- f e. _V
72		vex 3203	. . . . . 6 |- s e. _V
73	32, 18, 64, 56, 20, 1, 70, 5, 33, 6, 2, 23, 71, 72	dihopelvalc 36538	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( .1. e. ( Base ` K ) /\ -. .1. ( le ` K ) W ) /\ ( ( ( ( oc ` K ) ` W ) e. ( Atoms ` K ) /\ -. ( ( oc ` K ) ` W ) ( le ` K ) W ) /\ ( ( ( oc ` K ) ` W ) ( join ` K ) ( .1. ( meet ` K ) W ) ) = .1. ) ) -> ( <. f , s >. e. ( I ` .1. ) <-> ( ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) /\ ( ( ( trL ` K ) ` W ) ` ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) ) ( le ` K ) .1. ) ) )
74	12, 46, 54, 21, 69, 73	syl122anc 1335	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( <. f , s >. e. ( I ` .1. ) <-> ( ( f e. ( ( LTrn ` K ) ` W ) /\ s e. ( ( TEndo ` K ) ` W ) ) /\ ( ( ( trL ` K ) ` W ) ` ( f o. `' ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) ) ) ) ( le ` K ) .1. ) ) )
75	40, 43, 74	3bitr4rd 301	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( <. f , s >. e. ( I ` .1. ) <-> <. f , s >. e. V ) )
76	75	eqrelrdv2 5219	. 2 |- ( ( ( Rel ( I ` .1. ) /\ Rel V ) /\ ( K e. HL /\ W e. H ) ) -> ( I ` .1. ) = V )
77	3, 11, 12, 76	syl21anc 1325	1 |- ( ( K e. HL /\ W e. H ) -> ( I ` .1. ) = V )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   o. ccom 5118   Rel wrel 5119   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   occoc 15949   Posetcpo 16940   joincjn 16944   meetcmee 16945   1.cp1 17038   Latclat 17045   OPcops 34459   OLcol 34461   <o ccvr 34549   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   TEndoctendo 36040   DVecHcdvh 36367   DIsoHcdih 36517

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-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  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-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  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-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043  df-edring 36045  df-disoa 36318  df-dvech 36368  df-dib 36428  df-dic 36462  df-dih 36518

This theorem is referenced by:  dih1rn  36576  dih1cnv  36577  dihglb2  36631  doch0  36647  dochocss  36655