Theorem issubrng2 20535

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	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
issubrng2	⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦   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 20529	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		issubrng2.t	. . . . . 6 ⊢ · = (.r‘𝑅)
3	2	subrngmcl 20534	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
4	3	3expb 1121	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
5	4	ralrimivva 3180	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
6	1, 5	jca 511	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))
7		simpl 482	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Rng)
8		simprl 771	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
9		eqid 2736	. . . . . . 7 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
10	9	subgbas 19106	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅 ↾s 𝐴)))
11	8, 10	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅 ↾s 𝐴)))
12		eqid 2736	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
13	9, 12	ressplusg 17254	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝐴)))
14	8, 13	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝐴)))
15	9, 2	ressmulr 17270	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → · = (.r‘(𝑅 ↾s 𝐴)))
16	8, 15	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (.r‘(𝑅 ↾s 𝐴)))
17		rngabl 20136	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
18	9	subgabl 19811	. . . . . 6 ⊢ ((𝑅 ∈ Abel ∧ 𝐴 ∈ (SubGrp‘𝑅)) → (𝑅 ↾s 𝐴) ∈ Abel)
19	17, 8, 18	syl2an2r 686	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Abel)
20		simprr 773	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
21		oveq1 7374	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
22	21	eleq1d 2821	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
23		oveq2 7375	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
24	23	eleq1d 2821	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
25	22, 24	rspc2v 3575	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
26	20, 25	syl5com 31	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
27	26	3impib 1117	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
28		issubrng2.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
29	28	subgss 19103	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ⊆ 𝐵)
30	8, 29	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ⊆ 𝐵)
31	30	sseld 3920	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵))
32	30	sseld 3920	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝐵))
33	30	sseld 3920	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
34	31, 32, 33	3anim123d 1446	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
35	34	imp 406	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))
36	28, 2	rngass 20140	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
37	36	adantlr 716	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
38	35, 37	syldan 592	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
39	28, 12, 2	rngdi 20141	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
40	39	adantlr 716	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
41	35, 40	syldan 592	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
42	28, 12, 2	rngdir 20142	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
43	42	adantlr 716	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
44	35, 43	syldan 592	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
45	11, 14, 16, 19, 27, 38, 41, 44	isrngd 20154	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Rng)
46	28	issubrng 20524	. . . 4 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
47	7, 45, 30, 46	syl3anbrc 1345	. . 3 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRng‘𝑅))
48	47	ex 412	. 2 ⊢ (𝑅 ∈ Rng → ((𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRng‘𝑅)))
49	6, 48	impbid2 226	1 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  ‘cfv 6498  (class class class)co 7367  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  ._rcmulr 17221  SubGrpcsubg 19096  Abelcabl 19756  Rngcrng 20133  SubRngcsubrng 20522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-mgm 18608  df-sgrp 18687  df-grp 18912  df-subg 19099  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-subrng 20523

This theorem is referenced by:  opprsubrng  20536  subrngint  20537  rhmimasubrng  20543  cntzsubrng  20544  pzriprnglem5  21465