Theorem subrngpropd 14350

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 1024	. . . . 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 1024	. . . . 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 14088	. . . . 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 3421	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
12	11	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
13		eqidd 2233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠))
14		eqidd 2233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (Base‘𝐾) = (Base‘𝐾))
15		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝐾 ∈ Rng)
16		vex 2815	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
17	16	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝑠 ∈ V)
18	13, 14, 15, 17	ressbasd 13269	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝑠)))
19	12, 18	eqtrd 2265	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (Base‘(𝐾 ↾s 𝑠)))
20	5	ineq2d 3421	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
21	20	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
22		eqidd 2233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠))
23		eqidd 2233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (Base‘𝐿) = (Base‘𝐿))
24	8	biimpa 296	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → 𝐿 ∈ Rng)
25	22, 23, 24, 17	ressbasd 13269	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿 ↾s 𝑠)))
26	21, 25	eqtrd 2265	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (𝑠 ∩ 𝐵) = (Base‘(𝐿 ↾s 𝑠)))
27		elinel2 3405	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑠 ∩ 𝐵) → 𝑥 ∈ 𝐵)
28		elinel2 3405	. . . . . . . . 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 2233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐾) = (+g‘𝐾))
32	13, 31, 17, 15	ressplusgd 13331	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐾) = (+g‘(𝐾 ↾s 𝑠)))
33	32	oveqdr 6077	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(𝐾 ↾s 𝑠))𝑦))
34		eqidd 2233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐿) = (+g‘𝐿))
35	22, 34, 17, 24	ressplusgd 13331	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → (+g‘𝐿) = (+g‘(𝐿 ↾s 𝑠)))
36	35	oveqdr 6077	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(𝐿 ↾s 𝑠))𝑦))
37	30, 33, 36	3eqtr3d 2273	. . . . . . . 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 2232	. . . . . . . . . . . 12 ⊢ (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠)
41		eqid 2232	. . . . . . . . . . . 12 ⊢ (.r‘𝐾) = (.r‘𝐾)
42	40, 41	ressmulrg 13347	. . . . . . . . . . 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 6077	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘(𝐾 ↾s 𝑠))𝑦))
45		eqid 2232	. . . . . . . . . . . 12 ⊢ (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠)
46		eqid 2232	. . . . . . . . . . . 12 ⊢ (.r‘𝐿) = (.r‘𝐿)
47	45, 46	ressmulrg 13347	. . . . . . . . . . 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 6077	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Rng) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐿)𝑦) = (𝑥(.r‘(𝐿 ↾s 𝑠))𝑦))
50	39, 44, 49	3eqtr3d 2273	. . . . . . . 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 14088	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Rng) → ((𝐾 ↾s 𝑠) ∈ Rng ↔ (𝐿 ↾s 𝑠) ∈ Rng))
53	4, 5	eqtr3d 2267	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
54	53	sseq2d 3267	. . . . . . 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 1349	. . . . 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 713	. . 3 ⊢ (𝜑 → ((𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)) ↔ (𝐿 ∈ Rng ∧ (𝐿 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐿))))
59		eqid 2232	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
60	59	issubrng 14333	. . 3 ⊢ (𝑠 ∈ (SubRng‘𝐾) ↔ (𝐾 ∈ Rng ∧ (𝐾 ↾s 𝑠) ∈ Rng ∧ 𝑠 ⊆ (Base‘𝐾)))
61		eqid 2232	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
62	61	issubrng 14333	. . 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 2230	1 ⊢ (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ∩ cin 3209   ⊆ wss 3210  ‘cfv 5351  (class class class)co 6049  Basecbs 13201   ↾_s cress 13202  +_gcplusg 13279  ._rcmulr 13280  Rngcrng 14065  SubRngcsubrng 14331

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-pre-ltirr 8235  ax-pre-lttrn 8237  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8306  df-mnf 8307  df-ltxr 8309  df-inn 9234  df-2 9292  df-3 9293  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-mulr 13293  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-cmn 13992  df-abl 13993  df-mgp 14054  df-rng 14066  df-subrng 14332

This theorem is referenced by: (None)