Theorem submarchi 29740

Description: A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)

Assertion

Ref	Expression
submarchi	|- ( ( ( W e. Toset /\ W e. Archi ) /\ A e. (SubMnd ` W ) ) -> ( W ↾s A ) e. Archi )

Proof of Theorem submarchi

Dummy variables x n y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		submrcl 17346	. . . . . 6 |- ( A e. (SubMnd ` W ) -> W e. Mnd )
2		eqid 2622	. . . . . . 7 |- ( Base ` W ) = ( Base ` W )
3		eqid 2622	. . . . . . 7 |- ( 0g ` W ) = ( 0g ` W )
4		eqid 2622	. . . . . . 7 |- (.g ` W ) = (.g ` W )
5		eqid 2622	. . . . . . 7 |- ( le ` W ) = ( le ` W )
6		eqid 2622	. . . . . . 7 |- ( lt ` W ) = ( lt ` W )
7	2, 3, 4, 5, 6	isarchi2 29739	. . . . . 6 |- ( ( W e. Toset /\ W e. Mnd ) -> ( W e. Archi <-> A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) ) )
8	1, 7	sylan2 491	. . . . 5 |- ( ( W e. Toset /\ A e. (SubMnd ` W ) ) -> ( W e. Archi <-> A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) ) )
9	8	biimpa 501	. . . 4 |- ( ( ( W e. Toset /\ A e. (SubMnd ` W ) ) /\ W e. Archi ) -> A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) )
10	9	an32s 846	. . 3 |- ( ( ( W e. Toset /\ W e. Archi ) /\ A e. (SubMnd ` W ) ) -> A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) )
11		eqid 2622	. . . . . . . 8 |- ( W ↾s A ) = ( W ↾s A )
12	11	submbas 17355	. . . . . . 7 |- ( A e. (SubMnd ` W ) -> A = ( Base ` ( W ↾s A ) ) )
13	2	submss 17350	. . . . . . 7 |- ( A e. (SubMnd ` W ) -> A C_ ( Base ` W ) )
14	12, 13	eqsstr3d 3640	. . . . . 6 |- ( A e. (SubMnd ` W ) -> ( Base ` ( W ↾s A ) ) C_ ( Base ` W ) )
15		ssralv 3666	. . . . . . . 8 |- ( ( Base ` ( W ↾s A ) ) C_ ( Base ` W ) -> ( A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) -> A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) ) )
16	15	ralimdv 2963	. . . . . . 7 |- ( ( Base ` ( W ↾s A ) ) C_ ( Base ` W ) -> ( A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) -> A. x e. ( Base ` W ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) ) )
17		ssralv 3666	. . . . . . 7 |- ( ( Base ` ( W ↾s A ) ) C_ ( Base ` W ) -> ( A. x e. ( Base ` W ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) -> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) ) )
18	16, 17	syld 47	. . . . . 6 |- ( ( Base ` ( W ↾s A ) ) C_ ( Base ` W ) -> ( A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) -> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) ) )
19	14, 18	syl 17	. . . . 5 |- ( A e. (SubMnd ` W ) -> ( A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) -> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) ) )
20	19	adantl 482	. . . 4 |- ( ( ( W e. Toset /\ W e. Archi ) /\ A e. (SubMnd ` W ) ) -> ( A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) -> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) ) )
21	11, 3	subm0 17356	. . . . . . . . . 10 |- ( A e. (SubMnd ` W ) -> ( 0g ` W ) = ( 0g ` ( W ↾s A ) ) )
22	21	ad2antrr 762	. . . . . . . . 9 |- ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) -> ( 0g ` W ) = ( 0g ` ( W ↾s A ) ) )
23	11, 5	ressle 16059	. . . . . . . . . . . 12 |- ( A e. (SubMnd ` W ) -> ( le ` W ) = ( le ` ( W ↾s A ) ) )
24	23	difeq1d 3727	. . . . . . . . . . 11 |- ( A e. (SubMnd ` W ) -> ( ( le ` W ) \ _I ) = ( ( le ` ( W ↾s A ) ) \ _I ) )
25	5, 6	pltfval 16959	. . . . . . . . . . . 12 |- ( W e. Mnd -> ( lt ` W ) = ( ( le ` W ) \ _I ) )
26	1, 25	syl 17	. . . . . . . . . . 11 |- ( A e. (SubMnd ` W ) -> ( lt ` W ) = ( ( le ` W ) \ _I ) )
27	11	submmnd 17354	. . . . . . . . . . . 12 |- ( A e. (SubMnd ` W ) -> ( W ↾s A ) e. Mnd )
28		eqid 2622	. . . . . . . . . . . . 13 |- ( le ` ( W ↾s A ) ) = ( le ` ( W ↾s A ) )
29		eqid 2622	. . . . . . . . . . . . 13 |- ( lt ` ( W ↾s A ) ) = ( lt ` ( W ↾s A ) )
30	28, 29	pltfval 16959	. . . . . . . . . . . 12 |- ( ( W ↾s A ) e. Mnd -> ( lt ` ( W ↾s A ) ) = ( ( le ` ( W ↾s A ) ) \ _I ) )
31	27, 30	syl 17	. . . . . . . . . . 11 |- ( A e. (SubMnd ` W ) -> ( lt ` ( W ↾s A ) ) = ( ( le ` ( W ↾s A ) ) \ _I ) )
32	24, 26, 31	3eqtr4d 2666	. . . . . . . . . 10 |- ( A e. (SubMnd ` W ) -> ( lt ` W ) = ( lt ` ( W ↾s A ) ) )
33	32	ad2antrr 762	. . . . . . . . 9 |- ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) -> ( lt ` W ) = ( lt ` ( W ↾s A ) ) )
34		eqidd 2623	. . . . . . . . 9 |- ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) -> x = x )
35	22, 33, 34	breq123d 4667	. . . . . . . 8 |- ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) -> ( ( 0g ` W ) ( lt ` W ) x <-> ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x ) )
36		eqidd 2623	. . . . . . . . . 10 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> y = y )
37	23	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> ( le ` W ) = ( le ` ( W ↾s A ) ) )
38		simplll 798	. . . . . . . . . . 11 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> A e. (SubMnd ` W ) )
39		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> n e. NN )
40	39	nnnn0d 11351	. . . . . . . . . . 11 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> n e. NN0 )
41		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> x e. ( Base ` ( W ↾s A ) ) )
42	38, 12	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> A = ( Base ` ( W ↾s A ) ) )
43	41, 42	eleqtrrd 2704	. . . . . . . . . . 11 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> x e. A )
44		eqid 2622	. . . . . . . . . . . 12 |- (.g ` ( W ↾s A ) ) = (.g ` ( W ↾s A ) )
45	4, 11, 44	submmulg 17586	. . . . . . . . . . 11 |- ( ( A e. (SubMnd ` W ) /\ n e. NN0 /\ x e. A ) -> ( n (.g ` W ) x ) = ( n (.g ` ( W ↾s A ) ) x ) )
46	38, 40, 43, 45	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> ( n (.g ` W ) x ) = ( n (.g ` ( W ↾s A ) ) x ) )
47	36, 37, 46	breq123d 4667	. . . . . . . . 9 |- ( ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) /\ n e. NN ) -> ( y ( le ` W ) ( n (.g ` W ) x ) <-> y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) )
48	47	rexbidva 3049	. . . . . . . 8 |- ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) -> ( E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) <-> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) )
49	35, 48	imbi12d 334	. . . . . . 7 |- ( ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) /\ y e. ( Base ` ( W ↾s A ) ) ) -> ( ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) <-> ( ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x -> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) ) )
50	49	ralbidva 2985	. . . . . 6 |- ( ( A e. (SubMnd ` W ) /\ x e. ( Base ` ( W ↾s A ) ) ) -> ( A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) <-> A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x -> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) ) )
51	50	ralbidva 2985	. . . . 5 |- ( A e. (SubMnd ` W ) -> ( A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) <-> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x -> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) ) )
52	51	adantl 482	. . . 4 |- ( ( ( W e. Toset /\ W e. Archi ) /\ A e. (SubMnd ` W ) ) -> ( A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) <-> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x -> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) ) )
53	20, 52	sylibd 229	. . 3 |- ( ( ( W e. Toset /\ W e. Archi ) /\ A e. (SubMnd ` W ) ) -> ( A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( 0g ` W ) ( lt ` W ) x -> E. n e. NN y ( le ` W ) ( n (.g ` W ) x ) ) -> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x -> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) ) )
54	10, 53	mpd 15	. 2 |- ( ( ( W e. Toset /\ W e. Archi ) /\ A e. (SubMnd ` W ) ) -> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x -> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) )
55		resstos 29660	. . . 4 |- ( ( W e. Toset /\ A e. (SubMnd ` W ) ) -> ( W ↾s A ) e. Toset )
56	27	adantl 482	. . . 4 |- ( ( W e. Toset /\ A e. (SubMnd ` W ) ) -> ( W ↾s A ) e. Mnd )
57		eqid 2622	. . . . 5 |- ( Base ` ( W ↾s A ) ) = ( Base ` ( W ↾s A ) )
58		eqid 2622	. . . . 5 |- ( 0g ` ( W ↾s A ) ) = ( 0g ` ( W ↾s A ) )
59	57, 58, 44, 28, 29	isarchi2 29739	. . . 4 |- ( ( ( W ↾s A ) e. Toset /\ ( W ↾s A ) e. Mnd ) -> ( ( W ↾s A ) e. Archi <-> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x -> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) ) )
60	55, 56, 59	syl2anc 693	. . 3 |- ( ( W e. Toset /\ A e. (SubMnd ` W ) ) -> ( ( W ↾s A ) e. Archi <-> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x -> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) ) )
61	60	adantlr 751	. 2 |- ( ( ( W e. Toset /\ W e. Archi ) /\ A e. (SubMnd ` W ) ) -> ( ( W ↾s A ) e. Archi <-> A. x e. ( Base ` ( W ↾s A ) ) A. y e. ( Base ` ( W ↾s A ) ) ( ( 0g ` ( W ↾s A ) ) ( lt ` ( W ↾s A ) ) x -> E. n e. NN y ( le ` ( W ↾s A ) ) ( n (.g ` ( W ↾s A ) ) x ) ) ) )
62	54, 61	mpbird 247	1 |- ( ( ( W e. Toset /\ W e. Archi ) /\ A e. (SubMnd ` W ) ) -> ( W ↾s A ) e. Archi )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   class class class wbr 4653   _I cid 5023   ` cfv 5888  (class class class)co 6650   NNcn 11020   NN0cn0 11292   Basecbs 15857   ↾_s cress 15858   lecple 15948   0gc0g 16100   ltcplt 16941  Tosetctos 17033   Mndcmnd 17294  SubMndcsubmnd 17334  ._gcmg 17540  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-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-seq 12802  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-ple 15961  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-submnd 17336  df-mulg 17541  df-inftm 29732  df-archi 29733

This theorem is referenced by:  nn0archi  29843