Theorem subrngpropd 13560

Description: If two structures have the same ring components (properties), they have the same set of subrings. (Contributed by AV, 17-Feb-2025.)

Hypotheses

Ref	Expression
subrngpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
subrngpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
subrngpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
subrngpropd.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))

Assertion

Ref	Expression
subrngpropd	⊢ (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝐿,𝑦

Proof of Theorem subrngpropd

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 999	. . . . 5 ⊢ ((𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)) → 𝐾 ∈ Rng)
2	1	a1i 9	. . . 4 ⊢ (𝜑 → ((𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)) → 𝐾 ∈ Rng))
3		simp1 999	. . . . 5 ⊢ ((𝐿 ∈ Rng ∧ (𝐿 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐿)) → 𝐿 ∈ Rng)
4		subrngpropd.1	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
5		subrngpropd.2	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
6		subrngpropd.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7		subrngpropd.4	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
8	4, 5, 6, 7	rngpropd 13306	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ Rng ↔ 𝐿 ∈ Rng))
9	3, 8	imbitrrid 156	. . . 4 ⊢ (𝜑 → ((𝐿 ∈ Rng ∧ (𝐿 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐿)) → 𝐾 ∈ Rng))
10	8	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝐾 ∈ Rng ↔ 𝐿 ∈ Rng))
11	4	ineq2d 3351	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
12	11	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
13		eqidd 2190	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠))
14		eqidd 2190	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (Base‘𝐾) = (Base‘𝐾))
15		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝐾 ∈ Rng)
16		vex 2755	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
17	16	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝑠 ∈ V)
18	13, 14, 15, 17	ressbasd 12576	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝑠)))
19	12, 18	eqtrd 2222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (Base‘(𝐾 ↾s 𝑠)))
20	5	ineq2d 3351	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
21	20	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
22		eqidd 2190	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠))
23		eqidd 2190	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (Base‘𝐿) = (Base‘𝐿))
24	8	biimpa 296	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝐿 ∈ Rng)
25	22, 23, 24, 17	ressbasd 12576	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿 ↾s 𝑠)))
26	21, 25	eqtrd 2222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (Base‘(𝐿 ↾s 𝑠)))
27		elinel2 3337	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑠 ∩ 𝐵) → 𝑥 ∈ 𝐵)
28		elinel2 3337	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑠 ∩ 𝐵) → 𝑦 ∈ 𝐵)
29	27, 28	anim12i 338	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑠 ∩ 𝐵) ∧ 𝑦 ∈ (𝑠 ∩ 𝐵)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
30	6	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
31		eqidd 2190	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐾) = (+g‘𝐾))
32	13, 31, 17, 15	ressplusgd 12637	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐾) = (+g‘(𝐾 ↾s 𝑠)))
33	32	oveqdr 5923	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(𝐾 ↾s 𝑠))𝑦))
34		eqidd 2190	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐿) = (+g‘𝐿))
35	22, 34, 17, 24	ressplusgd 12637	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐿) = (+g‘(𝐿 ↾s 𝑠)))
36	35	oveqdr 5923	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(𝐿 ↾s 𝑠))𝑦))
37	30, 33, 36	3eqtr3d 2230	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(𝐾 ↾s 𝑠))𝑦) = (𝑥(+g‘(𝐿 ↾s 𝑠))𝑦))
38	29, 37	sylan2 286	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ (𝑠 ∩ 𝐵) ∧ 𝑦 ∈ (𝑠 ∩ 𝐵))) → (𝑥(+g‘(𝐾 ↾s 𝑠))𝑦) = (𝑥(+g‘(𝐿 ↾s 𝑠))𝑦))
39	7	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
40		eqid 2189	. . . . . . . . . . . 12 ⊢ (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠)
41		eqid 2189	. . . . . . . . . . . 12 ⊢ (.r‘𝐾) = (.r‘𝐾)
42	40, 41	ressmulrg 12653	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ V ∧ 𝐾 ∈ Rng) → (.r‘𝐾) = (.r‘(𝐾 ↾s 𝑠)))
43	17, 15, 42	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (.r‘𝐾) = (.r‘(𝐾 ↾s 𝑠)))
44	43	oveqdr 5923	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘(𝐾 ↾s 𝑠))𝑦))
45		eqid 2189	. . . . . . . . . . . 12 ⊢ (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠)
46		eqid 2189	. . . . . . . . . . . 12 ⊢ (.r‘𝐿) = (.r‘𝐿)
47	45, 46	ressmulrg 12653	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ V ∧ 𝐿 ∈ Rng) → (.r‘𝐿) = (.r‘(𝐿 ↾s 𝑠)))
48	17, 24, 47	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (.r‘𝐿) = (.r‘(𝐿 ↾s 𝑠)))
49	48	oveqdr 5923	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐿)𝑦) = (𝑥(.r‘(𝐿 ↾s 𝑠))𝑦))
50	39, 44, 49	3eqtr3d 2230	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘(𝐾 ↾s 𝑠))𝑦) = (𝑥(.r‘(𝐿 ↾s 𝑠))𝑦))
51	29, 50	sylan2 286	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ (𝑠 ∩ 𝐵) ∧ 𝑦 ∈ (𝑠 ∩ 𝐵))) → (𝑥(.r‘(𝐾 ↾s 𝑠))𝑦) = (𝑥(.r‘(𝐿 ↾s 𝑠))𝑦))
52	19, 26, 38, 51	rngpropd 13306	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → ((𝐾 ↾s 𝑠) ∈ Rng ↔ (𝐿 ↾s 𝑠) ∈ Rng))
53	4, 5	eqtr3d 2224	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
54	53	sseq2d 3200	. . . . . . 7 ⊢ (𝜑 → (𝑠 ⊆ (Base‘𝐾) ↔ 𝑠 ⊆ (Base‘𝐿)))
55	54	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ⊆ (Base‘𝐾) ↔ 𝑠 ⊆ (Base‘𝐿)))
56	10, 52, 55	3anbi123d 1323	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → ((𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)) ↔ (𝐿 ∈ Rng ∧ (𝐿 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐿))))
57	56	ex 115	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Rng → ((𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)) ↔ (𝐿 ∈ Rng ∧ (𝐿 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐿)))))
58	2, 9, 57	pm5.21ndd 706	. . 3 ⊢ (𝜑 → ((𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)) ↔ (𝐿 ∈ Rng ∧ (𝐿 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐿))))
59		eqid 2189	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
60	59	issubrng 13543	. . 3 ⊢ (𝑠 ∈ (SubRng‘𝐾) ↔ (𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)))
61		eqid 2189	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
62	61	issubrng 13543	. . 3 ⊢ (𝑠 ∈ (SubRng‘𝐿) ↔ (𝐿 ∈ Rng ∧ (𝐿 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐿)))
63	58, 60, 62	3bitr4g 223	. 2 ⊢ (𝜑 → (𝑠 ∈ (SubRng‘𝐾) ↔ 𝑠 ∈ (SubRng‘𝐿)))
64	63	eqrdv 2187	1 ⊢ (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ∩ cin 3143   ⊆ wss 3144  ‘cfv 5235  (class class class)co 5895  Basecbs 12511   ↾_s cress 12512  +_gcplusg 12586  ._rcmulr 12587  Rngcrng 13283  SubRngcsubrng 13541

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-iress 12519  df-plusg 12599  df-mulr 12600  df-0g 12760  df-mgm 12829  df-sgrp 12862  df-mnd 12875  df-grp 12945  df-cmn 13222  df-abl 13223  df-mgp 13272  df-rng 13284  df-subrng 13542

This theorem is referenced by: (None)