Theorem ringidss 14272

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 2234	. 2 |- ( Base ` M ) = ( Base ` M )
2		eqid 2234	. 2 |- ( 0g ` M ) = ( 0g ` M )
3		eqid 2234	. 2 |- ( +g ` M ) = ( +g ` M )
4		simp3 1026	. . 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 2234	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
8		ringidss.b	. . . . . 6 |- B = ( Base ` R )
9	7, 8	mgpbasg 14165	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
10	9	3ad2ant1 1045	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> B = ( Base ` (mulGrp ` R ) ) )
11	7	mgpex 14164	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. _V )
12	11	3ad2ant1 1045	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> (mulGrp ` R ) e. _V )
13		simp2 1025	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A C_ B )
14	6, 10, 12, 13	ressbas2d 13365	. . 3 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A = ( Base ` M ) )
15	4, 14	eleqtrd 2313	. 2 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. e. ( Base ` M ) )
16	14, 13	eqsstrrd 3279	. . . 4 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( Base ` M ) C_ B )
17	16	sselda 3242	. . 3 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. ( Base ` M ) ) -> y e. B )
18		eqid 2234	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
19	7, 18	mgpplusgg 14163	. . . . . . . 8 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
20	19	3ad2ant1 1045	. . . . . . 7 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
21		basfn 13355	. . . . . . . . . 10 |- Base Fn _V
22		simp1 1024	. . . . . . . . . . 11 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> R e. Ring )
23	22	elexd 2829	. . . . . . . . . 10 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> R e. _V )
24		funfvex 5692	. . . . . . . . . . 11 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5463	. . . . . . . . . 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 2321	. . . . . . . 8 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> B e. _V )
28	27, 13	ssexd 4255	. . . . . . 7 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> A e. _V )
29	6, 20, 28, 12	ressplusgd 13426	. . . . . 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 6075	. . . 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 14266	. . . . 5 |- ( ( R e. Ring /\ y e. B ) -> ( .1. ( .r ` R ) y ) = y )
34	33	3ad2antl1 1186	. . . 4 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( .1. ( .r ` R ) y ) = y )
35	31, 34	eqtr3d 2269	. . 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 6075	. . . 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 14267	. . . . 5 |- ( ( R e. Ring /\ y e. B ) -> ( y ( .r ` R ) .1. ) = y )
39	38	3ad2antl1 1186	. . . 4 |- ( ( ( R e. Ring /\ A C_ B /\ .1. e. A ) /\ y e. B ) -> ( y ( .r ` R ) .1. ) = y )
40	37, 39	eqtr3d 2269	. . 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 13643	1 |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. = ( 0g ` M ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3214   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   +g cplusg 13374   .rcmulr 13375   0gc0g 13553  mulGrpcmgp 14159   1rcur 14202   Ringcrg 14239

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mgp 14160  df-ur 14203  df-ring 14241

This theorem is referenced by:  unitgrpid  14363