Theorem issubrg3 14392

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 2232	. . . 4 |- ( Base ` R ) = ( Base ` R )
2		eqid 2232	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
3		eqid 2232	. . . 4 |- ( .r ` R ) = ( .r ` R )
4	1, 2, 3	issubrg2 14386	. . 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 1009	. . 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 13891	. . . 4 |- ( S e. (SubGrp ` R ) -> S C_ ( Base ` R ) )
8		issubrg3.m	. . . . . . . . 9 |- M = (mulGrp ` R )
9	8	ringmgp 14146	. . . . . . . 8 |- ( R e. Ring -> M e. Mnd )
10		eqid 2232	. . . . . . . . 9 |- ( Base ` M ) = ( Base ` M )
11		eqid 2232	. . . . . . . . 9 |- ( 0g ` M ) = ( 0g ` M )
12		eqid 2232	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
13	10, 11, 12	issubm 13685	. . . . . . . 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 14070	. . . . . . . . 9 |- ( R e. Ring -> ( Base ` R ) = ( Base ` M ) )
16	15	sseq2d 3268	. . . . . . . 8 |- ( R e. Ring -> ( S C_ ( Base ` R ) <-> S C_ ( Base ` M ) ) )
17	8, 2	ringidvalg 14105	. . . . . . . . 9 |- ( R e. Ring -> ( 1r ` R ) = ( 0g ` M ) )
18	17	eleq1d 2301	. . . . . . . 8 |- ( R e. Ring -> ( ( 1r ` R ) e. S <-> ( 0g ` M ) e. S ) )
19	8, 3	mgpplusgg 14068	. . . . . . . . . . 11 |- ( R e. Ring -> ( .r ` R ) = ( +g ` M ) )
20	19	oveqd 6067	. . . . . . . . . 10 |- ( R e. Ring -> ( x ( .r ` R ) y ) = ( x ( +g ` M ) y ) )
21	20	eleq1d 2301	. . . . . . . . 9 |- ( R e. Ring -> ( ( x ( .r ` R ) y ) e. S <-> ( x ( +g ` M ) y ) e. S ) )
22	21	2ralbidv 2566	. . . . . . . 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 1349	. . . . . . 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 1009	. . . . . 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 931	. . . 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 1005   = wceq 1398   e. wcel 2203   A.wral 2520   C_ wss 3211   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291   0gc0g 13469   Mndcmnd 13629  SubMndcsubmnd 13671  SubGrpcsubg 13884  mulGrpcmgp 14064   1rcur 14103   Ringcrg 14140  SubRingcsubrg 14362

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-submnd 13673  df-subg 13887  df-mgp 14065  df-ur 14104  df-ring 14142  df-subrg 14364

This theorem is referenced by:  rhmeql  14395  rhmima  14396