Theorem ringidss 13585

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 13482	. . . . 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 13481	. . . . 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 12746	. . 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 3220	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( Base ` M ) C_ B )
17	16	sselda 3183	. . 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 13480	. . . . . . . 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 12736	. . . . . . . . . 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 5575	. . . . . . . . . . 11 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5358	. . . . . . . . . 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 4173	. . . . . . 7 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A e. _V )
29	6, 20, 28, 12	ressplusgd 12806	. . . . . 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 5939	. . . 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 13579	. . . . 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 5939	. . . 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 13580	. . . . 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 13023	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 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   .rcmulr 12756   0gc0g 12927  mulGrpcmgp 13476   1rcur 13515   Ringcrg 13552

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

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 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mgp 13477  df-ur 13516  df-ring 13554

This theorem is referenced by:  unitgrpid  13674