Theorem issubrg3 14493

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 2234	. . . 4 |- ( Base ` R ) = ( Base ` R )
2		eqid 2234	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
3		eqid 2234	. . . 4 |- ( .r ` R ) = ( .r ` R )
4	1, 2, 3	issubrg2 14487	. . 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 13927	. . . 4 |- ( S e. (SubGrp ` R ) -> S C_ ( Base ` R ) )
8		issubrg3.m	. . . . . . . . 9 |- M = (mulGrp ` R )
9	8	ringmgp 14245	. . . . . . . 8 |- ( R e. Ring -> M e. Mnd )
10		eqid 2234	. . . . . . . . 9 |- ( Base ` M ) = ( Base ` M )
11		eqid 2234	. . . . . . . . 9 |- ( 0g ` M ) = ( 0g ` M )
12		eqid 2234	. . . . . . . . 9 |- ( +g ` M ) = ( +g ` M )
13	10, 11, 12	issubm 13727	. . . . . . . 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 14165	. . . . . . . . 9 |- ( R e. Ring -> ( Base ` R ) = ( Base ` M ) )
16	15	sseq2d 3272	. . . . . . . 8 |- ( R e. Ring -> ( S C_ ( Base ` R ) <-> S C_ ( Base ` M ) ) )
17	8, 2	ringidvalg 14204	. . . . . . . . 9 |- ( R e. Ring -> ( 1r ` R ) = ( 0g ` M ) )
18	17	eleq1d 2303	. . . . . . . 8 |- ( R e. Ring -> ( ( 1r ` R ) e. S <-> ( 0g ` M ) e. S ) )
19	8, 3	mgpplusgg 14163	. . . . . . . . . . 11 |- ( R e. Ring -> ( .r ` R ) = ( +g ` M ) )
20	19	oveqd 6075	. . . . . . . . . 10 |- ( R e. Ring -> ( x ( .r ` R ) y ) = ( x ( +g ` M ) y ) )
21	20	eleq1d 2303	. . . . . . . . 9 |- ( R e. Ring -> ( ( x ( .r ` R ) y ) e. S <-> ( x ( +g ` M ) y ) e. S ) )
22	21	2ralbidv 2568	. . . . . . . 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 2205   A.wral 2522   C_ wss 3214   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   .rcmulr 13375   0gc0g 13553   Mndcmnd 13677  SubMndcsubmnd 13713  SubGrpcsubg 13920  mulGrpcmgp 14159   1rcur 14202   Ringcrg 14239  SubRingcsubrg 14463

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-submnd 13715  df-subg 13923  df-mgp 14160  df-ur 14203  df-ring 14241  df-subrg 14465

This theorem is referenced by:  rhmeql  14496  rhmima  14497