Theorem subrgsubm 14029

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 2205	. . 3 |- ( Base ` R ) = ( Base ` R )
2	1	subrgss 14017	. 2 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
3		eqid 2205	. . 3 |- ( 1r ` R ) = ( 1r ` R )
4	3	subrg1cl 14024	. 2 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. A )
5		subrgrcl 14021	. . . 4 |- ( A e. (SubRing ` R ) -> R e. Ring )
6		eqid 2205	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
7		subrgsubm.1	. . . . 5 |- M = (mulGrp ` R )
8	6, 7	mgpress 13726	. . . 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 14019	. . . 4 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Ring )
11		eqid 2205	. . . . 5 |- (mulGrp ` ( R ↾s A ) ) = (mulGrp ` ( R ↾s A ) )
12	11	ringmgp 13797	. . . 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 2282	. 2 |- ( A e. (SubRing ` R ) -> ( M ↾s A ) e. Mnd )
15	7	ringmgp 13797	. . . . 5 |- ( R e. Ring -> M e. Mnd )
16		eqid 2205	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
17		eqid 2205	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
18		eqid 2205	. . . . . 6 |- ( M ↾s A ) = ( M ↾s A )
19	16, 17, 18	issubm2 13338	. . . . 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 13721	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` M ) )
23	22	sseq2d 3223	. . . . . 6 |- ( R e. Ring -> ( A C_ ( Base ` R ) <-> A C_ ( Base ` M ) ) )
24	7, 3	ringidvalg 13756	. . . . . . 7 |- ( R e. Ring -> ( 1r ` R ) = ( 0g ` M ) )
25	24	eleq1d 2274	. . . . . 6 |- ( R e. Ring -> ( ( 1r ` R ) e. A <-> ( 0g ` M ) e. A ) )
26	23, 25	3anbi12d 1326	. . . . 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 1183	1 |- ( A e. (SubRing ` R ) -> A e. (SubMnd ` M ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   C_ wss 3166   ` cfv 5272  (class class class)co 5946   Basecbs 12865   ↾_s cress 12866   0gc0g 13121   Mndcmnd 13281  SubMndcsubmnd 13323  mulGrpcmgp 13715   1rcur 13754   Ringcrg 13791  SubRingcsubrg 14012

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-base 12871  df-sets 12872  df-iress 12873  df-plusg 12955  df-mulr 12956  df-0g 13123  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-submnd 13325  df-mgp 13716  df-ur 13755  df-ring 13793  df-subrg 14014

This theorem is referenced by:  resrhm  14043  resrhm2b  14044  rhmima  14046  lgseisenlem4  15583