Theorem subrgsubm 13360

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 2177	. . 3 |- ( Base ` R ) = ( Base ` R )
2	1	subrgss 13348	. 2 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
3		eqid 2177	. . 3 |- ( 1r ` R ) = ( 1r ` R )
4	3	subrg1cl 13355	. 2 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) e. A )
5		subrgrcl 13352	. . . 4 |- ( A e. (SubRing ` R ) -> R e. Ring )
6		eqid 2177	. . . . 5 |- ( R ↾s A ) = ( R ↾s A )
7		subrgsubm.1	. . . . 5 |- M = (mulGrp ` R )
8	6, 7	mgpress 13146	. . . 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 13350	. . . 4 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Ring )
11		eqid 2177	. . . . 5 |- (mulGrp ` ( R ↾s A ) ) = (mulGrp ` ( R ↾s A ) )
12	11	ringmgp 13190	. . . 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 2254	. 2 |- ( A e. (SubRing ` R ) -> ( M ↾s A ) e. Mnd )
15	7	ringmgp 13190	. . . . 5 |- ( R e. Ring -> M e. Mnd )
16		eqid 2177	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
17		eqid 2177	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
18		eqid 2177	. . . . . 6 |- ( M ↾s A ) = ( M ↾s A )
19	16, 17, 18	issubm2 12869	. . . . 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 13141	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` M ) )
23	22	sseq2d 3187	. . . . . 6 |- ( R e. Ring -> ( A C_ ( Base ` R ) <-> A C_ ( Base ` M ) ) )
24	7, 3	ringidvalg 13149	. . . . . . 7 |- ( R e. Ring -> ( 1r ` R ) = ( 0g ` M ) )
25	24	eleq1d 2246	. . . . . 6 |- ( R e. Ring -> ( ( 1r ` R ) e. A <-> ( 0g ` M ) e. A ) )
26	23, 25	3anbi12d 1313	. . . . 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 1180	1 |- ( A e. (SubRing ` R ) -> A e. (SubMnd ` M ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   C_ wss 3131   ` cfv 5218  (class class class)co 5877   Basecbs 12464   ↾_s cress 12465   0gc0g 12710   Mndcmnd 12822  SubMndcsubmnd 12855  mulGrpcmgp 13135   1rcur 13147   Ringcrg 13184  SubRingcsubrg 13343

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-submnd 12857  df-mgp 13136  df-ur 13148  df-ring 13186  df-subrg 13345

This theorem is referenced by: (None)