Theorem subrgsubm 13730

Description: A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)

Hypothesis

Ref	Expression
subrgsubm.1	|- M = (mulGrp ` R )

Assertion

Ref	Expression
subrgsubm	|- ( A e. (SubRing ` R ) -> A e. (SubMnd ` M ) )

Proof of Theorem subrgsubm

Step	Hyp	Ref	Expression
1		eqid 2193	. . 3 |- ( Base ` R ) = ( Base ` R )
2	1	subrgss 13718	. 2 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
3		eqid 2193	. . 3 |- ( 1r ` R ) = ( 1r ` R )
4	3	subrg1cl 13725	. 2 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. A )
5		subrgrcl 13722	. . . 4 |- ( A e. (SubRing ` R ) -> R e. Ring )
6		eqid 2193	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
7		subrgsubm.1	. . . . 5 |- M = (mulGrp ` R )
8	6, 7	mgpress 13427	. . . 4 |- ( ( R e. Ring /\ A e. (SubRing ` R ) ) -> ( M ↾s A ) = (mulGrp ` ( R ↾s A ) ) )
9	5, 8	mpancom 422	. . 3 |- ( A e. (SubRing ` R ) -> ( M ↾s A ) = (mulGrp ` ( R ↾s A ) ) )
10	6	subrgring 13720	. . . 4 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Ring )
11		eqid 2193	. . . . 5 |- (mulGrp ` ( R ↾s A ) ) = (mulGrp ` ( R ↾s A ) )
12	11	ringmgp 13498	. . . 4 |- ( ( R ↾s A ) e. Ring -> (mulGrp ` ( R ↾s A ) ) e. Mnd )
13	10, 12	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> (mulGrp ` ( R ↾s A ) ) e. Mnd )
14	9, 13	eqeltrd 2270	. 2 |- ( A e. (SubRing ` R ) -> ( M ↾s A ) e. Mnd )
15	7	ringmgp 13498	. . . . 5 |- ( R e. Ring -> M e. Mnd )
16		eqid 2193	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
17		eqid 2193	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
18		eqid 2193	. . . . . 6 |- ( M ↾s A ) = ( M ↾s A )
19	16, 17, 18	issubm2 13045	. . . . 5 |- ( M e. Mnd -> ( A e. (SubMnd ` M ) <-> ( A C_ ( Base ` M ) /\ ( 0g ` M ) e. A /\ ( M ↾s A ) e. Mnd ) ) )
20	15, 19	syl 14	. . . 4 |- ( R e. Ring -> ( A e. (SubMnd ` M ) <-> ( A C_ ( Base ` M ) /\ ( 0g ` M ) e. A /\ ( M ↾s A ) e. Mnd ) ) )
21	5, 20	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> ( A e. (SubMnd ` M ) <-> ( A C_ ( Base ` M ) /\ ( 0g ` M ) e. A /\ ( M ↾s A ) e. Mnd ) ) )
22	7, 1	mgpbasg 13422	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` M ) )
23	22	sseq2d 3209	. . . . . 6 |- ( R e. Ring -> ( A C_ ( Base ` R ) <-> A C_ ( Base ` M ) ) )
24	7, 3	ringidvalg 13457	. . . . . . 7 |- ( R e. Ring -> ( 1r ` R ) = ( 0g ` M ) )
25	24	eleq1d 2262	. . . . . 6 |- ( R e. Ring -> ( ( 1r ` R ) e. A <-> ( 0g ` M ) e. A ) )
26	23, 25	3anbi12d 1324	. . . . 5 |- ( R e. Ring -> ( ( A C_ ( Base ` R ) /\ ( 1r ` R ) e. A /\ ( M ↾s A ) e. Mnd ) <-> ( A C_ ( Base ` M ) /\ ( 0g ` M ) e. A /\ ( M ↾s A ) e. Mnd ) ) )
27	26	bibi2d 232	. . . 4 |- ( R e. Ring -> ( ( A e. (SubMnd ` M ) <-> ( A C_ ( Base ` R ) /\ ( 1r ` R ) e. A /\ ( M ↾s A ) e. Mnd ) ) <-> ( A e. (SubMnd ` M ) <-> ( A C_ ( Base ` M ) /\ ( 0g ` M ) e. A /\ ( M ↾s A ) e. Mnd ) ) ) )
28	5, 27	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> ( ( A e. (SubMnd ` M ) <-> ( A C_ ( Base ` R ) /\ ( 1r ` R ) e. A /\ ( M ↾s A ) e. Mnd ) ) <-> ( A e. (SubMnd ` M ) <-> ( A C_ ( Base ` M ) /\ ( 0g ` M ) e. A /\ ( M ↾s A ) e. Mnd ) ) ) )
29	21, 28	mpbird 167	. 2 |- ( A e. (SubRing ` R ) -> ( A e. (SubMnd ` M ) <-> ( A C_ ( Base ` R ) /\ ( 1r ` R ) e. A /\ ( M ↾s A ) e. Mnd ) ) )
30	2, 4, 14, 29	mpbir3and 1182	1 |- ( A e. (SubRing ` R ) -> A e. (SubMnd ` M ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   C_ wss 3153   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   0gc0g 12867   Mndcmnd 12997  SubMndcsubmnd 13030  mulGrpcmgp 13416   1rcur 13455   Ringcrg 13492  SubRingcsubrg 13713

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-lttrn 7986  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-submnd 13032  df-mgp 13417  df-ur 13456  df-ring 13494  df-subrg 13715

This theorem is referenced by:  resrhm  13744  resrhm2b  13745  rhmima  13747  lgseisenlem4  15189