Theorem ringidss 13525

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 2193	. 2 |- ( Base ` M ) = ( Base ` M )
2		eqid 2193	. 2 |- ( 0g ` M ) = ( 0g ` M )
3		eqid 2193	. 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 2193	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
8		ringidss.b	. . . . . 6 |- B = ( Base ` R )
9	7, 8	mgpbasg 13422	. . . . 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 13421	. . . . 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 12686	. . 3 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A = ( Base ` M ) )
15	4, 14	eleqtrd 2272	. 2 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. e. ( Base ` M ) )
16	14, 13	eqsstrrd 3216	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( Base ` M ) C_ B )
17	16	sselda 3179	. . 3 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. ( Base ` M ) ) -> y e. B )
18		eqid 2193	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
19	7, 18	mgpplusgg 13420	. . . . . . . 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 12676	. . . . . . . . . 10 |- Base Fn _V
22		simp1 999	. . . . . . . . . . 11 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> R e. Ring )
23	22	elexd 2773	. . . . . . . . . 10 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> R e. _V )
24		funfvex 5571	. . . . . . . . . . 11 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5354	. . . . . . . . . 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 2280	. . . . . . . 8 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> B e. _V )
28	27, 13	ssexd 4169	. . . . . . 7 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A e. _V )
29	6, 20, 28, 12	ressplusgd 12746	. . . . . 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 5935	. . . 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 13519	. . . . 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 2228	. . 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 5935	. . . 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 13520	. . . . 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 2228	. . 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 12963	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 2164   _Vcvv 2760   C_ wss 3153   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   +g cplusg 12695   .rcmulr 12696   0gc0g 12867  mulGrpcmgp 13416   1rcur 13455   Ringcrg 13492

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mgp 13417  df-ur 13456  df-ring 13494

This theorem is referenced by:  unitgrpid  13614