Theorem mapdheq4lem 37020

Description: Lemma for mapdheq4 37021. Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)

Hypotheses

Ref	Expression
mapdh.q	|- Q = ( 0g ` C )
mapdh.i	|- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
mapdh.h	|- H = ( LHyp ` K )
mapdh.m	|- M = ( (mapd ` K ) ` W )
mapdh.u	|- U = ( ( DVecH ` K ) ` W )
mapdh.v	|- V = ( Base ` U )
mapdh.s	|- .- = ( -g ` U )
mapdhc.o	|- .0. = ( 0g ` U )
mapdh.n	|- N = ( LSpan ` U )
mapdh.c	|- C = ( (LCDual ` K ) ` W )
mapdh.d	|- D = ( Base ` C )
mapdh.r	|- R = ( -g ` C )
mapdh.j	|- J = ( LSpan ` C )
mapdh.k	|- ( ph -> ( K e. HL /\ W e. H ) )
mapdhc.f	|- ( ph -> F e. D )
mapdh.mn	|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
mapdhcl.x	|- ( ph -> X e. ( V \ { .0. } ) )
mapdhe4.y	|- ( ph -> Y e. ( V \ { .0. } ) )
mapdhe.z	|- ( ph -> Z e. ( V \ { .0. } ) )
mapdh.xn	|- ( ph -> -. X e. ( N ` { Y , Z } ) )
mapdh.yz	|- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
mapdh.eg	|- ( ph -> ( I ` <. X , F , Y >. ) = G )
mapdh.ee	|- ( ph -> ( I ` <. X , F , Z >. ) = E )

Assertion

Ref	Expression
mapdheq4lem	|- ( ph -> ( M ` ( N ` { ( Y .- Z ) } ) ) = ( J ` { ( G R E ) } ) )

Distinct variable groups:   x, D, h   h, F, x   x, J   x, M   x, N   x, .0.   x, Q   x, R   x, .-   h, X, x   h, Y, x   ph, h   .0. , h   C, h   D, h   h, J   h, M   h, N   R, h   U, h   .- , h   h, G, x   h, E   h, Z, x

Allowed substitution hints:   ph( x)   C( x)   Q( h)   U( x)   E( x)   H( x, h)   I( x, h)   K( x, h)   V( x, h)   W( x, h)

Proof of Theorem mapdheq4lem

Step	Hyp	Ref	Expression
1		mapdh.h	. . . 4 |- H = ( LHyp ` K )
2		mapdh.m	. . . 4 |- M = ( (mapd ` K ) ` W )
3		mapdh.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
4		eqid 2622	. . . 4 |- ( LSubSp ` U ) = ( LSubSp ` U )
5		mapdh.k	. . . 4 |- ( ph -> ( K e. HL /\ W e. H ) )
6	1, 3, 5	dvhlmod 36399	. . . . 5 |- ( ph -> U e. LMod )
7		mapdhe4.y	. . . . . . 7 |- ( ph -> Y e. ( V \ { .0. } ) )
8	7	eldifad 3586	. . . . . 6 |- ( ph -> Y e. V )
9		mapdh.v	. . . . . . 7 |- V = ( Base ` U )
10		mapdh.n	. . . . . . 7 |- N = ( LSpan ` U )
11	9, 4, 10	lspsncl 18977	. . . . . 6 |- ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
12	6, 8, 11	syl2anc 693	. . . . 5 |- ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
13		mapdhe.z	. . . . . . 7 |- ( ph -> Z e. ( V \ { .0. } ) )
14	13	eldifad 3586	. . . . . 6 |- ( ph -> Z e. V )
15	9, 4, 10	lspsncl 18977	. . . . . 6 |- ( ( U e. LMod /\ Z e. V ) -> ( N ` { Z } ) e. ( LSubSp ` U ) )
16	6, 14, 15	syl2anc 693	. . . . 5 |- ( ph -> ( N ` { Z } ) e. ( LSubSp ` U ) )
17		eqid 2622	. . . . . 6 |- ( LSSum ` U ) = ( LSSum ` U )
18	4, 17	lsmcl 19083	. . . . 5 |- ( ( U e. LMod /\ ( N ` { Y } ) e. ( LSubSp ` U ) /\ ( N ` { Z } ) e. ( LSubSp ` U ) ) -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) e. ( LSubSp ` U ) )
19	6, 12, 16, 18	syl3anc 1326	. . . 4 |- ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) e. ( LSubSp ` U ) )
20		mapdhcl.x	. . . . . . . 8 |- ( ph -> X e. ( V \ { .0. } ) )
21	20	eldifad 3586	. . . . . . 7 |- ( ph -> X e. V )
22		mapdh.s	. . . . . . . 8 |- .- = ( -g ` U )
23	9, 22	lmodvsubcl 18908	. . . . . . 7 |- ( ( U e. LMod /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V )
24	6, 21, 8, 23	syl3anc 1326	. . . . . 6 |- ( ph -> ( X .- Y ) e. V )
25	9, 4, 10	lspsncl 18977	. . . . . 6 |- ( ( U e. LMod /\ ( X .- Y ) e. V ) -> ( N ` { ( X .- Y ) } ) e. ( LSubSp ` U ) )
26	6, 24, 25	syl2anc 693	. . . . 5 |- ( ph -> ( N ` { ( X .- Y ) } ) e. ( LSubSp ` U ) )
27	9, 22	lmodvsubcl 18908	. . . . . . 7 |- ( ( U e. LMod /\ X e. V /\ Z e. V ) -> ( X .- Z ) e. V )
28	6, 21, 14, 27	syl3anc 1326	. . . . . 6 |- ( ph -> ( X .- Z ) e. V )
29	9, 4, 10	lspsncl 18977	. . . . . 6 |- ( ( U e. LMod /\ ( X .- Z ) e. V ) -> ( N ` { ( X .- Z ) } ) e. ( LSubSp ` U ) )
30	6, 28, 29	syl2anc 693	. . . . 5 |- ( ph -> ( N ` { ( X .- Z ) } ) e. ( LSubSp ` U ) )
31	4, 17	lsmcl 19083	. . . . 5 |- ( ( U e. LMod /\ ( N ` { ( X .- Y ) } ) e. ( LSubSp ` U ) /\ ( N ` { ( X .- Z ) } ) e. ( LSubSp ` U ) ) -> ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) e. ( LSubSp ` U ) )
32	6, 26, 30, 31	syl3anc 1326	. . . 4 |- ( ph -> ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) e. ( LSubSp ` U ) )
33	1, 2, 3, 4, 5, 19, 32	mapdin 36951	. . 3 |- ( ph -> ( M ` ( ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) i^i ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) ) ) = ( ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) i^i ( M ` ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) ) ) )
34		mapdh.c	. . . . . 6 |- C = ( (LCDual ` K ) ` W )
35		eqid 2622	. . . . . 6 |- ( LSSum ` C ) = ( LSSum ` C )
36	1, 2, 3, 4, 17, 34, 35, 5, 12, 16	mapdlsm 36953	. . . . 5 |- ( ph -> ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) = ( ( M ` ( N ` { Y } ) ) ( LSSum ` C ) ( M ` ( N ` { Z } ) ) ) )
37		mapdh.eg	. . . . . . . 8 |- ( ph -> ( I ` <. X , F , Y >. ) = G )
38		mapdh.q	. . . . . . . . 9 |- Q = ( 0g ` C )
39		mapdh.i	. . . . . . . . 9 |- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
40		mapdhc.o	. . . . . . . . 9 |- .0. = ( 0g ` U )
41		mapdh.d	. . . . . . . . 9 |- D = ( Base ` C )
42		mapdh.r	. . . . . . . . 9 |- R = ( -g ` C )
43		mapdh.j	. . . . . . . . 9 |- J = ( LSpan ` C )
44		mapdhc.f	. . . . . . . . 9 |- ( ph -> F e. D )
45		mapdh.mn	. . . . . . . . 9 |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
46	1, 3, 5	dvhlvec 36398	. . . . . . . . . . . . 13 |- ( ph -> U e. LVec )
47		mapdh.yz	. . . . . . . . . . . . 13 |- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
48		mapdh.xn	. . . . . . . . . . . . 13 |- ( ph -> -. X e. ( N ` { Y , Z } ) )
49	9, 40, 10, 46, 8, 13, 21, 47, 48	lspindp2 19135	. . . . . . . . . . . 12 |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ -. Z e. ( N ` { X , Y } ) ) )
50	49	simpld 475	. . . . . . . . . . 11 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
51	38, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 8, 50	mapdhcl 37016	. . . . . . . . . 10 |- ( ph -> ( I ` <. X , F , Y >. ) e. D )
52	37, 51	eqeltrrd 2702	. . . . . . . . 9 |- ( ph -> G e. D )
53	38, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 7, 52, 50	mapdheq 37017	. . . . . . . 8 |- ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )
54	37, 53	mpbid 222	. . . . . . 7 |- ( ph -> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) )
55	54	simpld 475	. . . . . 6 |- ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
56		mapdh.ee	. . . . . . . 8 |- ( ph -> ( I ` <. X , F , Z >. ) = E )
57	9, 40, 10, 46, 7, 14, 21, 47, 48	lspindp1 19133	. . . . . . . . . . . 12 |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Z } ) /\ -. Y e. ( N ` { X , Z } ) ) )
58	57	simpld 475	. . . . . . . . . . 11 |- ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
59	38, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 14, 58	mapdhcl 37016	. . . . . . . . . 10 |- ( ph -> ( I ` <. X , F , Z >. ) e. D )
60	56, 59	eqeltrrd 2702	. . . . . . . . 9 |- ( ph -> E e. D )
61	38, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 13, 60, 58	mapdheq 37017	. . . . . . . 8 |- ( ph -> ( ( I ` <. X , F , Z >. ) = E <-> ( ( M ` ( N ` { Z } ) ) = ( J ` { E } ) /\ ( M ` ( N ` { ( X .- Z ) } ) ) = ( J ` { ( F R E ) } ) ) ) )
62	56, 61	mpbid 222	. . . . . . 7 |- ( ph -> ( ( M ` ( N ` { Z } ) ) = ( J ` { E } ) /\ ( M ` ( N ` { ( X .- Z ) } ) ) = ( J ` { ( F R E ) } ) ) )
63	62	simpld 475	. . . . . 6 |- ( ph -> ( M ` ( N ` { Z } ) ) = ( J ` { E } ) )
64	55, 63	oveq12d 6668	. . . . 5 |- ( ph -> ( ( M ` ( N ` { Y } ) ) ( LSSum ` C ) ( M ` ( N ` { Z } ) ) ) = ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) )
65	36, 64	eqtrd 2656	. . . 4 |- ( ph -> ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) = ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) )
66	1, 2, 3, 4, 17, 34, 35, 5, 26, 30	mapdlsm 36953	. . . . 5 |- ( ph -> ( M ` ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) ) = ( ( M ` ( N ` { ( X .- Y ) } ) ) ( LSSum ` C ) ( M ` ( N ` { ( X .- Z ) } ) ) ) )
67	54	simprd 479	. . . . . 6 |- ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) )
68	62	simprd 479	. . . . . 6 |- ( ph -> ( M ` ( N ` { ( X .- Z ) } ) ) = ( J ` { ( F R E ) } ) )
69	67, 68	oveq12d 6668	. . . . 5 |- ( ph -> ( ( M ` ( N ` { ( X .- Y ) } ) ) ( LSSum ` C ) ( M ` ( N ` { ( X .- Z ) } ) ) ) = ( ( J ` { ( F R G ) } ) ( LSSum ` C ) ( J ` { ( F R E ) } ) ) )
70	66, 69	eqtrd 2656	. . . 4 |- ( ph -> ( M ` ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) ) = ( ( J ` { ( F R G ) } ) ( LSSum ` C ) ( J ` { ( F R E ) } ) ) )
71	65, 70	ineq12d 3815	. . 3 |- ( ph -> ( ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) i^i ( M ` ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) ) ) = ( ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) i^i ( ( J ` { ( F R G ) } ) ( LSSum ` C ) ( J ` { ( F R E ) } ) ) ) )
72	33, 71	eqtrd 2656	. 2 |- ( ph -> ( M ` ( ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) i^i ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) ) ) = ( ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) i^i ( ( J ` { ( F R G ) } ) ( LSSum ` C ) ( J ` { ( F R E ) } ) ) ) )
73	9, 22, 40, 17, 10, 46, 21, 48, 47, 7, 13	baerlem3 37002	. . 3 |- ( ph -> ( N ` { ( Y .- Z ) } ) = ( ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) i^i ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) ) )
74	73	fveq2d 6195	. 2 |- ( ph -> ( M ` ( N ` { ( Y .- Z ) } ) ) = ( M ` ( ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) i^i ( ( N ` { ( X .- Y ) } ) ( LSSum ` U ) ( N ` { ( X .- Z ) } ) ) ) ) )
75		eqid 2622	. . 3 |- ( 0g ` C ) = ( 0g ` C )
76	1, 34, 5	lcdlvec 36880	. . 3 |- ( ph -> C e. LVec )
77	1, 2, 3, 9, 10, 34, 41, 43, 5, 44, 45, 21, 8, 52, 55, 14, 60, 63, 48	mapdindp 36960	. . 3 |- ( ph -> -. F e. ( J ` { G , E } ) )
78	1, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 8, 14, 60, 63, 47	mapdncol 36959	. . 3 |- ( ph -> ( J ` { G } ) =/= ( J ` { E } ) )
79	1, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 40, 75, 7	mapdn0 36958	. . 3 |- ( ph -> G e. ( D \ { ( 0g ` C ) } ) )
80	1, 2, 3, 9, 10, 34, 41, 43, 5, 60, 63, 40, 75, 13	mapdn0 36958	. . 3 |- ( ph -> E e. ( D \ { ( 0g ` C ) } ) )
81	41, 42, 75, 35, 43, 76, 44, 77, 78, 79, 80	baerlem3 37002	. 2 |- ( ph -> ( J ` { ( G R E ) } ) = ( ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) i^i ( ( J ` { ( F R G ) } ) ( LSSum ` C ) ( J ` { ( F R E ) } ) ) ) )
82	72, 74, 81	3eqtr4d 2666	1 |- ( ph -> ( M ` ( N ` { ( Y .- Z ) } ) ) = ( J ` { ( G R E ) } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   i^i cin 3573   ifcif 4086   {csn 4177   {cpr 4179   <.cotp 4185   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   0gc0g 16100   -gcsg 17424   LSSumclsm 18049   LModclmod 18863   LSubSpclss 18932   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-ot 4186  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:  mapdheq4  37021