Theorem subrngpropd 13772

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 13511	. . . . 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 3364	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
12	11	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
13		eqidd 2197	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠))
14		eqidd 2197	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (Base‘𝐾) = (Base‘𝐾))
15		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝐾 ∈ Rng)
16		vex 2766	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
17	16	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝑠 ∈ V)
18	13, 14, 15, 17	ressbasd 12745	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝑠)))
19	12, 18	eqtrd 2229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (Base‘(𝐾 ↾s 𝑠)))
20	5	ineq2d 3364	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
21	20	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
22		eqidd 2197	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠))
23		eqidd 2197	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (Base‘𝐿) = (Base‘𝐿))
24	8	biimpa 296	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝐿 ∈ Rng)
25	22, 23, 24, 17	ressbasd 12745	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿 ↾s 𝑠)))
26	21, 25	eqtrd 2229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (Base‘(𝐿 ↾s 𝑠)))
27		elinel2 3350	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑠 ∩ 𝐵) → 𝑥 ∈ 𝐵)
28		elinel2 3350	. . . . . . . . 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 2197	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐾) = (+g‘𝐾))
32	13, 31, 17, 15	ressplusgd 12806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐾) = (+g‘(𝐾 ↾s 𝑠)))
33	32	oveqdr 5950	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(𝐾 ↾s 𝑠))𝑦))
34		eqidd 2197	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐿) = (+g‘𝐿))
35	22, 34, 17, 24	ressplusgd 12806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐿) = (+g‘(𝐿 ↾s 𝑠)))
36	35	oveqdr 5950	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(𝐿 ↾s 𝑠))𝑦))
37	30, 33, 36	3eqtr3d 2237	. . . . . . . 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 2196	. . . . . . . . . . . 12 ⊢ (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠)
41		eqid 2196	. . . . . . . . . . . 12 ⊢ (.r‘𝐾) = (.r‘𝐾)
42	40, 41	ressmulrg 12822	. . . . . . . . . . 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 5950	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘(𝐾 ↾s 𝑠))𝑦))
45		eqid 2196	. . . . . . . . . . . 12 ⊢ (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠)
46		eqid 2196	. . . . . . . . . . . 12 ⊢ (.r‘𝐿) = (.r‘𝐿)
47	45, 46	ressmulrg 12822	. . . . . . . . . . 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 5950	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐿)𝑦) = (𝑥(.r‘(𝐿 ↾s 𝑠))𝑦))
50	39, 44, 49	3eqtr3d 2237	. . . . . . . 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 13511	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → ((𝐾 ↾s 𝑠) ∈ Rng ↔ (𝐿 ↾s 𝑠) ∈ Rng))
53	4, 5	eqtr3d 2231	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
54	53	sseq2d 3213	. . . . . . 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 2196	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
60	59	issubrng 13755	. . 3 ⊢ (𝑠 ∈ (SubRng‘𝐾) ↔ (𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)))
61		eqid 2196	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
62	61	issubrng 13755	. . 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 2194	1 ⊢ (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157  ‘cfv 5258  (class class class)co 5922  Basecbs 12678   ↾_s cress 12679  +_gcplusg 12755  ._rcmulr 12756  Rngcrng 13488  SubRngcsubrng 13753

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-cmn 13416  df-abl 13417  df-mgp 13477  df-rng 13489  df-subrng 13754

This theorem is referenced by: (None)