Theorem subrgsubm 14206

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 2229	. . 3 |- ( Base ` R ) = ( Base ` R )
2	1	subrgss 14194	. 2 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
3		eqid 2229	. . 3 |- ( 1r ` R ) = ( 1r ` R )
4	3	subrg1cl 14201	. 2 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. A )
5		subrgrcl 14198	. . . 4 |- ( A e. (SubRing ` R ) -> R e. Ring )
6		eqid 2229	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
7		subrgsubm.1	. . . . 5 |- M = (mulGrp ` R )
8	6, 7	mgpress 13902	. . . 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 14196	. . . 4 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Ring )
11		eqid 2229	. . . . 5 |- (mulGrp ` ( R ↾s A ) ) = (mulGrp ` ( R ↾s A ) )
12	11	ringmgp 13973	. . . 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 2306	. 2 |- ( A e. (SubRing ` R ) -> ( M ↾s A ) e. Mnd )
15	7	ringmgp 13973	. . . . 5 |- ( R e. Ring -> M e. Mnd )
16		eqid 2229	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
17		eqid 2229	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
18		eqid 2229	. . . . . 6 |- ( M ↾s A ) = ( M ↾s A )
19	16, 17, 18	issubm2 13514	. . . . 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 13897	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` M ) )
23	22	sseq2d 3254	. . . . . 6 |- ( R e. Ring -> ( A C_ ( Base ` R ) <-> A C_ ( Base ` M ) ) )
24	7, 3	ringidvalg 13932	. . . . . . 7 |- ( R e. Ring -> ( 1r ` R ) = ( 0g ` M ) )
25	24	eleq1d 2298	. . . . . 6 |- ( R e. Ring -> ( ( 1r ` R ) e. A <-> ( 0g ` M ) e. A ) )
26	23, 25	3anbi12d 1347	. . . . 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 1204	1 |- ( A e. (SubRing ` R ) -> A e. (SubMnd ` M ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   ` cfv 5318  (class class class)co 6007   Basecbs 13040   ↾_s cress 13041   0gc0g 13297   Mndcmnd 13457  SubMndcsubmnd 13499  mulGrpcmgp 13891   1rcur 13930   Ringcrg 13967  SubRingcsubrg 14189

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-submnd 13501  df-mgp 13892  df-ur 13931  df-ring 13969  df-subrg 14191

This theorem is referenced by:  resrhm  14220  resrhm2b  14221  rhmima  14223  lgseisenlem4  15760