Theorem mapdpglem30 36991

Description: Lemma for mapdpg 36995. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 36990, using lvecindp2 19139) that v = 1 and v = u...". TODO: would it be shorter to have only the v = ( 1r ` A ) part and use mapdpglem28.u2 in mapdpglem31 36992? (Contributed by NM, 22-Mar-2015.)

Hypotheses

Ref	Expression
mapdpg.h	|- H = ( LHyp ` K )
mapdpg.m	|- M = ( (mapd ` K ) ` W )
mapdpg.u	|- U = ( ( DVecH ` K ) ` W )
mapdpg.v	|- V = ( Base ` U )
mapdpg.s	|- .- = ( -g ` U )
mapdpg.z	|- .0. = ( 0g ` U )
mapdpg.n	|- N = ( LSpan ` U )
mapdpg.c	|- C = ( (LCDual ` K ) ` W )
mapdpg.f	|- F = ( Base ` C )
mapdpg.r	|- R = ( -g ` C )
mapdpg.j	|- J = ( LSpan ` C )
mapdpg.k	|- ( ph -> ( K e. HL /\ W e. H ) )
mapdpg.x	|- ( ph -> X e. ( V \ { .0. } ) )
mapdpg.y	|- ( ph -> Y e. ( V \ { .0. } ) )
mapdpg.g	|- ( ph -> G e. F )
mapdpg.ne	|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
mapdpg.e	|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
mapdpgem25.h1	|- ( ph -> ( h e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) ) )
mapdpgem25.i1	|- ( ph -> ( i e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) )
mapdpglem26.a	|- A = (Scalar ` U )
mapdpglem26.b	|- B = ( Base ` A )
mapdpglem26.t	|- .x. = ( .s ` C )
mapdpglem26.o	|- O = ( 0g ` A )
mapdpglem28.ve	|- ( ph -> v e. B )
mapdpglem28.u1	|- ( ph -> h = ( u .x. i ) )
mapdpglem28.u2	|- ( ph -> ( G R h ) = ( v .x. ( G R i ) ) )
mapdpglem28.ue	|- ( ph -> u e. B )

Assertion

Ref	Expression
mapdpglem30	|- ( ph -> ( v = ( 1r ` A ) /\ v = u ) )

Distinct variable groups:   h, i, u, v   u, B, v   u, C, v   u, O, v   u, .x. , v   v, G   v, R

Allowed substitution hints:   ph( v, u, h, i)   A( v, u, h, i)   B( h, i)   C( h, i)   R( u, h, i)   .x. ( h, i)   U( v, u, h, i)   F( v, u, h, i)   G( u, h, i)   H( v, u, h, i)   J( v, u, h, i)   K( v, u, h, i)   M( v, u, h, i)   .- ( v, u, h, i)   N( v, u, h, i)   O( h, i)   V( v, u, h, i)   W( v, u, h, i)   X( v, u, h, i)   Y( v, u, h, i)   .0. ( v, u, h, i)

Proof of Theorem mapdpglem30

Step	Hyp	Ref	Expression
1		mapdpg.f	. . 3 |- F = ( Base ` C )
2		eqid 2622	. . 3 |- ( +g ` C ) = ( +g ` C )
3		eqid 2622	. . 3 |- (Scalar ` C ) = (Scalar ` C )
4		eqid 2622	. . 3 |- ( Base ` (Scalar ` C ) ) = ( Base ` (Scalar ` C ) )
5		mapdpglem26.t	. . 3 |- .x. = ( .s ` C )
6		eqid 2622	. . 3 |- ( 0g ` C ) = ( 0g ` C )
7		mapdpg.j	. . 3 |- J = ( LSpan ` C )
8		mapdpg.h	. . . 4 |- H = ( LHyp ` K )
9		mapdpg.c	. . . 4 |- C = ( (LCDual ` K ) ` W )
10		mapdpg.k	. . . 4 |- ( ph -> ( K e. HL /\ W e. H ) )
11	8, 9, 10	lcdlvec 36880	. . 3 |- ( ph -> C e. LVec )
12		mapdpg.g	. . . 4 |- ( ph -> G e. F )
13		mapdpg.m	. . . . 5 |- M = ( (mapd ` K ) ` W )
14		mapdpg.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
15		mapdpg.v	. . . . 5 |- V = ( Base ` U )
16		mapdpg.s	. . . . 5 |- .- = ( -g ` U )
17		mapdpg.z	. . . . 5 |- .0. = ( 0g ` U )
18		mapdpg.n	. . . . 5 |- N = ( LSpan ` U )
19		mapdpg.r	. . . . 5 |- R = ( -g ` C )
20		mapdpg.x	. . . . 5 |- ( ph -> X e. ( V \ { .0. } ) )
21		mapdpg.y	. . . . 5 |- ( ph -> Y e. ( V \ { .0. } ) )
22		mapdpg.ne	. . . . 5 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
23		mapdpg.e	. . . . 5 |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
24	8, 13, 14, 15, 16, 17, 18, 9, 1, 19, 7, 10, 20, 21, 12, 22, 23	mapdpglem30a 36984	. . . 4 |- ( ph -> G =/= ( 0g ` C ) )
25		eldifsn 4317	. . . 4 |- ( G e. ( F \ { ( 0g ` C ) } ) <-> ( G e. F /\ G =/= ( 0g ` C ) ) )
26	12, 24, 25	sylanbrc 698	. . 3 |- ( ph -> G e. ( F \ { ( 0g ` C ) } ) )
27		mapdpgem25.i1	. . . . 5 |- ( ph -> ( i e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) )
28	27	simpld 475	. . . 4 |- ( ph -> i e. F )
29		mapdpgem25.h1	. . . . 5 |- ( ph -> ( h e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) ) )
30	8, 13, 14, 15, 16, 17, 18, 9, 1, 19, 7, 10, 20, 21, 12, 22, 23, 29, 27	mapdpglem30b 36985	. . . 4 |- ( ph -> i =/= ( 0g ` C ) )
31		eldifsn 4317	. . . 4 |- ( i e. ( F \ { ( 0g ` C ) } ) <-> ( i e. F /\ i =/= ( 0g ` C ) ) )
32	28, 30, 31	sylanbrc 698	. . 3 |- ( ph -> i e. ( F \ { ( 0g ` C ) } ) )
33		mapdpglem28.ve	. . . 4 |- ( ph -> v e. B )
34		mapdpglem26.a	. . . . 5 |- A = (Scalar ` U )
35		mapdpglem26.b	. . . . 5 |- B = ( Base ` A )
36	8, 14, 34, 35, 9, 3, 4, 10	lcdsbase 36889	. . . 4 |- ( ph -> ( Base ` (Scalar ` C ) ) = B )
37	33, 36	eleqtrrd 2704	. . 3 |- ( ph -> v e. ( Base ` (Scalar ` C ) ) )
38	8, 14, 10	dvhlmod 36399	. . . . . 6 |- ( ph -> U e. LMod )
39	34	lmodring 18871	. . . . . 6 |- ( U e. LMod -> A e. Ring )
40	38, 39	syl 17	. . . . 5 |- ( ph -> A e. Ring )
41		ringgrp 18552	. . . . . . 7 |- ( A e. Ring -> A e. Grp )
42	40, 41	syl 17	. . . . . 6 |- ( ph -> A e. Grp )
43		eqid 2622	. . . . . . . 8 |- ( 1r ` A ) = ( 1r ` A )
44	35, 43	ringidcl 18568	. . . . . . 7 |- ( A e. Ring -> ( 1r ` A ) e. B )
45	40, 44	syl 17	. . . . . 6 |- ( ph -> ( 1r ` A ) e. B )
46		eqid 2622	. . . . . . 7 |- ( invg ` A ) = ( invg ` A )
47	35, 46	grpinvcl 17467	. . . . . 6 |- ( ( A e. Grp /\ ( 1r ` A ) e. B ) -> ( ( invg ` A ) ` ( 1r ` A ) ) e. B )
48	42, 45, 47	syl2anc 693	. . . . 5 |- ( ph -> ( ( invg ` A ) ` ( 1r ` A ) ) e. B )
49		eqid 2622	. . . . . 6 |- ( .r ` A ) = ( .r ` A )
50	35, 49	ringcl 18561	. . . . 5 |- ( ( A e. Ring /\ v e. B /\ ( ( invg ` A ) ` ( 1r ` A ) ) e. B ) -> ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. B )
51	40, 33, 48, 50	syl3anc 1326	. . . 4 |- ( ph -> ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. B )
52	51, 36	eleqtrrd 2704	. . 3 |- ( ph -> ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. ( Base ` (Scalar ` C ) ) )
53	45, 36	eleqtrrd 2704	. . 3 |- ( ph -> ( 1r ` A ) e. ( Base ` (Scalar ` C ) ) )
54		mapdpglem28.ue	. . . . 5 |- ( ph -> u e. B )
55	35, 49	ringcl 18561	. . . . 5 |- ( ( A e. Ring /\ u e. B /\ ( ( invg ` A ) ` ( 1r ` A ) ) e. B ) -> ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. B )
56	40, 54, 48, 55	syl3anc 1326	. . . 4 |- ( ph -> ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. B )
57	56, 36	eleqtrrd 2704	. . 3 |- ( ph -> ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. ( Base ` (Scalar ` C ) ) )
58		mapdpglem26.o	. . . 4 |- O = ( 0g ` A )
59		mapdpglem28.u1	. . . 4 |- ( ph -> h = ( u .x. i ) )
60		mapdpglem28.u2	. . . 4 |- ( ph -> ( G R h ) = ( v .x. ( G R i ) ) )
61	8, 13, 14, 15, 16, 17, 18, 9, 1, 19, 7, 10, 20, 21, 12, 22, 23, 29, 27, 34, 35, 5, 58, 33, 59, 60	mapdpglem29 36989	. . 3 |- ( ph -> ( J ` { G } ) =/= ( J ` { i } ) )
62	8, 14, 34, 35, 49, 9, 1, 5, 10, 48, 54, 28	lcdvsass 36896	. . . . 5 |- ( ph -> ( ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) = ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( u .x. i ) ) )
63	62	oveq2d 6666	. . . 4 |- ( ph -> ( ( ( 1r ` A ) .x. G ) ( +g ` C ) ( ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) = ( ( ( 1r ` A ) .x. G ) ( +g ` C ) ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( u .x. i ) ) ) )
64	8, 14, 34, 35, 9, 1, 5, 10, 45, 12	lcdvscl 36894	. . . . 5 |- ( ph -> ( ( 1r ` A ) .x. G ) e. F )
65	8, 14, 34, 35, 9, 1, 5, 10, 54, 28	lcdvscl 36894	. . . . 5 |- ( ph -> ( u .x. i ) e. F )
66	8, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 64, 65	lcdvsub 36906	. . . 4 |- ( ph -> ( ( ( 1r ` A ) .x. G ) R ( u .x. i ) ) = ( ( ( 1r ` A ) .x. G ) ( +g ` C ) ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( u .x. i ) ) ) )
67	8, 14, 34, 35, 49, 9, 1, 5, 10, 48, 33, 28	lcdvsass 36896	. . . . . 6 |- ( ph -> ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) = ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( v .x. i ) ) )
68	67	oveq2d 6666	. . . . 5 |- ( ph -> ( ( v .x. G ) ( +g ` C ) ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) = ( ( v .x. G ) ( +g ` C ) ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( v .x. i ) ) ) )
69	8, 14, 34, 35, 9, 1, 5, 10, 33, 12	lcdvscl 36894	. . . . . 6 |- ( ph -> ( v .x. G ) e. F )
70	8, 14, 34, 35, 9, 1, 5, 10, 33, 28	lcdvscl 36894	. . . . . 6 |- ( ph -> ( v .x. i ) e. F )
71	8, 14, 34, 46, 43, 9, 1, 2, 5, 19, 10, 69, 70	lcdvsub 36906	. . . . 5 |- ( ph -> ( ( v .x. G ) R ( v .x. i ) ) = ( ( v .x. G ) ( +g ` C ) ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( v .x. i ) ) ) )
72	8, 13, 14, 15, 16, 17, 18, 9, 1, 19, 7, 10, 20, 21, 12, 22, 23, 29, 27, 34, 35, 5, 58, 33, 59, 60	mapdpglem28 36990	. . . . . 6 |- ( ph -> ( ( v .x. G ) R ( v .x. i ) ) = ( G R ( u .x. i ) ) )
73		eqid 2622	. . . . . . . . . 10 |- ( 1r ` (Scalar ` C ) ) = ( 1r ` (Scalar ` C ) )
74	8, 14, 34, 43, 9, 3, 73, 10	lcd1 36898	. . . . . . . . 9 |- ( ph -> ( 1r ` (Scalar ` C ) ) = ( 1r ` A ) )
75	74	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( 1r ` (Scalar ` C ) ) .x. G ) = ( ( 1r ` A ) .x. G ) )
76	8, 9, 10	lcdlmod 36881	. . . . . . . . 9 |- ( ph -> C e. LMod )
77	1, 3, 5, 73	lmodvs1 18891	. . . . . . . . 9 |- ( ( C e. LMod /\ G e. F ) -> ( ( 1r ` (Scalar ` C ) ) .x. G ) = G )
78	76, 12, 77	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( 1r ` (Scalar ` C ) ) .x. G ) = G )
79	75, 78	eqtr3d 2658	. . . . . . 7 |- ( ph -> ( ( 1r ` A ) .x. G ) = G )
80	79	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( 1r ` A ) .x. G ) R ( u .x. i ) ) = ( G R ( u .x. i ) ) )
81	72, 80	eqtr4d 2659	. . . . 5 |- ( ph -> ( ( v .x. G ) R ( v .x. i ) ) = ( ( ( 1r ` A ) .x. G ) R ( u .x. i ) ) )
82	68, 71, 81	3eqtr2rd 2663	. . . 4 |- ( ph -> ( ( ( 1r ` A ) .x. G ) R ( u .x. i ) ) = ( ( v .x. G ) ( +g ` C ) ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) )
83	63, 66, 82	3eqtr2rd 2663	. . 3 |- ( ph -> ( ( v .x. G ) ( +g ` C ) ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) = ( ( ( 1r ` A ) .x. G ) ( +g ` C ) ( ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) )
84	1, 2, 3, 4, 5, 6, 7, 11, 26, 32, 37, 52, 53, 57, 61, 83	lvecindp2 19139	. 2 |- ( ph -> ( v = ( 1r ` A ) /\ ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) ) )
85	35, 49, 43, 46, 40, 33	rngnegr 18595	. . . . 5 |- ( ph -> ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( ( invg ` A ) ` v ) )
86	35, 49, 43, 46, 40, 54	rngnegr 18595	. . . . 5 |- ( ph -> ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( ( invg ` A ) ` u ) )
87	85, 86	eqeq12d 2637	. . . 4 |- ( ph -> ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) <-> ( ( invg ` A ) ` v ) = ( ( invg ` A ) ` u ) ) )
88	35, 46, 42, 33, 54	grpinv11 17484	. . . 4 |- ( ph -> ( ( ( invg ` A ) ` v ) = ( ( invg ` A ) ` u ) <-> v = u ) )
89	87, 88	bitrd 268	. . 3 |- ( ph -> ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) <-> v = u ) )
90	89	anbi2d 740	. 2 |- ( ph -> ( ( v = ( 1r ` A ) /\ ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) ) <-> ( v = ( 1r ` A ) /\ v = u ) ) )
91	84, 90	mpbid 222	1 |- ( ph -> ( v = ( 1r ` A ) /\ v = u ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424   1rcur 18501   Ringcrg 18547   LModclmod 18863   LSpanclspn 18971   HLchlt 34637   LHypclh 35270   DVecHcdvh 36367  LCDualclcd 36875  mapdcmpd 36913

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-of 6897  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-mre 16246  df-mrc 16247  df-acs 16249  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-oppg 17776  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-lsatoms 34263  df-lshyp 34264  df-lcv 34306  df-lfl 34345  df-lkr 34373  df-ldual 34411  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-tgrp 36031  df-tendo 36043  df-edring 36045  df-dveca 36291  df-disoa 36318  df-dvech 36368  df-dib 36428  df-dic 36462  df-dih 36518  df-doch 36637  df-djh 36684  df-lcdual 36876  df-mapd 36914

This theorem is referenced by:  mapdpglem31  36992