Theorem submomnd 29710

Description: A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)

Assertion

Ref	Expression
submomnd	|- ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) -> ( M ↾s A ) e. oMnd )

Proof of Theorem submomnd

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. 2 |- ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) -> ( M ↾s A ) e. Mnd )
2		omndtos 29705	. . . 4 |- ( M e. oMnd -> M e. Toset )
3	2	adantr 481	. . 3 |- ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) -> M e. Toset )
4		reldmress 15926	. . . . . . . 8 |- Rel dom ↾s
5	4	ovprc2 6685	. . . . . . 7 |- ( -. A e. _V -> ( M ↾s A ) = (/) )
6	5	fveq2d 6195	. . . . . 6 |- ( -. A e. _V -> ( Base ` ( M ↾s A ) ) = ( Base ` (/) ) )
7	6	adantl 482	. . . . 5 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ -. A e. _V ) -> ( Base ` ( M ↾s A ) ) = ( Base ` (/) ) )
8		base0 15912	. . . . 5 |- (/) = ( Base ` (/) )
9	7, 8	syl6eqr 2674	. . . 4 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ -. A e. _V ) -> ( Base ` ( M ↾s A ) ) = (/) )
10		eqid 2622	. . . . . . . 8 |- ( Base ` ( M ↾s A ) ) = ( Base ` ( M ↾s A ) )
11		eqid 2622	. . . . . . . 8 |- ( 0g ` ( M ↾s A ) ) = ( 0g ` ( M ↾s A ) )
12	10, 11	mndidcl 17308	. . . . . . 7 |- ( ( M ↾s A ) e. Mnd -> ( 0g ` ( M ↾s A ) ) e. ( Base ` ( M ↾s A ) ) )
13		ne0i 3921	. . . . . . 7 |- ( ( 0g ` ( M ↾s A ) ) e. ( Base ` ( M ↾s A ) ) -> ( Base ` ( M ↾s A ) ) =/= (/) )
14	12, 13	syl 17	. . . . . 6 |- ( ( M ↾s A ) e. Mnd -> ( Base ` ( M ↾s A ) ) =/= (/) )
15	14	ad2antlr 763	. . . . 5 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ -. A e. _V ) -> ( Base ` ( M ↾s A ) ) =/= (/) )
16	15	neneqd 2799	. . . 4 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ -. A e. _V ) -> -. ( Base ` ( M ↾s A ) ) = (/) )
17	9, 16	condan 835	. . 3 |- ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) -> A e. _V )
18		resstos 29660	. . 3 |- ( ( M e. Toset /\ A e. _V ) -> ( M ↾s A ) e. Toset )
19	3, 17, 18	syl2anc 693	. 2 |- ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) -> ( M ↾s A ) e. Toset )
20		simplll 798	. . . . . 6 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> M e. oMnd )
21		eqid 2622	. . . . . . . . . . 11 |- ( M ↾s A ) = ( M ↾s A )
22		eqid 2622	. . . . . . . . . . 11 |- ( Base ` M ) = ( Base ` M )
23	21, 22	ressbas 15930	. . . . . . . . . 10 |- ( A e. _V -> ( A i^i ( Base ` M ) ) = ( Base ` ( M ↾s A ) ) )
24		inss2 3834	. . . . . . . . . 10 |- ( A i^i ( Base ` M ) ) C_ ( Base ` M )
25	23, 24	syl6eqssr 3656	. . . . . . . . 9 |- ( A e. _V -> ( Base ` ( M ↾s A ) ) C_ ( Base ` M ) )
26	17, 25	syl 17	. . . . . . . 8 |- ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) -> ( Base ` ( M ↾s A ) ) C_ ( Base ` M ) )
27	26	ad2antrr 762	. . . . . . 7 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> ( Base ` ( M ↾s A ) ) C_ ( Base ` M ) )
28		simplr1 1103	. . . . . . 7 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> a e. ( Base ` ( M ↾s A ) ) )
29	27, 28	sseldd 3604	. . . . . 6 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> a e. ( Base ` M ) )
30		simplr2 1104	. . . . . . 7 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> b e. ( Base ` ( M ↾s A ) ) )
31	27, 30	sseldd 3604	. . . . . 6 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> b e. ( Base ` M ) )
32		simplr3 1105	. . . . . . 7 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> c e. ( Base ` ( M ↾s A ) ) )
33	27, 32	sseldd 3604	. . . . . 6 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> c e. ( Base ` M ) )
34		eqid 2622	. . . . . . . . . . 11 |- ( le ` M ) = ( le ` M )
35	21, 34	ressle 16059	. . . . . . . . . 10 |- ( A e. _V -> ( le ` M ) = ( le ` ( M ↾s A ) ) )
36	17, 35	syl 17	. . . . . . . . 9 |- ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) -> ( le ` M ) = ( le ` ( M ↾s A ) ) )
37	36	adantr 481	. . . . . . . 8 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) -> ( le ` M ) = ( le ` ( M ↾s A ) ) )
38	37	breqd 4664	. . . . . . 7 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) -> ( a ( le ` M ) b <-> a ( le ` ( M ↾s A ) ) b ) )
39	38	biimpar 502	. . . . . 6 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> a ( le ` M ) b )
40		eqid 2622	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
41	22, 34, 40	omndadd 29706	. . . . . 6 |- ( ( M e. oMnd /\ ( a e. ( Base ` M ) /\ b e. ( Base ` M ) /\ c e. ( Base ` M ) ) /\ a ( le ` M ) b ) -> ( a ( +g ` M ) c ) ( le ` M ) ( b ( +g ` M ) c ) )
42	20, 29, 31, 33, 39, 41	syl131anc 1339	. . . . 5 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> ( a ( +g ` M ) c ) ( le ` M ) ( b ( +g ` M ) c ) )
43	17	adantr 481	. . . . . . . . 9 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) -> A e. _V )
44	21, 40	ressplusg 15993	. . . . . . . . 9 |- ( A e. _V -> ( +g ` M ) = ( +g ` ( M ↾s A ) ) )
45	43, 44	syl 17	. . . . . . . 8 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) -> ( +g ` M ) = ( +g ` ( M ↾s A ) ) )
46	45	oveqd 6667	. . . . . . 7 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) -> ( a ( +g ` M ) c ) = ( a ( +g ` ( M ↾s A ) ) c ) )
47	43, 35	syl 17	. . . . . . 7 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) -> ( le ` M ) = ( le ` ( M ↾s A ) ) )
48	45	oveqd 6667	. . . . . . 7 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) -> ( b ( +g ` M ) c ) = ( b ( +g ` ( M ↾s A ) ) c ) )
49	46, 47, 48	breq123d 4667	. . . . . 6 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) -> ( ( a ( +g ` M ) c ) ( le ` M ) ( b ( +g ` M ) c ) <-> ( a ( +g ` ( M ↾s A ) ) c ) ( le ` ( M ↾s A ) ) ( b ( +g ` ( M ↾s A ) ) c ) ) )
50	49	adantr 481	. . . . 5 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> ( ( a ( +g ` M ) c ) ( le ` M ) ( b ( +g ` M ) c ) <-> ( a ( +g ` ( M ↾s A ) ) c ) ( le ` ( M ↾s A ) ) ( b ( +g ` ( M ↾s A ) ) c ) ) )
51	42, 50	mpbid 222	. . . 4 |- ( ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) /\ a ( le ` ( M ↾s A ) ) b ) -> ( a ( +g ` ( M ↾s A ) ) c ) ( le ` ( M ↾s A ) ) ( b ( +g ` ( M ↾s A ) ) c ) )
52	51	ex 450	. . 3 |- ( ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) /\ ( a e. ( Base ` ( M ↾s A ) ) /\ b e. ( Base ` ( M ↾s A ) ) /\ c e. ( Base ` ( M ↾s A ) ) ) ) -> ( a ( le ` ( M ↾s A ) ) b -> ( a ( +g ` ( M ↾s A ) ) c ) ( le ` ( M ↾s A ) ) ( b ( +g ` ( M ↾s A ) ) c ) ) )
53	52	ralrimivvva 2972	. 2 |- ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) -> A. a e. ( Base ` ( M ↾s A ) ) A. b e. ( Base ` ( M ↾s A ) ) A. c e. ( Base ` ( M ↾s A ) ) ( a ( le ` ( M ↾s A ) ) b -> ( a ( +g ` ( M ↾s A ) ) c ) ( le ` ( M ↾s A ) ) ( b ( +g ` ( M ↾s A ) ) c ) ) )
54		eqid 2622	. . 3 |- ( +g ` ( M ↾s A ) ) = ( +g ` ( M ↾s A ) )
55		eqid 2622	. . 3 |- ( le ` ( M ↾s A ) ) = ( le ` ( M ↾s A ) )
56	10, 54, 55	isomnd 29701	. 2 |- ( ( M ↾s A ) e. oMnd <-> ( ( M ↾s A ) e. Mnd /\ ( M ↾s A ) e. Toset /\ A. a e. ( Base ` ( M ↾s A ) ) A. b e. ( Base ` ( M ↾s A ) ) A. c e. ( Base ` ( M ↾s A ) ) ( a ( le ` ( M ↾s A ) ) b -> ( a ( +g ` ( M ↾s A ) ) c ) ( le ` ( M ↾s A ) ) ( b ( +g ` ( M ↾s A ) ) c ) ) ) )
57	1, 19, 53, 56	syl3anbrc 1246	1 |- ( ( M e. oMnd /\ ( M ↾s A ) e. Mnd ) -> ( M ↾s A ) e. oMnd )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   lecple 15948   0gc0g 16100  Tosetctos 17033   Mndcmnd 17294  oMndcomnd 29697

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-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

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-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-dec 11494  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-poset 16946  df-toset 17034  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-omnd 29699

This theorem is referenced by:  suborng  29815  nn0omnd  29841