Theorem issubrng2 14087

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 14081	. . 3 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
2		issubrng2.t	. . . . . 6 |- .x. = ( .r ` R )
3	2	subrngmcl 14086	. . . . 5 |- ( ( A e. (SubRng ` R ) /\ x e. A /\ y e. A ) -> ( x .x. y ) e. A )
4	3	3expb 1207	. . . 4 |- ( ( A e. (SubRng ` R ) /\ ( x e. A /\ y e. A ) ) -> ( x .x. y ) e. A )
5	4	ralrimivva 2590	. . 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 529	. . . . . 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 2207	. . . . . . 7 |- ( R ↾s A ) = ( R ↾s A )
10	9	subgbas 13629	. . . . . 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 2208	. . . . . . 7 |- ( A e. (SubGrp ` R ) -> ( R ↾s A ) = ( R ↾s A ) )
13		eqidd 2208	. . . . . . 7 |- ( A e. (SubGrp ` R ) -> ( +g ` R ) = ( +g ` R ) )
14		id 19	. . . . . . 7 |- ( A e. (SubGrp ` R ) -> A e. (SubGrp ` R ) )
15		subgrcl 13630	. . . . . . 7 |- ( A e. (SubGrp ` R ) -> R e. Grp )
16	12, 13, 14, 15	ressplusgd 13076	. . . . . 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 13092	. . . . . 6 |- ( ( A e. (SubGrp ` R ) /\ R e. Grp ) -> .x. = ( .r ` ( R ↾s A ) ) )
19	8, 15, 18	syl2anc2 412	. . . . 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 13812	. . . . . 6 |- ( R e. Rng -> R e. Abel )
21	9	subgabl 13783	. . . . . 6 |- ( ( R e. Abel /\ A e. (SubGrp ` R ) ) -> ( R ↾s A ) e. Abel )
22	20, 8, 21	syl2an2r 595	. . . . 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 531	. . . . . . 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 5974	. . . . . . . . 9 |- ( x = u -> ( x .x. y ) = ( u .x. y ) )
25	24	eleq1d 2276	. . . . . . . 8 |- ( x = u -> ( ( x .x. y ) e. A <-> ( u .x. y ) e. A ) )
26		oveq2 5975	. . . . . . . . 9 |- ( y = v -> ( u .x. y ) = ( u .x. v ) )
27	26	eleq1d 2276	. . . . . . . 8 |- ( y = v -> ( ( u .x. y ) e. A <-> ( u .x. v ) e. A ) )
28	25, 27	rspc2v 2897	. . . . . . 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 1204	. . . . 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 13625	. . . . . . . . . 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 3200	. . . . . . . 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 3200	. . . . . . . 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 3200	. . . . . . . 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 1332	. . . . . . 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 13816	. . . . . . 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 477	. . . . . 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 2207	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
43	31, 42, 2	rngdi 13817	. . . . . . 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 477	. . . . . 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 13818	. . . . . . 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 477	. . . . . 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 13830	. . . 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 14076	. . . 4 |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )
51	7, 49, 33, 50	syl3anbrc 1184	. . 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 981   = wceq 1373   e. wcel 2178   A.wral 2486   C_ wss 3174   ` cfv 5290  (class class class)co 5967   Basecbs 12947   ↾_s cress 12948   +g cplusg 13024   .rcmulr 13025   Grpcgrp 13447  SubGrpcsubg 13618   Abelcabl 13736  Rngcrng 13809  SubRngcsubrng 14074

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-mgm 13303  df-sgrp 13349  df-grp 13450  df-subg 13621  df-cmn 13737  df-abl 13738  df-mgp 13798  df-rng 13810  df-subrng 14075

This theorem is referenced by:  opprsubrngg  14088  subrngintm  14089