Theorem subrngpropd 14507

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 1028	. . . . 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 1028	. . . . 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 14237	. . . . 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 3432	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
12	11	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
13		eqidd 2239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠))
14		eqidd 2239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (Base‘𝐾) = (Base‘𝐾))
15		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝐾 ∈ Rng)
16		vex 2824	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
17	16	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝑠 ∈ V)
18	13, 14, 15, 17	ressbasd 13404	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝑠)))
19	12, 18	eqtrd 2271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (Base‘(𝐾 ↾s 𝑠)))
20	5	ineq2d 3432	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
21	20	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
22		eqidd 2239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠))
23		eqidd 2239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (Base‘𝐿) = (Base‘𝐿))
24	8	biimpa 296	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝐿 ∈ Rng)
25	22, 23, 24, 17	ressbasd 13404	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿 ↾s 𝑠)))
26	21, 25	eqtrd 2271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (Base‘(𝐿 ↾s 𝑠)))
27		elinel2 3416	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑠 ∩ 𝐵) → 𝑥 ∈ 𝐵)
28		elinel2 3416	. . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
31		eqidd 2239	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐾) = (+g‘𝐾))
32	13, 31, 17, 15	ressplusgd 13466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐾) = (+g‘(𝐾 ↾s 𝑠)))
33	32	oveqdr 6107	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(𝐾 ↾s 𝑠))𝑦))
34		eqidd 2239	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐿) = (+g‘𝐿))
35	22, 34, 17, 24	ressplusgd 13466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐿) = (+g‘(𝐿 ↾s 𝑠)))
36	35	oveqdr 6107	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(𝐿 ↾s 𝑠))𝑦))
37	30, 33, 36	3eqtr3d 2279	. . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
40		eqid 2238	. . . . . . . . . . . 12 ⊢ (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠)
41		eqid 2238	. . . . . . . . . . . 12 ⊢ (.r‘𝐾) = (.r‘𝐾)
42	40, 41	ressmulrg 13482	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ V ∧ 𝐾 ∈ Rng) → (.r‘𝐾) = (.r‘(𝐾 ↾s 𝑠)))
43	17, 15, 42	syl2anc 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (.r‘𝐾) = (.r‘(𝐾 ↾s 𝑠)))
44	43	oveqdr 6107	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘(𝐾 ↾s 𝑠))𝑦))
45		eqid 2238	. . . . . . . . . . . 12 ⊢ (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠)
46		eqid 2238	. . . . . . . . . . . 12 ⊢ (.r‘𝐿) = (.r‘𝐿)
47	45, 46	ressmulrg 13482	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ V ∧ 𝐿 ∈ Rng) → (.r‘𝐿) = (.r‘(𝐿 ↾s 𝑠)))
48	17, 24, 47	syl2anc 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (.r‘𝐿) = (.r‘(𝐿 ↾s 𝑠)))
49	48	oveqdr 6107	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐿)𝑦) = (𝑥(.r‘(𝐿 ↾s 𝑠))𝑦))
50	39, 44, 49	3eqtr3d 2279	. . . . . . . 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 14237	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → ((𝐾 ↾s 𝑠) ∈ Rng ↔ (𝐿 ↾s 𝑠) ∈ Rng))
53	4, 5	eqtr3d 2273	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
54	53	sseq2d 3278	. . . . . . 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 1353	. . . . 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 717	. . 3 ⊢ (𝜑 → ((𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)) ↔ (𝐿 ∈ Rng ∧ (𝐿 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐿))))
59		eqid 2238	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
60	59	issubrng 14490	. . 3 ⊢ (𝑠 ∈ (SubRng‘𝐾) ↔ (𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)))
61		eqid 2238	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
62	61	issubrng 14490	. . 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 2236	1 ⊢ (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  ‘cfv 5375  (class class class)co 6079  Basecbs 13335   ↾_s cress 13336  +_gcplusg 13414  ._rcmulr 13415  Rngcrng 14214  SubRngcsubrng 14488

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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-cmn 14072  df-abl 14073  df-mgp 14201  df-rng 14215  df-subrng 14489

This theorem is referenced by: (None)