Theorem djhcvat42 31605

Description: A covering property. (cvrat42 29633 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 445	. . 3 |- ( ph -> K e. HL )
3		djhcvat42.s	. . . 4 |- ( ph -> S e. ran I )
4		eqid 2283	. . . . 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 31461	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ S e. ran I ) -> ( `' I ` S ) e. ( Base ` K ) )
8	1, 3, 7	syl2anc 642	. . 3 |- ( ph -> ( `' I ` S ) e. ( Base ` K ) )
9		djhcvat42.x	. . . . 5 |- ( ph -> X e. ( V \ { .0. } ) )
10		eldifi 3298	. . . . 5 |- ( X e. ( V \ { .0. } ) -> X e. V )
11	9, 10	syl 15	. . . 4 |- ( ph -> X e. V )
12		eldifsni 3750	. . . . 5 |- ( X e. ( V \ { .0. } ) -> X =/= .0. )
13	9, 12	syl 15	. . . 4 |- ( ph -> X =/= .0. )
14		eqid 2283	. . . . 5 |- ( Atoms ` K ) = ( Atoms ` K )
15		djhcvat42.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
16		djhcvat42.v	. . . . 5 |- V = ( Base ` U )
17		djhcvat42.o	. . . . 5 |- .0. = ( 0g ` U )
18		djhcvat42.n	. . . . 5 |- N = ( LSpan ` U )
19	14, 5, 15, 16, 17, 18, 6	dihlspsnat 31523	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ X e. V /\ X =/= .0. ) -> ( `' I ` ( N ` { X } ) ) e. ( Atoms ` K ) )
20	1, 11, 13, 19	syl3anc 1182	. . 3 |- ( ph -> ( `' I ` ( N ` { X } ) ) e. ( Atoms ` K ) )
21		djhcvat42.y	. . . . 5 |- ( ph -> Y e. ( V \ { .0. } ) )
22		eldifi 3298	. . . . 5 |- ( Y e. ( V \ { .0. } ) -> Y e. V )
23	21, 22	syl 15	. . . 4 |- ( ph -> Y e. V )
24		eldifsni 3750	. . . . 5 |- ( Y e. ( V \ { .0. } ) -> Y =/= .0. )
25	21, 24	syl 15	. . . 4 |- ( ph -> Y =/= .0. )
26	14, 5, 15, 16, 17, 18, 6	dihlspsnat 31523	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ Y e. V /\ Y =/= .0. ) -> ( `' I ` ( N ` { Y } ) ) e. ( Atoms ` K ) )
27	1, 23, 25, 26	syl3anc 1182	. . 3 |- ( ph -> ( `' I ` ( N ` { Y } ) ) e. ( Atoms ` K ) )
28		eqid 2283	. . . 4 |- ( le ` K ) = ( le ` K )
29		eqid 2283	. . . 4 |- ( join ` K ) = ( join ` K )
30		eqid 2283	. . . 4 |- ( 0. ` K ) = ( 0. ` K )
31	4, 28, 29, 30, 14	cvrat42 29633	. . 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 } ) ) ) ) ) )
32	2, 8, 20, 27, 31	syl13anc 1184	. 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 } ) ) ) ) ) )
33	5, 30, 6, 15, 16, 17, 18, 1, 3	dih0sb 31475	. . . 4 |- ( ph -> ( S = { .0. } <-> ( `' I ` S ) = ( 0. ` K ) ) )
34	33	necon3bid 2481	. . 3 |- ( ph -> ( S =/= { .0. } <-> ( `' I ` S ) =/= ( 0. ` K ) ) )
35	5, 15, 16, 18, 6	dihlsprn 31521	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ X e. V ) -> ( N ` { X } ) e. ran I )
36	1, 11, 35	syl2anc 642	. . . . 5 |- ( ph -> ( N ` { X } ) e. ran I )
37	5, 15, 6, 16	dihrnss 31468	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ S e. ran I ) -> S C_ V )
38	1, 3, 37	syl2anc 642	. . . . . 6 |- ( ph -> S C_ V )
39	5, 15, 16, 18, 6	dihlsprn 31521	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ Y e. V ) -> ( N ` { Y } ) e. ran I )
40	1, 23, 39	syl2anc 642	. . . . . . 7 |- ( ph -> ( N ` { Y } ) e. ran I )
41	5, 15, 6, 16	dihrnss 31468	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( N ` { Y } ) e. ran I ) -> ( N ` { Y } ) C_ V )
42	1, 40, 41	syl2anc 642	. . . . . 6 |- ( ph -> ( N ` { Y } ) C_ V )
43		djhcvat42.j	. . . . . . 7 |- .\/ = ( (joinH ` K ) ` W )
44	5, 6, 15, 16, 43	djhcl 31590	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ V /\ ( N ` { Y } ) C_ V ) ) -> ( S .\/ ( N ` { Y } ) ) e. ran I )
45	1, 38, 42, 44	syl12anc 1180	. . . . 5 |- ( ph -> ( S .\/ ( N ` { Y } ) ) e. ran I )
46	28, 5, 6, 1, 36, 45	dihcnvord 31464	. . . 4 |- ( ph -> ( ( `' I ` ( N ` { X } ) ) ( le ` K ) ( `' I ` ( S .\/ ( N ` { Y } ) ) ) <-> ( N ` { X } ) C_ ( S .\/ ( N ` { Y } ) ) ) )
47	29, 5, 6, 43, 1, 3, 40	djhj 31594	. . . . 5 |- ( ph -> ( `' I ` ( S .\/ ( N ` { Y } ) ) ) = ( ( `' I ` S ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) )
48	47	breq2d 4035	. . . 4 |- ( ph -> ( ( `' I ` ( N ` { X } ) ) ( le ` K ) ( `' I ` ( S .\/ ( N ` { Y } ) ) ) <-> ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` S ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) )
49	46, 48	bitr3d 246	. . 3 |- ( ph -> ( ( N ` { X } ) C_ ( S .\/ ( N ` { Y } ) ) <-> ( `' I ` ( N ` { X } ) ) ( le ` K ) ( ( `' I ` S ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) )
50	34, 49	anbi12d 691	. 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 } ) ) ) ) ) )
51	1	adantr 451	. . . . 5 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( K e. HL /\ W e. H ) )
52		eldifi 3298	. . . . . 6 |- ( z e. ( V \ { .0. } ) -> z e. V )
53	52	adantl 452	. . . . 5 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> z e. V )
54		eldifsni 3750	. . . . . 6 |- ( z e. ( V \ { .0. } ) -> z =/= .0. )
55	54	adantl 452	. . . . 5 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> z =/= .0. )
56	14, 5, 15, 16, 17, 18, 6	dihlspsnat 31523	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ z e. V /\ z =/= .0. ) -> ( `' I ` ( N ` { z } ) ) e. ( Atoms ` K ) )
57	51, 53, 55, 56	syl3anc 1182	. . . 4 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( `' I ` ( N ` { z } ) ) e. ( Atoms ` K ) )
58	14, 5, 15, 16, 17, 18, 6, 1	dihatexv2 31529	. . . 4 |- ( ph -> ( r e. ( Atoms ` K ) <-> E. z e. ( V \ { .0. } ) r = ( `' I ` ( N ` { z } ) ) ) )
59		breq1 4026	. . . . . 6 |- ( r = ( `' I ` ( N ` { z } ) ) -> ( r ( le ` K ) ( `' I ` S ) <-> ( `' I ` ( N ` { z } ) ) ( le ` K ) ( `' I ` S ) ) )
60		oveq1 5865	. . . . . . 7 |- ( r = ( `' I ` ( N ` { z } ) ) -> ( r ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) = ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) )
61	60	breq2d 4035	. . . . . 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 } ) ) ) ) )
62	59, 61	anbi12d 691	. . . . 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 } ) ) ) ) ) )
63	62	adantl 452	. . . 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 } ) ) ) ) ) )
64	57, 58, 63	rexxfr2d 4551	. . 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 } ) ) ) ) ) )
65	5, 15, 16, 18, 6	dihlsprn 31521	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ z e. V ) -> ( N ` { z } ) e. ran I )
66	51, 53, 65	syl2anc 642	. . . . . 6 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { z } ) e. ran I )
67	3	adantr 451	. . . . . 6 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> S e. ran I )
68	28, 5, 6, 51, 66, 67	dihcnvord 31464	. . . . 5 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( ( `' I ` ( N ` { z } ) ) ( le ` K ) ( `' I ` S ) <-> ( N ` { z } ) C_ S ) )
69	40	adantr 451	. . . . . . . 8 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { Y } ) e. ran I )
70	29, 5, 6, 43, 51, 66, 69	djhj 31594	. . . . . . 7 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( `' I ` ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) = ( ( `' I ` ( N ` { z } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) )
71	70	breq2d 4035	. . . . . 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 } ) ) ) ) )
72	11	adantr 451	. . . . . . . 8 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> X e. V )
73	51, 72, 35	syl2anc 642	. . . . . . 7 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { X } ) e. ran I )
74	5, 15, 6, 16	dihrnss 31468	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( N ` { z } ) e. ran I ) -> ( N ` { z } ) C_ V )
75	51, 66, 74	syl2anc 642	. . . . . . . 8 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { z } ) C_ V )
76	42	adantr 451	. . . . . . . 8 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( N ` { Y } ) C_ V )
77	5, 6, 15, 16, 43	djhcl 31590	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( N ` { z } ) C_ V /\ ( N ` { Y } ) C_ V ) ) -> ( ( N ` { z } ) .\/ ( N ` { Y } ) ) e. ran I )
78	51, 75, 76, 77	syl12anc 1180	. . . . . . 7 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( ( N ` { z } ) .\/ ( N ` { Y } ) ) e. ran I )
79	28, 5, 6, 51, 73, 78	dihcnvord 31464	. . . . . 6 |- ( ( ph /\ z e. ( V \ { .0. } ) ) -> ( ( `' I ` ( N ` { X } ) ) ( le ` K ) ( `' I ` ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) <-> ( N ` { X } ) C_ ( ( N ` { z } ) .\/ ( N ` { Y } ) ) ) )
80	71, 79	bitr3d 246	. . . . 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 } ) ) ) )
81	68, 80	anbi12d 691	. . . 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 } ) ) ) ) )
82	81	rexbidva 2560	. . 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 } ) ) ) ) )
83	64, 82	bitr2d 245	. 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 } ) ) ) ) ) )
84	32, 50, 83	3imtr4d 259	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 176   /\ wa 358   = wceq 1623   e. wcel 1684   =/= wne 2446   E.wrex 2544   \ cdif 3149   C_ wss 3152   {csn 3640   class class class wbr 4023   `'ccnv 4688   ran crn 4690   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   0gc0g 13400   joincjn 14078   0.cp0 14143   LSpanclspn 15728   Atomscatm 29453   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268   DIsoHcdih 31418  joinHcdjh 31584

This theorem is referenced by:  dihjat1lem  31618

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-edring 30946  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585