Theorem ringidss 13165

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 2177	. 2 |- ( Base ` M ) = ( Base ` M )
2		eqid 2177	. 2 |- ( 0g ` M ) = ( 0g ` M )
3		eqid 2177	. 2 |- ( +g ` M ) = ( +g ` M )
4		simp3 999	. . 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 2177	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
8		ringidss.b	. . . . . 6 |- B = ( Base ` R )
9	7, 8	mgpbasg 13089	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
10	9	3ad2ant1 1018	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> B = ( Base ` (mulGrp ` R ) ) )
11	7	mgpex 13088	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
12	11	3ad2ant1 1018	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> (mulGrp ` R ) e. _V )
13		simp2 998	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A C_ B )
14	6, 10, 12, 13	ressbas2d 12522	. . 3 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A = ( Base ` M ) )
15	4, 14	eleqtrd 2256	. 2 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. e. ( Base ` M ) )
16	14, 13	eqsstrrd 3192	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( Base ` M ) C_ B )
17	16	sselda 3155	. . 3 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. ( Base ` M ) ) -> y e. B )
18		eqid 2177	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
19	7, 18	mgpplusgg 13087	. . . . . . . 8 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
20	19	3ad2ant1 1018	. . . . . . 7 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
21		basfn 12514	. . . . . . . . . 10 |- Base Fn _V
22		simp1 997	. . . . . . . . . . 11 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> R e. Ring )
23	22	elexd 2750	. . . . . . . . . 10 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> R e. _V )
24		funfvex 5532	. . . . . . . . . . 11 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5316	. . . . . . . . . 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 2264	. . . . . . . 8 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> B e. _V )
28	27, 13	ssexd 4143	. . . . . . 7 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A e. _V )
29	6, 20, 28, 12	ressplusgd 12581	. . . . . 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 5891	. . . 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 13159	. . . . 5 |- ( ( R e. Ring /\ y e. B ) -> ( .1. ( .r ` R ) y ) = y )
34	33	3ad2antl1 1159	. . . 4 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( .1. ( .r ` R ) y ) = y )
35	31, 34	eqtr3d 2212	. . 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 5891	. . . 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 13160	. . . . 5 |- ( ( R e. Ring /\ y e. B ) -> ( y ( .r ` R ) .1. ) = y )
39	38	3ad2antl1 1159	. . . 4 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( y ( .r ` R ) .1. ) = y )
40	37, 39	eqtr3d 2212	. . 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 12753	1 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. = ( 0g ` M ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   _Vcvv 2737   C_ wss 3129   Fn wfn 5211   ` cfv 5216  (class class class)co 5874   Basecbs 12456   ↾_s cress 12457   +g cplusg 12530   .rcmulr 12531   0gc0g 12695  mulGrpcmgp 13083   1rcur 13095   Ringcrg 13132

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-iress 12464  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-mgp 13084  df-ur 13096  df-ring 13134

This theorem is referenced by:  unitgrpid  13240