Theorem archiabllem1a 29745

Description: Lemma for archiabl 29752: In case an archimedean group W admits a smallest positive element U, then any positive element X of W can be written as ( n .x. U ) with n e. NN. Since the reciprocal holds for negative elements, W is then isomorphic to ZZ. (Contributed by Thierry Arnoux, 12-Apr-2018.)

Hypotheses

Ref	Expression
archiabllem.b	|- B = ( Base ` W )
archiabllem.0	|- .0. = ( 0g ` W )
archiabllem.e	|- .<_ = ( le ` W )
archiabllem.t	|- .< = ( lt ` W )
archiabllem.m	|- .x. = (.g ` W )
archiabllem.g	|- ( ph -> W e. oGrp )
archiabllem.a	|- ( ph -> W e. Archi )
archiabllem1.u	|- ( ph -> U e. B )
archiabllem1.p	|- ( ph -> .0. .< U )
archiabllem1.s	|- ( ( ph /\ x e. B /\ .0. .< x ) -> U .<_ x )
archiabllem1a.x	|- ( ph -> X e. B )
archiabllem1a.c	|- ( ph -> .0. .< X )

Assertion

Ref	Expression
archiabllem1a	|- ( ph -> E. n e. NN X = ( n .x. U ) )

Distinct variable groups:   x, n, B   U, n, x   n, W, x   n, X, x   ph, n, x   .x. , n, x   .0. , n, x   .< , n, x   x, .<_

Allowed substitution hint:   .<_ ( n)

Proof of Theorem archiabllem1a

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 792	. . . 4 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> m e. NN0 )
2		nn0p1nn 11332	. . . 4 |- ( m e. NN0 -> ( m + 1 ) e. NN )
3	1, 2	syl 17	. . 3 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( m + 1 ) e. NN )
4		archiabllem1.u	. . . . . . . 8 |- ( ph -> U e. B )
5	4	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> U e. B )
6		archiabllem.b	. . . . . . . 8 |- B = ( Base ` W )
7		archiabllem.m	. . . . . . . 8 |- .x. = (.g ` W )
8	6, 7	mulg1 17548	. . . . . . 7 |- ( U e. B -> ( 1 .x. U ) = U )
9	5, 8	syl 17	. . . . . 6 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( 1 .x. U ) = U )
10	9	oveq1d 6665	. . . . 5 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( 1 .x. U ) ( +g ` W ) ( m .x. U ) ) = ( U ( +g ` W ) ( m .x. U ) ) )
11		archiabllem.g	. . . . . . . 8 |- ( ph -> W e. oGrp )
12	11	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> W e. oGrp )
13		ogrpgrp 29703	. . . . . . 7 |- ( W e. oGrp -> W e. Grp )
14	12, 13	syl 17	. . . . . 6 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> W e. Grp )
15		1zzd 11408	. . . . . 6 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> 1 e. ZZ )
16	1	nn0zd 11480	. . . . . 6 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> m e. ZZ )
17		eqid 2622	. . . . . . 7 |- ( +g ` W ) = ( +g ` W )
18	6, 7, 17	mulgdir 17573	. . . . . 6 |- ( ( W e. Grp /\ ( 1 e. ZZ /\ m e. ZZ /\ U e. B ) ) -> ( ( 1 + m ) .x. U ) = ( ( 1 .x. U ) ( +g ` W ) ( m .x. U ) ) )
19	14, 15, 16, 5, 18	syl13anc 1328	. . . . 5 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( 1 + m ) .x. U ) = ( ( 1 .x. U ) ( +g ` W ) ( m .x. U ) ) )
20		isogrp 29702	. . . . . . . . . 10 |- ( W e. oGrp <-> ( W e. Grp /\ W e. oMnd ) )
21	20	simprbi 480	. . . . . . . . 9 |- ( W e. oGrp -> W e. oMnd )
22		omndtos 29705	. . . . . . . . 9 |- ( W e. oMnd -> W e. Toset )
23		tospos 29658	. . . . . . . . 9 |- ( W e. Toset -> W e. Poset )
24	21, 22, 23	3syl 18	. . . . . . . 8 |- ( W e. oGrp -> W e. Poset )
25	12, 24	syl 17	. . . . . . 7 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> W e. Poset )
26		archiabllem1a.x	. . . . . . . . 9 |- ( ph -> X e. B )
27	26	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> X e. B )
28	6, 7	mulgcl 17559	. . . . . . . . 9 |- ( ( W e. Grp /\ m e. ZZ /\ U e. B ) -> ( m .x. U ) e. B )
29	14, 16, 5, 28	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( m .x. U ) e. B )
30		eqid 2622	. . . . . . . . 9 |- ( -g ` W ) = ( -g ` W )
31	6, 30	grpsubcl 17495	. . . . . . . 8 |- ( ( W e. Grp /\ X e. B /\ ( m .x. U ) e. B ) -> ( X ( -g ` W ) ( m .x. U ) ) e. B )
32	14, 27, 29, 31	syl3anc 1326	. . . . . . 7 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( X ( -g ` W ) ( m .x. U ) ) e. B )
33	16	peano2zd 11485	. . . . . . . . . 10 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( m + 1 ) e. ZZ )
34	6, 7	mulgcl 17559	. . . . . . . . . 10 |- ( ( W e. Grp /\ ( m + 1 ) e. ZZ /\ U e. B ) -> ( ( m + 1 ) .x. U ) e. B )
35	14, 33, 5, 34	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( m + 1 ) .x. U ) e. B )
36		simprr 796	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> X .<_ ( ( m + 1 ) .x. U ) )
37		archiabllem.e	. . . . . . . . . 10 |- .<_ = ( le ` W )
38	6, 37, 30	ogrpsub 29717	. . . . . . . . 9 |- ( ( W e. oGrp /\ ( X e. B /\ ( ( m + 1 ) .x. U ) e. B /\ ( m .x. U ) e. B ) /\ X .<_ ( ( m + 1 ) .x. U ) ) -> ( X ( -g ` W ) ( m .x. U ) ) .<_ ( ( ( m + 1 ) .x. U ) ( -g ` W ) ( m .x. U ) ) )
39	12, 27, 35, 29, 36, 38	syl131anc 1339	. . . . . . . 8 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( X ( -g ` W ) ( m .x. U ) ) .<_ ( ( ( m + 1 ) .x. U ) ( -g ` W ) ( m .x. U ) ) )
40	1	nn0cnd 11353	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> m e. CC )
41		1cnd 10056	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> 1 e. CC )
42	40, 41	pncan2d 10394	. . . . . . . . . 10 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( m + 1 ) - m ) = 1 )
43	42	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( ( m + 1 ) - m ) .x. U ) = ( 1 .x. U ) )
44	6, 7, 30	mulgsubdir 17582	. . . . . . . . . 10 |- ( ( W e. Grp /\ ( ( m + 1 ) e. ZZ /\ m e. ZZ /\ U e. B ) ) -> ( ( ( m + 1 ) - m ) .x. U ) = ( ( ( m + 1 ) .x. U ) ( -g ` W ) ( m .x. U ) ) )
45	14, 33, 16, 5, 44	syl13anc 1328	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( ( m + 1 ) - m ) .x. U ) = ( ( ( m + 1 ) .x. U ) ( -g ` W ) ( m .x. U ) ) )
46	43, 45, 9	3eqtr3d 2664	. . . . . . . 8 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( ( m + 1 ) .x. U ) ( -g ` W ) ( m .x. U ) ) = U )
47	39, 46	breqtrd 4679	. . . . . . 7 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( X ( -g ` W ) ( m .x. U ) ) .<_ U )
48		archiabllem1.s	. . . . . . . . . . 11 |- ( ( ph /\ x e. B /\ .0. .< x ) -> U .<_ x )
49	48	3expia 1267	. . . . . . . . . 10 |- ( ( ph /\ x e. B ) -> ( .0. .< x -> U .<_ x ) )
50	49	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. x e. B ( .0. .< x -> U .<_ x ) )
51	50	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> A. x e. B ( .0. .< x -> U .<_ x ) )
52		archiabllem.0	. . . . . . . . . . 11 |- .0. = ( 0g ` W )
53	6, 52, 30	grpsubid 17499	. . . . . . . . . 10 |- ( ( W e. Grp /\ ( m .x. U ) e. B ) -> ( ( m .x. U ) ( -g ` W ) ( m .x. U ) ) = .0. )
54	14, 29, 53	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( m .x. U ) ( -g ` W ) ( m .x. U ) ) = .0. )
55		simprl 794	. . . . . . . . . 10 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( m .x. U ) .< X )
56		archiabllem.t	. . . . . . . . . . 11 |- .< = ( lt ` W )
57	6, 56, 30	ogrpsublt 29722	. . . . . . . . . 10 |- ( ( W e. oGrp /\ ( ( m .x. U ) e. B /\ X e. B /\ ( m .x. U ) e. B ) /\ ( m .x. U ) .< X ) -> ( ( m .x. U ) ( -g ` W ) ( m .x. U ) ) .< ( X ( -g ` W ) ( m .x. U ) ) )
58	12, 29, 27, 29, 55, 57	syl131anc 1339	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( m .x. U ) ( -g ` W ) ( m .x. U ) ) .< ( X ( -g ` W ) ( m .x. U ) ) )
59	54, 58	eqbrtrrd 4677	. . . . . . . 8 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> .0. .< ( X ( -g ` W ) ( m .x. U ) ) )
60		breq2 4657	. . . . . . . . . 10 |- ( x = ( X ( -g ` W ) ( m .x. U ) ) -> ( .0. .< x <-> .0. .< ( X ( -g ` W ) ( m .x. U ) ) ) )
61		breq2 4657	. . . . . . . . . 10 |- ( x = ( X ( -g ` W ) ( m .x. U ) ) -> ( U .<_ x <-> U .<_ ( X ( -g ` W ) ( m .x. U ) ) ) )
62	60, 61	imbi12d 334	. . . . . . . . 9 |- ( x = ( X ( -g ` W ) ( m .x. U ) ) -> ( ( .0. .< x -> U .<_ x ) <-> ( .0. .< ( X ( -g ` W ) ( m .x. U ) ) -> U .<_ ( X ( -g ` W ) ( m .x. U ) ) ) ) )
63	62	rspcv 3305	. . . . . . . 8 |- ( ( X ( -g ` W ) ( m .x. U ) ) e. B -> ( A. x e. B ( .0. .< x -> U .<_ x ) -> ( .0. .< ( X ( -g ` W ) ( m .x. U ) ) -> U .<_ ( X ( -g ` W ) ( m .x. U ) ) ) ) )
64	32, 51, 59, 63	syl3c 66	. . . . . . 7 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> U .<_ ( X ( -g ` W ) ( m .x. U ) ) )
65	6, 37	posasymb 16952	. . . . . . . 8 |- ( ( W e. Poset /\ ( X ( -g ` W ) ( m .x. U ) ) e. B /\ U e. B ) -> ( ( ( X ( -g ` W ) ( m .x. U ) ) .<_ U /\ U .<_ ( X ( -g ` W ) ( m .x. U ) ) ) <-> ( X ( -g ` W ) ( m .x. U ) ) = U ) )
66	65	biimpa 501	. . . . . . 7 |- ( ( ( W e. Poset /\ ( X ( -g ` W ) ( m .x. U ) ) e. B /\ U e. B ) /\ ( ( X ( -g ` W ) ( m .x. U ) ) .<_ U /\ U .<_ ( X ( -g ` W ) ( m .x. U ) ) ) ) -> ( X ( -g ` W ) ( m .x. U ) ) = U )
67	25, 32, 5, 47, 64, 66	syl32anc 1334	. . . . . 6 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( X ( -g ` W ) ( m .x. U ) ) = U )
68	67	oveq1d 6665	. . . . 5 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( X ( -g ` W ) ( m .x. U ) ) ( +g ` W ) ( m .x. U ) ) = ( U ( +g ` W ) ( m .x. U ) ) )
69	10, 19, 68	3eqtr4rd 2667	. . . 4 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( X ( -g ` W ) ( m .x. U ) ) ( +g ` W ) ( m .x. U ) ) = ( ( 1 + m ) .x. U ) )
70	6, 17, 30	grpnpcan 17507	. . . . 5 |- ( ( W e. Grp /\ X e. B /\ ( m .x. U ) e. B ) -> ( ( X ( -g ` W ) ( m .x. U ) ) ( +g ` W ) ( m .x. U ) ) = X )
71	14, 27, 29, 70	syl3anc 1326	. . . 4 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( X ( -g ` W ) ( m .x. U ) ) ( +g ` W ) ( m .x. U ) ) = X )
72	41, 40	addcomd 10238	. . . . 5 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( 1 + m ) = ( m + 1 ) )
73	72	oveq1d 6665	. . . 4 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> ( ( 1 + m ) .x. U ) = ( ( m + 1 ) .x. U ) )
74	69, 71, 73	3eqtr3d 2664	. . 3 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> X = ( ( m + 1 ) .x. U ) )
75		oveq1 6657	. . . . 5 |- ( n = ( m + 1 ) -> ( n .x. U ) = ( ( m + 1 ) .x. U ) )
76	75	eqeq2d 2632	. . . 4 |- ( n = ( m + 1 ) -> ( X = ( n .x. U ) <-> X = ( ( m + 1 ) .x. U ) ) )
77	76	rspcev 3309	. . 3 |- ( ( ( m + 1 ) e. NN /\ X = ( ( m + 1 ) .x. U ) ) -> E. n e. NN X = ( n .x. U ) )
78	3, 74, 77	syl2anc 693	. 2 |- ( ( ( ph /\ m e. NN0 ) /\ ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) ) -> E. n e. NN X = ( n .x. U ) )
79		archiabllem.a	. . 3 |- ( ph -> W e. Archi )
80		archiabllem1.p	. . 3 |- ( ph -> .0. .< U )
81		archiabllem1a.c	. . 3 |- ( ph -> .0. .< X )
82	6, 52, 56, 37, 7, 11, 79, 4, 26, 80, 81	archirng 29742	. 2 |- ( ph -> E. m e. NN0 ( ( m .x. U ) .< X /\ X .<_ ( ( m + 1 ) .x. U ) ) )
83	78, 82	r19.29a 3078	1 |- ( ph -> E. n e. NN X = ( n .x. U ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   Basecbs 15857   +g cplusg 15941   lecple 15948   0gc0g 16100   Posetcpo 16940   ltcplt 16941  Tosetctos 17033   Grpcgrp 17422   -gcsg 17424  ._gcmg 17540  oMndcomnd 29697  oGrpcogrp 29698  Archicarchi 29731

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-inf2 8538  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iun 4522  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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-0g 16102  df-preset 16928  df-poset 16946  df-plt 16958  df-toset 17034  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-omnd 29699  df-ogrp 29700  df-inftm 29732  df-archi 29733

This theorem is referenced by:  archiabllem1b  29746