Theorem issubrg3 14251

Description: A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)

Hypothesis

Ref	Expression
issubrg3.m	|- M = (mulGrp ` R )

Assertion

Ref	Expression
issubrg3	|- ( R e. Ring -> ( S e. (SubRing ` R ) <-> ( S e. (SubGrp ` R ) /\ S e. (SubMnd ` M ) ) ) )

Proof of Theorem issubrg3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2229	. . . 4 |- ( Base ` R ) = ( Base ` R )
2		eqid 2229	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
3		eqid 2229	. . . 4 |- ( .r ` R ) = ( .r ` R )
4	1, 2, 3	issubrg2 14245	. . 3 |- ( R e. Ring -> ( S e. (SubRing ` R ) <-> ( S e. (SubGrp ` R ) /\ ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) ) )
5		3anass 1006	. . 3 |- ( ( S e. (SubGrp ` R ) /\ ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) <-> ( S e. (SubGrp ` R ) /\ ( ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) ) )
6	4, 5	bitrdi 196	. 2 |- ( R e. Ring -> ( S e. (SubRing ` R ) <-> ( S e. (SubGrp ` R ) /\ ( ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) ) ) )
7	1	subgss 13751	. . . 4 |- ( S e. (SubGrp ` R ) -> S C_ ( Base ` R ) )
8		issubrg3.m	. . . . . . . . 9 |- M = (mulGrp ` R )
9	8	ringmgp 14005	. . . . . . . 8 |- ( R e. Ring -> M e. Mnd )
10		eqid 2229	. . . . . . . . 9 |- ( Base ` M ) = ( Base ` M )
11		eqid 2229	. . . . . . . . 9 |- ( 0g ` M ) = ( 0g ` M )
12		eqid 2229	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
13	10, 11, 12	issubm 13545	. . . . . . . 8 |- ( M e. Mnd -> ( S e. (SubMnd ` M ) <-> ( S C_ ( Base ` M ) /\ ( 0g ` M ) e. S /\ A. x e. S A. y e. S ( x ( +g ` M ) y ) e. S ) ) )
14	9, 13	syl 14	. . . . . . 7 |- ( R e. Ring -> ( S e. (SubMnd ` M ) <-> ( S C_ ( Base ` M ) /\ ( 0g ` M ) e. S /\ A. x e. S A. y e. S ( x ( +g ` M ) y ) e. S ) ) )
15	8, 1	mgpbasg 13929	. . . . . . . . 9 |- ( R e. Ring -> ( Base ` R ) = ( Base ` M ) )
16	15	sseq2d 3255	. . . . . . . 8 |- ( R e. Ring -> ( S C_ ( Base ` R ) <-> S C_ ( Base ` M ) ) )
17	8, 2	ringidvalg 13964	. . . . . . . . 9 |- ( R e. Ring -> ( 1r ` R ) = ( 0g ` M ) )
18	17	eleq1d 2298	. . . . . . . 8 |- ( R e. Ring -> ( ( 1r ` R ) e. S <-> ( 0g ` M ) e. S ) )
19	8, 3	mgpplusgg 13927	. . . . . . . . . . 11 |- ( R e. Ring -> ( .r ` R ) = ( +g ` M ) )
20	19	oveqd 6030	. . . . . . . . . 10 |- ( R e. Ring -> ( x ( .r ` R ) y ) = ( x ( +g ` M ) y ) )
21	20	eleq1d 2298	. . . . . . . . 9 |- ( R e. Ring -> ( ( x ( .r ` R ) y ) e. S <-> ( x ( +g ` M ) y ) e. S ) )
22	21	2ralbidv 2554	. . . . . . . 8 |- ( R e. Ring -> ( A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S <-> A. x e. S A. y e. S ( x ( +g ` M ) y ) e. S ) )
23	16, 18, 22	3anbi123d 1346	. . . . . . 7 |- ( R e. Ring -> ( ( S C_ ( Base ` R ) /\ ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) <-> ( S C_ ( Base ` M ) /\ ( 0g ` M ) e. S /\ A. x e. S A. y e. S ( x ( +g ` M ) y ) e. S ) ) )
24	14, 23	bitr4d 191	. . . . . 6 |- ( R e. Ring -> ( S e. (SubMnd ` M ) <-> ( S C_ ( Base ` R ) /\ ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) ) )
25		3anass 1006	. . . . . 6 |- ( ( S C_ ( Base ` R ) /\ ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) <-> ( S C_ ( Base ` R ) /\ ( ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) ) )
26	24, 25	bitrdi 196	. . . . 5 |- ( R e. Ring -> ( S e. (SubMnd ` M ) <-> ( S C_ ( Base ` R ) /\ ( ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) ) ) )
27	26	baibd 928	. . . 4 |- ( ( R e. Ring /\ S C_ ( Base ` R ) ) -> ( S e. (SubMnd ` M ) <-> ( ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) ) )
28	7, 27	sylan2 286	. . 3 |- ( ( R e. Ring /\ S e. (SubGrp ` R ) ) -> ( S e. (SubMnd ` M ) <-> ( ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) ) )
29	28	pm5.32da 452	. 2 |- ( R e. Ring -> ( ( S e. (SubGrp ` R ) /\ S e. (SubMnd ` M ) ) <-> ( S e. (SubGrp ` R ) /\ ( ( 1r ` R ) e. S /\ A. x e. S A. y e. S ( x ( .r ` R ) y ) e. S ) ) ) )
30	6, 29	bitr4d 191	1 |- ( R e. Ring -> ( S e. (SubRing ` R ) <-> ( S e. (SubGrp ` R ) /\ S e. (SubMnd ` M ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3198   ` cfv 5324  (class class class)co 6013   Basecbs 13072   +g cplusg 13150   .rcmulr 13151   0gc0g 13329   Mndcmnd 13489  SubMndcsubmnd 13531  SubGrpcsubg 13744  mulGrpcmgp 13923   1rcur 13962   Ringcrg 13999  SubRingcsubrg 14221

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-submnd 13533  df-subg 13747  df-mgp 13924  df-ur 13963  df-ring 14001  df-subrg 14223

This theorem is referenced by:  rhmeql  14254  rhmima  14255