Theorem dvhvaddass 36386

Description: Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)

Hypotheses

Ref	Expression
dvhvaddcl.h	|- H = ( LHyp ` K )
dvhvaddcl.t	|- T = ( ( LTrn ` K ) ` W )
dvhvaddcl.e	|- E = ( ( TEndo ` K ) ` W )
dvhvaddcl.u	|- U = ( ( DVecH ` K ) ` W )
dvhvaddcl.d	|- D = (Scalar ` U )
dvhvaddcl.p	|- .+^ = ( +g ` D )
dvhvaddcl.a	|- .+ = ( +g ` U )

Assertion

Ref	Expression
dvhvaddass	|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( F .+ G ) .+ I ) = ( F .+ ( G .+ I ) ) )

Proof of Theorem dvhvaddass

Step	Hyp	Ref	Expression
1		coass 5654	. . . 4 |- ( ( ( 1st ` F ) o. ( 1st ` G ) ) o. ( 1st ` I ) ) = ( ( 1st ` F ) o. ( ( 1st ` G ) o. ( 1st ` I ) ) )
2		dvhvaddcl.h	. . . . . . . . 9 |- H = ( LHyp ` K )
3		dvhvaddcl.t	. . . . . . . . 9 |- T = ( ( LTrn ` K ) ` W )
4		dvhvaddcl.e	. . . . . . . . 9 |- E = ( ( TEndo ` K ) ` W )
5		dvhvaddcl.u	. . . . . . . . 9 |- U = ( ( DVecH ` K ) ` W )
6		dvhvaddcl.d	. . . . . . . . 9 |- D = (Scalar ` U )
7		dvhvaddcl.a	. . . . . . . . 9 |- .+ = ( +g ` U )
8		dvhvaddcl.p	. . . . . . . . 9 |- .+^ = ( +g ` D )
9	2, 3, 4, 5, 6, 7, 8	dvhvadd 36381	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( F .+ G ) = <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. )
10	9	3adantr3 1222	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( F .+ G ) = <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. )
11	10	fveq2d 6195	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( 1st ` ( F .+ G ) ) = ( 1st ` <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. ) )
12		fvex 6201	. . . . . . . 8 |- ( 1st ` F ) e. _V
13		fvex 6201	. . . . . . . 8 |- ( 1st ` G ) e. _V
14	12, 13	coex 7118	. . . . . . 7 |- ( ( 1st ` F ) o. ( 1st ` G ) ) e. _V
15		ovex 6678	. . . . . . 7 |- ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) e. _V
16	14, 15	op1st 7176	. . . . . 6 |- ( 1st ` <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. ) = ( ( 1st ` F ) o. ( 1st ` G ) )
17	11, 16	syl6eq 2672	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( 1st ` ( F .+ G ) ) = ( ( 1st ` F ) o. ( 1st ` G ) ) )
18	17	coeq1d 5283	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( 1st ` ( F .+ G ) ) o. ( 1st ` I ) ) = ( ( ( 1st ` F ) o. ( 1st ` G ) ) o. ( 1st ` I ) ) )
19	2, 3, 4, 5, 6, 7, 8	dvhvadd 36381	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( G .+ I ) = <. ( ( 1st ` G ) o. ( 1st ` I ) ) , ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) >. )
20	19	3adantr1 1220	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( G .+ I ) = <. ( ( 1st ` G ) o. ( 1st ` I ) ) , ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) >. )
21	20	fveq2d 6195	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( 1st ` ( G .+ I ) ) = ( 1st ` <. ( ( 1st ` G ) o. ( 1st ` I ) ) , ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) >. ) )
22		fvex 6201	. . . . . . . 8 |- ( 1st ` I ) e. _V
23	13, 22	coex 7118	. . . . . . 7 |- ( ( 1st ` G ) o. ( 1st ` I ) ) e. _V
24		ovex 6678	. . . . . . 7 |- ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) e. _V
25	23, 24	op1st 7176	. . . . . 6 |- ( 1st ` <. ( ( 1st ` G ) o. ( 1st ` I ) ) , ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) >. ) = ( ( 1st ` G ) o. ( 1st ` I ) )
26	21, 25	syl6eq 2672	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( 1st ` ( G .+ I ) ) = ( ( 1st ` G ) o. ( 1st ` I ) ) )
27	26	coeq2d 5284	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( 1st ` F ) o. ( 1st ` ( G .+ I ) ) ) = ( ( 1st ` F ) o. ( ( 1st ` G ) o. ( 1st ` I ) ) ) )
28	1, 18, 27	3eqtr4a 2682	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( 1st ` ( F .+ G ) ) o. ( 1st ` I ) ) = ( ( 1st ` F ) o. ( 1st ` ( G .+ I ) ) ) )
29		xp2nd 7199	. . . . . 6 |- ( F e. ( T X. E ) -> ( 2nd ` F ) e. E )
30		xp2nd 7199	. . . . . 6 |- ( G e. ( T X. E ) -> ( 2nd ` G ) e. E )
31		xp2nd 7199	. . . . . 6 |- ( I e. ( T X. E ) -> ( 2nd ` I ) e. E )
32	29, 30, 31	3anim123i 1247	. . . . 5 |- ( ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) -> ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) )
33		eqid 2622	. . . . . . . . . 10 |- ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
34	2, 33, 5, 6	dvhsca 36371	. . . . . . . . 9 |- ( ( K e. HL /\ W e. H ) -> D = ( ( EDRing ` K ) ` W ) )
35	2, 33	erngdv 36281	. . . . . . . . 9 |- ( ( K e. HL /\ W e. H ) -> ( ( EDRing ` K ) ` W ) e. DivRing )
36	34, 35	eqeltrd 2701	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> D e. DivRing )
37		drnggrp 18755	. . . . . . . 8 |- ( D e. DivRing -> D e. Grp )
38	36, 37	syl 17	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> D e. Grp )
39	38	adantr 481	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) ) -> D e. Grp )
40		simpr1 1067	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) ) -> ( 2nd ` F ) e. E )
41		eqid 2622	. . . . . . . . 9 |- ( Base ` D ) = ( Base ` D )
42	2, 4, 5, 6, 41	dvhbase 36372	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> ( Base ` D ) = E )
43	42	adantr 481	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) ) -> ( Base ` D ) = E )
44	40, 43	eleqtrrd 2704	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) ) -> ( 2nd ` F ) e. ( Base ` D ) )
45		simpr2 1068	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) ) -> ( 2nd ` G ) e. E )
46	45, 43	eleqtrrd 2704	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) ) -> ( 2nd ` G ) e. ( Base ` D ) )
47		simpr3 1069	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) ) -> ( 2nd ` I ) e. E )
48	47, 43	eleqtrrd 2704	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) ) -> ( 2nd ` I ) e. ( Base ` D ) )
49	41, 8	grpass 17431	. . . . . 6 |- ( ( D e. Grp /\ ( ( 2nd ` F ) e. ( Base ` D ) /\ ( 2nd ` G ) e. ( Base ` D ) /\ ( 2nd ` I ) e. ( Base ` D ) ) ) -> ( ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) .+^ ( 2nd ` I ) ) = ( ( 2nd ` F ) .+^ ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) ) )
50	39, 44, 46, 48, 49	syl13anc 1328	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` F ) e. E /\ ( 2nd ` G ) e. E /\ ( 2nd ` I ) e. E ) ) -> ( ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) .+^ ( 2nd ` I ) ) = ( ( 2nd ` F ) .+^ ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) ) )
51	32, 50	sylan2 491	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) .+^ ( 2nd ` I ) ) = ( ( 2nd ` F ) .+^ ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) ) )
52	10	fveq2d 6195	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( 2nd ` ( F .+ G ) ) = ( 2nd ` <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. ) )
53	14, 15	op2nd 7177	. . . . . 6 |- ( 2nd ` <. ( ( 1st ` F ) o. ( 1st ` G ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) >. ) = ( ( 2nd ` F ) .+^ ( 2nd ` G ) )
54	52, 53	syl6eq 2672	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( 2nd ` ( F .+ G ) ) = ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) )
55	54	oveq1d 6665	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( 2nd ` ( F .+ G ) ) .+^ ( 2nd ` I ) ) = ( ( ( 2nd ` F ) .+^ ( 2nd ` G ) ) .+^ ( 2nd ` I ) ) )
56	20	fveq2d 6195	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( 2nd ` ( G .+ I ) ) = ( 2nd ` <. ( ( 1st ` G ) o. ( 1st ` I ) ) , ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) >. ) )
57	23, 24	op2nd 7177	. . . . . 6 |- ( 2nd ` <. ( ( 1st ` G ) o. ( 1st ` I ) ) , ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) >. ) = ( ( 2nd ` G ) .+^ ( 2nd ` I ) )
58	56, 57	syl6eq 2672	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( 2nd ` ( G .+ I ) ) = ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) )
59	58	oveq2d 6666	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( 2nd ` F ) .+^ ( 2nd ` ( G .+ I ) ) ) = ( ( 2nd ` F ) .+^ ( ( 2nd ` G ) .+^ ( 2nd ` I ) ) ) )
60	51, 55, 59	3eqtr4d 2666	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( 2nd ` ( F .+ G ) ) .+^ ( 2nd ` I ) ) = ( ( 2nd ` F ) .+^ ( 2nd ` ( G .+ I ) ) ) )
61	28, 60	opeq12d 4410	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> <. ( ( 1st ` ( F .+ G ) ) o. ( 1st ` I ) ) , ( ( 2nd ` ( F .+ G ) ) .+^ ( 2nd ` I ) ) >. = <. ( ( 1st ` F ) o. ( 1st ` ( G .+ I ) ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` ( G .+ I ) ) ) >. )
62		simpl 473	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( K e. HL /\ W e. H ) )
63	2, 3, 4, 5, 6, 8, 7	dvhvaddcl 36384	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) ) ) -> ( F .+ G ) e. ( T X. E ) )
64	63	3adantr3 1222	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( F .+ G ) e. ( T X. E ) )
65		simpr3 1069	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> I e. ( T X. E ) )
66	2, 3, 4, 5, 6, 7, 8	dvhvadd 36381	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( F .+ G ) e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( F .+ G ) .+ I ) = <. ( ( 1st ` ( F .+ G ) ) o. ( 1st ` I ) ) , ( ( 2nd ` ( F .+ G ) ) .+^ ( 2nd ` I ) ) >. )
67	62, 64, 65, 66	syl12anc 1324	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( F .+ G ) .+ I ) = <. ( ( 1st ` ( F .+ G ) ) o. ( 1st ` I ) ) , ( ( 2nd ` ( F .+ G ) ) .+^ ( 2nd ` I ) ) >. )
68		simpr1 1067	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> F e. ( T X. E ) )
69	2, 3, 4, 5, 6, 8, 7	dvhvaddcl 36384	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( G .+ I ) e. ( T X. E ) )
70	69	3adantr1 1220	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( G .+ I ) e. ( T X. E ) )
71	2, 3, 4, 5, 6, 7, 8	dvhvadd 36381	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ ( G .+ I ) e. ( T X. E ) ) ) -> ( F .+ ( G .+ I ) ) = <. ( ( 1st ` F ) o. ( 1st ` ( G .+ I ) ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` ( G .+ I ) ) ) >. )
72	62, 68, 70, 71	syl12anc 1324	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( F .+ ( G .+ I ) ) = <. ( ( 1st ` F ) o. ( 1st ` ( G .+ I ) ) ) , ( ( 2nd ` F ) .+^ ( 2nd ` ( G .+ I ) ) ) >. )
73	61, 67, 72	3eqtr4d 2666	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. ( T X. E ) /\ G e. ( T X. E ) /\ I e. ( T X. E ) ) ) -> ( ( F .+ G ) .+ I ) = ( F .+ ( G .+ I ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   Grpcgrp 17422   DivRingcdr 18747   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   EDRingcedring 36041   DVecHcdvh 36367

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-grp 17425  df-minusg 17426  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-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-dvech 36368

This theorem is referenced by:  dvhgrp  36396