Theorem issubrng2 14491

Description: Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.)

Hypotheses

Ref	Expression
issubrng2.b	|- B = ( Base ` R )
issubrng2.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
issubrng2	|- ( R e. Rng -> ( A e. (SubRng ` R ) <-> ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) )

Distinct variable groups:   x, y, A   x, R, y   x, .x. , y

Allowed substitution hints:   B( x, y)

Proof of Theorem issubrng2

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrngsubg 14485	. . 3 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
2		issubrng2.t	. . . . . 6 |- .x. = ( .r ` R )
3	2	subrngmcl 14490	. . . . 5 |- ( ( A e. (SubRng ` R ) /\ x e. A /\ y e. A ) -> ( x .x. y ) e. A )
4	3	3expb 1235	. . . 4 |- ( ( A e. (SubRng ` R ) /\ ( x e. A /\ y e. A ) ) -> ( x .x. y ) e. A )
5	4	ralrimivva 2632	. . 3 |- ( A e. (SubRng ` R ) -> A. x e. A A. y e. A ( x .x. y ) e. A )
6	1, 5	jca 306	. 2 |- ( A e. (SubRng ` R ) -> ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) )
7		simpl 109	. . . 4 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> R e. Rng )
8		simprl 535	. . . . . 6 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> A e. (SubGrp ` R ) )
9		eqid 2238	. . . . . . 7 |- ( R ↾s A ) = ( R ↾s A )
10	9	subgbas 13958	. . . . . 6 |- ( A e. (SubGrp ` R ) -> A = ( Base ` ( R ↾s A ) ) )
11	8, 10	syl 14	. . . . 5 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> A = ( Base ` ( R ↾s A ) ) )
12		eqidd 2239	. . . . . . 7 |- ( A e. (SubGrp ` R ) -> ( R ↾s A ) = ( R ↾s A ) )
13		eqidd 2239	. . . . . . 7 |- ( A e. (SubGrp ` R ) -> ( +g ` R ) = ( +g ` R ) )
14		id 19	. . . . . . 7 |- ( A e. (SubGrp ` R ) -> A e. (SubGrp ` R ) )
15		subgrcl 13959	. . . . . . 7 |- ( A e. (SubGrp ` R ) -> R e. Grp )
16	12, 13, 14, 15	ressplusgd 13460	. . . . . 6 |- ( A e. (SubGrp ` R ) -> ( +g ` R ) = ( +g ` ( R ↾s A ) ) )
17	8, 16	syl 14	. . . . 5 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> ( +g ` R ) = ( +g ` ( R ↾s A ) ) )
18	9, 2	ressmulrg 13476	. . . . . 6 |- ( ( A e. (SubGrp ` R ) /\ R e. Grp ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	8, 15, 18	syl2anc2 416	. . . . 5 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> .x. = ( .r ` ( R ↾s A ) ) )
20		rngabl 14209	. . . . . 6 |- ( R e. Rng -> R e. Abel )
21	9	subgabl 14113	. . . . . 6 |- ( ( R e. Abel /\ A e. (SubGrp ` R ) ) -> ( R ↾s A ) e. Abel )
22	20, 8, 21	syl2an2r 603	. . . . 5 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> ( R ↾s A ) e. Abel )
23		simprr 537	. . . . . . 7 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> A. x e. A A. y e. A ( x .x. y ) e. A )
24		oveq1 6082	. . . . . . . . 9 |- ( x = u -> ( x .x. y ) = ( u .x. y ) )
25	24	eleq1d 2307	. . . . . . . 8 |- ( x = u -> ( ( x .x. y ) e. A <-> ( u .x. y ) e. A ) )
26		oveq2 6083	. . . . . . . . 9 |- ( y = v -> ( u .x. y ) = ( u .x. v ) )
27	26	eleq1d 2307	. . . . . . . 8 |- ( y = v -> ( ( u .x. y ) e. A <-> ( u .x. v ) e. A ) )
28	25, 27	rspc2v 2943	. . . . . . 7 |- ( ( u e. A /\ v e. A ) -> ( A. x e. A A. y e. A ( x .x. y ) e. A -> ( u .x. v ) e. A ) )
29	23, 28	syl5com 29	. . . . . 6 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> ( ( u e. A /\ v e. A ) -> ( u .x. v ) e. A ) )
30	29	3impib 1232	. . . . 5 |- ( ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) /\ u e. A /\ v e. A ) -> ( u .x. v ) e. A )
31		issubrng2.b	. . . . . . . . . . 11 |- B = ( Base ` R )
32	31	subgss 13954	. . . . . . . . . 10 |- ( A e. (SubGrp ` R ) -> A C_ B )
33	8, 32	syl 14	. . . . . . . . 9 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> A C_ B )
34	33	sseld 3247	. . . . . . . 8 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> ( u e. A -> u e. B ) )
35	33	sseld 3247	. . . . . . . 8 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> ( v e. A -> v e. B ) )
36	33	sseld 3247	. . . . . . . 8 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> ( w e. A -> w e. B ) )
37	34, 35, 36	3anim123d 1360	. . . . . . 7 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> ( ( u e. A /\ v e. A /\ w e. A ) -> ( u e. B /\ v e. B /\ w e. B ) ) )
38	37	imp 124	. . . . . 6 |- ( ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) /\ ( u e. A /\ v e. A /\ w e. A ) ) -> ( u e. B /\ v e. B /\ w e. B ) )
39	31, 2	rngass 14213	. . . . . . 7 |- ( ( R e. Rng /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u .x. v ) .x. w ) = ( u .x. ( v .x. w ) ) )
40	39	adantlr 481	. . . . . 6 |- ( ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u .x. v ) .x. w ) = ( u .x. ( v .x. w ) ) )
41	38, 40	syldan 282	. . . . 5 |- ( ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) /\ ( u e. A /\ v e. A /\ w e. A ) ) -> ( ( u .x. v ) .x. w ) = ( u .x. ( v .x. w ) ) )
42		eqid 2238	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
43	31, 42, 2	rngdi 14214	. . . . . . 7 |- ( ( R e. Rng /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( u .x. ( v ( +g ` R ) w ) ) = ( ( u .x. v ) ( +g ` R ) ( u .x. w ) ) )
44	43	adantlr 481	. . . . . 6 |- ( ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( u .x. ( v ( +g ` R ) w ) ) = ( ( u .x. v ) ( +g ` R ) ( u .x. w ) ) )
45	38, 44	syldan 282	. . . . 5 |- ( ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) /\ ( u e. A /\ v e. A /\ w e. A ) ) -> ( u .x. ( v ( +g ` R ) w ) ) = ( ( u .x. v ) ( +g ` R ) ( u .x. w ) ) )
46	31, 42, 2	rngdir 14215	. . . . . . 7 |- ( ( R e. Rng /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u ( +g ` R ) v ) .x. w ) = ( ( u .x. w ) ( +g ` R ) ( v .x. w ) ) )
47	46	adantlr 481	. . . . . 6 |- ( ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u ( +g ` R ) v ) .x. w ) = ( ( u .x. w ) ( +g ` R ) ( v .x. w ) ) )
48	38, 47	syldan 282	. . . . 5 |- ( ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) /\ ( u e. A /\ v e. A /\ w e. A ) ) -> ( ( u ( +g ` R ) v ) .x. w ) = ( ( u .x. w ) ( +g ` R ) ( v .x. w ) ) )
49	11, 17, 19, 22, 30, 41, 45, 48	isrngd 14227	. . . 4 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> ( R ↾s A ) e. Rng )
50	31	issubrng 14480	. . . 4 |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )
51	7, 49, 33, 50	syl3anbrc 1212	. . 3 |- ( ( R e. Rng /\ ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) -> A e. (SubRng ` R ) )
52	51	ex 115	. 2 |- ( R e. Rng -> ( ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) -> A e. (SubRng ` R ) ) )
53	6, 52	impbid2 143	1 |- ( R e. Rng -> ( A e. (SubRng ` R ) <-> ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   +g cplusg 13408   .rcmulr 13409   Grpcgrp 13782  SubGrpcsubg 13947   Abelcabl 14065  Rngcrng 14206  SubRngcsubrng 14478

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-mgm 13653  df-sgrp 13694  df-grp 13785  df-subg 13950  df-cmn 14066  df-abl 14067  df-mgp 14195  df-rng 14207  df-subrng 14479

This theorem is referenced by:  opprsubrngg  14492  subrngintm  14493