Theorem ringidss 13661

Description: A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
ringidss.g	|- M = ( (mulGrp ` R ) ↾s A )
ringidss.b	|- B = ( Base ` R )
ringidss.u	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
ringidss	|- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. = ( 0g ` M ) )

Proof of Theorem ringidss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2196	. 2 |- ( Base ` M ) = ( Base ` M )
2		eqid 2196	. 2 |- ( 0g ` M ) = ( 0g ` M )
3		eqid 2196	. 2 |- ( +g ` M ) = ( +g ` M )
4		simp3 1001	. . 3 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. e. A )
5		ringidss.g	. . . . 5 |- M = ( (mulGrp ` R ) ↾s A )
6	5	a1i 9	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> M = ( (mulGrp ` R ) ↾s A ) )
7		eqid 2196	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
8		ringidss.b	. . . . . 6 |- B = ( Base ` R )
9	7, 8	mgpbasg 13558	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
10	9	3ad2ant1 1020	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> B = ( Base ` (mulGrp ` R ) ) )
11	7	mgpex 13557	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
12	11	3ad2ant1 1020	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> (mulGrp ` R ) e. _V )
13		simp2 1000	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A C_ B )
14	6, 10, 12, 13	ressbas2d 12771	. . 3 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A = ( Base ` M ) )
15	4, 14	eleqtrd 2275	. 2 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. e. ( Base ` M ) )
16	14, 13	eqsstrrd 3221	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( Base ` M ) C_ B )
17	16	sselda 3184	. . 3 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. ( Base ` M ) ) -> y e. B )
18		eqid 2196	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
19	7, 18	mgpplusgg 13556	. . . . . . . 8 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
20	19	3ad2ant1 1020	. . . . . . 7 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
21		basfn 12761	. . . . . . . . . 10 |- Base Fn _V
22		simp1 999	. . . . . . . . . . 11 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> R e. Ring )
23	22	elexd 2776	. . . . . . . . . 10 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> R e. _V )
24		funfvex 5578	. . . . . . . . . . 11 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5361	. . . . . . . . . 10 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
26	21, 23, 25	sylancr 414	. . . . . . . . 9 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( Base ` R ) e. _V )
27	8, 26	eqeltrid 2283	. . . . . . . 8 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> B e. _V )
28	27, 13	ssexd 4174	. . . . . . 7 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A e. _V )
29	6, 20, 28, 12	ressplusgd 12831	. . . . . 6 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( .r ` R ) = ( +g ` M ) )
30	29	adantr 276	. . . . 5 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( .r ` R ) = ( +g ` M ) )
31	30	oveqd 5942	. . . 4 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( .1. ( .r ` R ) y ) = ( .1. ( +g ` M ) y ) )
32		ringidss.u	. . . . . 6 |- .1. = ( 1r ` R )
33	8, 18, 32	ringlidm 13655	. . . . 5 |- ( ( R e. Ring /\ y e. B ) -> ( .1. ( .r ` R ) y ) = y )
34	33	3ad2antl1 1161	. . . 4 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( .1. ( .r ` R ) y ) = y )
35	31, 34	eqtr3d 2231	. . 3 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( .1. ( +g ` M ) y ) = y )
36	17, 35	syldan 282	. 2 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. ( Base ` M ) ) -> ( .1. ( +g ` M ) y ) = y )
37	30	oveqd 5942	. . . 4 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( y ( .r ` R ) .1. ) = ( y ( +g ` M ) .1. ) )
38	8, 18, 32	ringridm 13656	. . . . 5 |- ( ( R e. Ring /\ y e. B ) -> ( y ( .r ` R ) .1. ) = y )
39	38	3ad2antl1 1161	. . . 4 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( y ( .r ` R ) .1. ) = y )
40	37, 39	eqtr3d 2231	. . 3 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( y ( +g ` M ) .1. ) = y )
41	17, 40	syldan 282	. 2 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. ( Base ` M ) ) -> ( y ( +g ` M ) .1. ) = y )
42	1, 2, 3, 15, 36, 41	ismgmid2 13082	1 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. = ( 0g ` M ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   .rcmulr 12781   0gc0g 12958  mulGrpcmgp 13552   1rcur 13591   Ringcrg 13628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-mgp 13553  df-ur 13592  df-ring 13630

This theorem is referenced by:  unitgrpid  13750