Theorem djhcvat42 32311

Description: A covering property. (cvrat42 30339 analog.) (Contributed by NM, 17-Aug-2014.)

Hypotheses

Ref	Expression
djhcvat42.h	|- H = ( LHyp ` K )
djhcvat42.u	|- U = ( ( DVecH ` K ) ` W )
djhcvat42.v	|- V = ( Base ` U )
djhcvat42.o	|- .0. = ( 0g ` U )
djhcvat42.n	|- N = ( LSpan ` U )
djhcvat42.i	|- I = ( ( DIsoH ` K ) ` W )
djhcvat42.j	|- .\/ = ( (joinH ` K ) ` W )
djhcvat42.k	|- ( ph -> ( K e. HL /\ W e. H ) )
djhcvat42.s	|- ( ph -> S e. ran I )
djhcvat42.x	|- ( ph -> X e. ( V \ { .0. } ) )
djhcvat42.y	|- ( ph -> Y e. ( V \ { .0. } ) )

Assertion

Ref	Expression
djhcvat42	|- ( ph -> ( ( S =/= { .0. } /\ ( N ` { X } ) C_ ( S .\/ ( N ` { Y } ) ) ) -> E. z e. ( V \ { .0. } ) ( ( N ` { z } ) C_ S /\ ( N ` { X } ) C_ ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) ) )

Distinct variable groups:   z, I   z, K   z, N   ph, z   z, W   z, S   z, V   z, X   z, Y

Allowed substitution hints:   U( z)   H( z)   .\/ ( z)   .0. ( z)

Proof of Theorem djhcvat42

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djhcvat42.k	. . . 4 |- ( ph -> ( K e. HL /\ W e. H ) )
2	1	simpld 447	. . 3 |- ( ph -> K e. HL )
3		djhcvat42.s	. . . 4 |- ( ph -> S e. ran I )
4		eqid 2442	. . . . 5 |- ( Base ` K ) = ( Base ` K )
5		djhcvat42.h	. . . . 5 |- H = ( LHyp ` K )
6		djhcvat42.i	. . . . 5 |- I = ( ( DIsoH ` K ) ` W )
7	4, 5, 6	dihcnvcl 32167	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ S e. ran I ) -> ( `' I ` S ) e. ( Base ` K ) )
8	1, 3, 7	syl2anc 644	. . 3 |- ( ph -> ( `' I ` S ) e. ( Base ` K ) )
9		djhcvat42.x	. . . . 5 |- ( ph -> X e. ( V \ { .0. } ) )
10	9	eldifad 3318	. . . 4 |- ( ph -> X e. V )
11		eldifsni 3952	. . . . 5 |- ( X e. ( V \ { .0. } ) -> X =/= .0. )
12	9, 11	syl 16	. . . 4 |- ( ph -> X =/= .0. )
13		eqid 2442	. . . . 5 |- ( Atoms ` K ) = ( Atoms ` K )
14		djhcvat42.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
15		djhcvat42.v	. . . . 5 |- V = ( Base ` U )
16		djhcvat42.o	. . . . 5 |- .0. = ( 0g ` U )
17		djhcvat42.n	. . . . 5 |- N = ( LSpan ` U )
18	13, 5, 14, 15, 16, 17, 6	dihlspsnat 32229	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ X e. V /\ X =/= .0. ) -> ( `' I ` ( N ` { X } ) ) e. ( Atoms ` K ) )
19	1, 10, 12, 18	syl3anc 1185	. . 3 |- ( ph -> ( `' I ` ( N ` { X } ) ) e. ( Atoms ` K ) )
20		djhcvat42.y	. . . . 5 |- ( ph -> Y e. ( V \ { .0. } ) )
21	20	eldifad 3318	. . . 4 |- ( ph -> Y e. V )
22		eldifsni 3952	. . . . 5 |- ( Y e. ( V \ { .0. } ) -> Y =/= .0. )
23	20, 22	syl 16	. . . 4 |- ( ph -> Y =/= .0. )
24	13, 5, 14, 15, 16, 17, 6	dihlspsnat 32229	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ Y e. V /\ Y =/= .0. ) -> ( `' I ` ( N ` { Y } ) ) e. ( Atoms ` K ) )
25	1, 21, 23, 24	syl3anc 1185	. . 3 |- ( ph -> ( `' I ` ( N ` { Y } ) ) e. ( Atoms ` K ) )
26		eqid 2442	. . . 4 |- ( le ` K ) = ( le ` K )
27		eqid 2442	. . . 4 |- ( join ` K ) = ( join ` K )
28		eqid 2442	. . . 4 |- ( 0. ` K ) = ( 0. ` K )
29	4, 26, 27, 28, 13	cvrat42 30339	. . 3 |- ( ( K e. HL /\ ( ( `' I ` S ) e. ( Base ` K ) /\ ( `' I ` ( N ` { X } ) ) e. ( Atoms ` K ) /\ ( `' I ` ( N ` { Y } ) ) e. ( Atoms ` K ) ) ) -> ( ( ( `' I ` S ) =/= ( 0. ` K ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` S ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) -> E. r e. ( Atoms ` K ) ( r ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( r ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) ) )
30	2, 8, 19, 25, 29	syl13anc 1187	. 2 |- ( ph -> ( ( ( `' I ` S ) =/= ( 0. ` K ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` S ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) -> E. r e. ( Atoms ` K ) ( r ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( r ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) ) )
31	5, 28, 6, 14, 15, 16, 17, 1, 3	dih0sb 32181	. . . 4 |- ( ph -> ( S = { .0. } <-> ( `' I ` S ) = ( 0. ` K ) ) )
32	31	necon3bid 2642	. . 3 |- ( ph -> ( S =/= { .0. } <-> ( `' I ` S ) =/= ( 0. ` K ) ) )
33	5, 14, 15, 17, 6	dihlsprn 32227	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ X e. V ) -> ( N ` { X } ) e. ran I )
34	1, 10, 33	syl2anc 644	. . . . 5 |- ( ph -> ( N ` { X } ) e. ran I )
35	5, 14, 6, 15	dihrnss 32174	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ S e. ran I ) -> S C_ V )
36	1, 3, 35	syl2anc 644	. . . . . 6 |- ( ph -> S C_ V )
37	5, 14, 15, 17, 6	dihlsprn 32227	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ Y e. V ) -> ( N ` { Y } ) e. ran I )
38	1, 21, 37	syl2anc 644	. . . . . . 7 |- ( ph -> ( N ` { Y } ) e. ran I )
39	5, 14, 6, 15	dihrnss 32174	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( N ` { Y } ) e. ran I ) -> ( N ` { Y } ) C_ V )
40	1, 38, 39	syl2anc 644	. . . . . 6 |- ( ph -> ( N ` { Y } ) C_ V )
41		djhcvat42.j	. . . . . . 7 |- .\/ = ( (joinH ` K ) ` W )
42	5, 6, 14, 15, 41	djhcl 32296	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ V /\ ( N ` { Y } ) C_ V ) ) -> ( S .\/ ( N ` { Y } ) ) e. ran I )
43	1, 36, 40, 42	syl12anc 1183	. . . . 5 |- ( ph -> ( S .\/ ( N ` { Y } ) ) e. ran I )
44	26, 5, 6, 1, 34, 43	dihcnvord 32170	. . . 4 |- ( ph -> ( ( `' I ` ( N ` { X } ) ) ( le ` K ) ( `' I ` ( S .\/ ( N ` { Y } ) ) ) <-> ( N ` { X } ) C_ ( S .\/ ( N ` { Y } ) ) ) )
45	27, 5, 6, 41, 1, 3, 38	djhj 32300	. . . . 5 |- ( ph -> ( `' I ` ( S .\/ ( N ` { Y } ) ) ) = ( ( `' I ` S ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) )
46	45	breq2d 4249	. . . 4 |- ( ph -> ( ( `' I ` ( N ` { X } ) ) ( le ` K ) ( `' I ` ( S .\/ ( N ` { Y } ) ) ) <-> ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` S ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) )
47	44, 46	bitr3d 248	. . 3 |- ( ph -> ( ( N ` { X } ) C_ ( S .\/ ( N ` { Y } ) ) <-> ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` S ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) )
48	32, 47	anbi12d 693	. 2 |- ( ph -> ( ( S =/= { .0. } /\ ( N ` { X } ) C_ ( S .\/ ( N ` { Y } ) ) ) <-> ( ( `' I ` S ) =/= ( 0. ` K ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` S ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) ) )
49	1	adantr 453	. . . . 5 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( K e. HL /\ W e. H ) )
50		eldifi 3455	. . . . . 6 |- ( z e. ( V \ { .0. } ) -> z e. V )
51	50	adantl 454	. . . . 5 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> z e. V )
52		eldifsni 3952	. . . . . 6 |- ( z e. ( V \ { .0. } ) -> z =/= .0. )
53	52	adantl 454	. . . . 5 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> z =/= .0. )
54	13, 5, 14, 15, 16, 17, 6	dihlspsnat 32229	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ z e. V /\ z =/= .0. ) -> ( `' I ` ( N ` { z } ) ) e. ( Atoms ` K ) )
55	49, 51, 53, 54	syl3anc 1185	. . . 4 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( `' I ` ( N ` { z } ) ) e. ( Atoms ` K ) )
56	13, 5, 14, 15, 16, 17, 6, 1	dihatexv2 32235	. . . 4 |- ( ph -> ( r e. ( Atoms ` K ) <-> E. z e. ( V \ { .0. } ) r = ( `' I ` ( N ` { z } ) ) ) )
57		breq1 4240	. . . . . 6 |- ( r = ( `' I ` ( N ` { z } ) ) -> ( r ( le ` K ) ( `' I ` S ) <-> ( `' I ` ( N ` { z } ) ) ( le ` K ) ( `' I ` S ) ) )
58		oveq1 6117	. . . . . . 7 |- ( r = ( `' I ` ( N ` { z } ) ) -> ( r ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) = ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) )
59	58	breq2d 4249	. . . . . 6 |- ( r = ( `' I ` ( N ` { z } ) ) -> ( ( `' I ` ( N ` { X } ) ) ( le ` K ) ( r ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) <-> ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) )
60	57, 59	anbi12d 693	. . . . 5 |- ( r = ( `' I ` ( N ` { z } ) ) -> ( ( r ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( r ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) <-> ( ( `' I ` ( N ` { z } ) ) ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) ) )
61	60	adantl 454	. . . 4 |- ( ( ph /\ r = ( `' I ` ( N ` { z } ) ) ) -> ( ( r ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( r ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) <-> ( ( `' I ` ( N ` { z } ) ) ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) ) )
62	55, 56, 61	rexxfr2d 4769	. . 3 |- ( ph -> ( E. r e. ( Atoms ` K ) ( r ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( r ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) <-> E. z e. ( V \ { .0. } ) ( ( `' I ` ( N ` { z } ) ) ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) ) )
63	5, 14, 15, 17, 6	dihlsprn 32227	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ z e. V ) -> ( N ` { z } ) e. ran I )
64	49, 51, 63	syl2anc 644	. . . . . 6 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { z } ) e. ran I )
65	3	adantr 453	. . . . . 6 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> S e. ran I )
66	26, 5, 6, 49, 64, 65	dihcnvord 32170	. . . . 5 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( ( `' I ` ( N ` { z } ) ) ( le ` K ) ( `' I ` S ) <-> ( N ` { z } ) C_ S ) )
67	38	adantr 453	. . . . . . . 8 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { Y } ) e. ran I )
68	27, 5, 6, 41, 49, 64, 67	djhj 32300	. . . . . . 7 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( `' I ` ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) = ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) )
69	68	breq2d 4249	. . . . . 6 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( ( `' I ` ( N ` { X } ) ) ( le ` K ) ( `' I ` ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) <-> ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) )
70	10	adantr 453	. . . . . . . 8 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> X e. V )
71	49, 70, 33	syl2anc 644	. . . . . . 7 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { X } ) e. ran I )
72	5, 14, 6, 15	dihrnss 32174	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( N ` { z } ) e. ran I ) -> ( N ` { z } ) C_ V )
73	49, 64, 72	syl2anc 644	. . . . . . . 8 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { z } ) C_ V )
74	40	adantr 453	. . . . . . . 8 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { Y } ) C_ V )
75	5, 6, 14, 15, 41	djhcl 32296	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( N ` { z } ) C_ V /\ ( N ` { Y } ) C_ V ) ) -> ( ( N ` { z } ) .\/ ( N ` { Y } ) ) e. ran I )
76	49, 73, 74, 75	syl12anc 1183	. . . . . . 7 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( ( N ` { z } ) .\/ ( N ` { Y } ) ) e. ran I )
77	26, 5, 6, 49, 71, 76	dihcnvord 32170	. . . . . 6 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( ( `' I ` ( N ` { X } ) ) ( le ` K ) ( `' I ` ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) <-> ( N ` { X } ) C_ ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) )
78	69, 77	bitr3d 248	. . . . 5 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) <-> ( N ` { X } ) C_ ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) )
79	66, 78	anbi12d 693	. . . 4 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( ( ( `' I ` ( N ` { z } ) ) ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) <-> ( ( N ` { z } ) C_ S /\ ( N ` { X } ) C_ ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) ) )
80	79	rexbidva 2728	. . 3 |- ( ph -> ( E. z e. ( V \ { .0. } ) ( ( `' I ` ( N ` { z } ) ) ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) <-> E. z e. ( V \ { .0. } ) ( ( N ` { z } ) C_ S /\ ( N ` { X } ) C_ ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) ) )
81	62, 80	bitr2d 247	. 2 |- ( ph -> ( E. z e. ( V \ { .0. } ) ( ( N ` { z } ) C_ S /\ ( N ` { X } ) C_ ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) <-> E. r e. ( Atoms ` K ) ( r ( le ` K ) ( `' I ` S ) /\ ( `' I ` ( N ` { X } ) ) ( le ` K ) ( r ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) ) )
82	30, 48, 81	3imtr4d 261	1 |- ( ph -> ( ( S =/= { .0. } /\ ( N ` { X } ) C_ ( S .\/ ( N ` { Y } ) ) ) -> E. z e. ( V \ { .0. } ) ( ( N ` { z } ) C_ S /\ ( N ` { X } ) C_ ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 178   /\ wa 360   = wceq 1653   e. wcel 1727   =/= wne 2605   E.wrex 2712   \ cdif 3303   C_ wss 3306   {csn 3838   class class class wbr 4237   `'ccnv 4906   ran crn 4908   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lecple 13567   0gc0g 13754   joincjn 14432   0.cp0 14497   LSpanclspn 16078   Atomscatm 30159   HLchlt 30246   LHypclh 30879   DVecHcdvh 31974   DIsoHcdih 32124  joinHcdjh 32290

This theorem is referenced by:  dihjat1lem  32324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-tpos 6508  df-undef 6572  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-0g 13758  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-p1 14500  df-lat 14506  df-clat 14568  df-mnd 14721  df-submnd 14770  df-grp 14843  df-minusg 14844  df-sbg 14845  df-subg 14972  df-cntz 15147  df-lsm 15301  df-cmn 15445  df-abl 15446  df-mgp 15680  df-rng 15694  df-ur 15696  df-oppr 15759  df-dvdsr 15777  df-unit 15778  df-invr 15808  df-dvr 15819  df-drng 15868  df-lmod 15983  df-lss 16040  df-lsp 16079  df-lvec 16206  df-lsatoms 29872  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-llines 30393  df-lplanes 30394  df-lvols 30395  df-lines 30396  df-psubsp 30398  df-pmap 30399  df-padd 30691  df-lhyp 30883  df-laut 30884  df-ldil 30999  df-ltrn 31000  df-trl 31054  df-tendo 31650  df-edring 31652  df-disoa 31925  df-dvech 31975  df-dib 32035  df-dic 32069  df-dih 32125  df-doch 32244  df-djh 32291