Theorem subrgpropd 14225

Description: If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypotheses

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

Assertion

Ref	Expression
subrgpropd	⊢ (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))

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

Proof of Theorem subrgpropd

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgrcl 14198	. . . 4 ⊢ (𝑠 ∈ (SubRing‘𝐾) → 𝐾 ∈ Ring)
2	1	a1i 9	. . 3 ⊢ (𝜑 → (𝑠 ∈ (SubRing‘𝐾) → 𝐾 ∈ Ring))
3		subrgrcl 14198	. . . 4 ⊢ (𝑠 ∈ (SubRing‘𝐿) → 𝐿 ∈ Ring)
4		subrgpropd.1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
5		subrgpropd.2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
6		subrgpropd.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7		subrgpropd.4	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
8	4, 5, 6, 7	ringpropd 14009	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
9	3, 8	imbitrrid 156	. . 3 ⊢ (𝜑 → (𝑠 ∈ (SubRing‘𝐿) → 𝐾 ∈ Ring))
10	8	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
11	4	ineq2d 3405	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
12	11	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐾)))
13		eqidd 2230	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠))
14		eqidd 2230	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (Base‘𝐾) = (Base‘𝐾))
15		simplr 528	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → 𝐾 ∈ Ring)
16		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → 𝑠 ∈ V)
17	13, 14, 15, 16	ressbasd 13108	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝑠)))
18	17	elvd 2804	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (𝑠 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝑠)))
19	12, 18	eqtrd 2262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (𝑠 ∩ 𝐵) = (Base‘(𝐾 ↾s 𝑠)))
20	5	ineq2d 3405	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
21	20	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (𝑠 ∩ 𝐵) = (𝑠 ∩ (Base‘𝐿)))
22		eqidd 2230	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠))
23		eqidd 2230	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (Base‘𝐿) = (Base‘𝐿))
24	8	biimpa 296	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐿 ∈ Ring)
25	24	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → 𝐿 ∈ Ring)
26	22, 23, 25, 16	ressbasd 13108	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿 ↾s 𝑠)))
27	26	elvd 2804	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (𝑠 ∩ (Base‘𝐿)) = (Base‘(𝐿 ↾s 𝑠)))
28	21, 27	eqtrd 2262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (𝑠 ∩ 𝐵) = (Base‘(𝐿 ↾s 𝑠)))
29		elinel2 3391	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑠 ∩ 𝐵) → 𝑥 ∈ 𝐵)
30		elinel2 3391	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑠 ∩ 𝐵) → 𝑦 ∈ 𝐵)
31	29, 30	anim12i 338	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑠 ∩ 𝐵) ∧ 𝑦 ∈ (𝑠 ∩ 𝐵)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
32	6	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
33		eqidd 2230	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (+g‘𝐾) = (+g‘𝐾))
34	13, 33, 16, 15	ressplusgd 13170	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (+g‘𝐾) = (+g‘(𝐾 ↾s 𝑠)))
35	34	elvd 2804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (+g‘𝐾) = (+g‘(𝐾 ↾s 𝑠)))
36	35	oveqdr 6035	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(𝐾 ↾s 𝑠))𝑦))
37		eqidd 2230	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (+g‘𝐿) = (+g‘𝐿))
38	22, 37, 16, 25	ressplusgd 13170	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ 𝑠 ∈ V) → (+g‘𝐿) = (+g‘(𝐿 ↾s 𝑠)))
39	38	elvd 2804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (+g‘𝐿) = (+g‘(𝐿 ↾s 𝑠)))
40	39	oveqdr 6035	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(𝐿 ↾s 𝑠))𝑦))
41	32, 36, 40	3eqtr3d 2270	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(𝐾 ↾s 𝑠))𝑦) = (𝑥(+g‘(𝐿 ↾s 𝑠))𝑦))
42	31, 41	sylan2 286	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ (𝑠 ∩ 𝐵) ∧ 𝑦 ∈ (𝑠 ∩ 𝐵))) → (𝑥(+g‘(𝐾 ↾s 𝑠))𝑦) = (𝑥(+g‘(𝐿 ↾s 𝑠))𝑦))
43	7	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
44		vex 2802	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
45		eqid 2229	. . . . . . . . . . . . . 14 ⊢ (𝐾 ↾s 𝑠) = (𝐾 ↾s 𝑠)
46		eqid 2229	. . . . . . . . . . . . . 14 ⊢ (.r‘𝐾) = (.r‘𝐾)
47	45, 46	ressmulrg 13186	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ V ∧ 𝐾 ∈ Ring) → (.r‘𝐾) = (.r‘(𝐾 ↾s 𝑠)))
48	44, 47	mpan 424	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (.r‘𝐾) = (.r‘(𝐾 ↾s 𝑠)))
49	48	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (.r‘𝐾) = (.r‘(𝐾 ↾s 𝑠)))
50	49	oveqdr 6035	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘(𝐾 ↾s 𝑠))𝑦))
51		eqid 2229	. . . . . . . . . . . . 13 ⊢ (𝐿 ↾s 𝑠) = (𝐿 ↾s 𝑠)
52		eqid 2229	. . . . . . . . . . . . 13 ⊢ (.r‘𝐿) = (.r‘𝐿)
53	51, 52	ressmulrg 13186	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ V ∧ 𝐿 ∈ Ring) → (.r‘𝐿) = (.r‘(𝐿 ↾s 𝑠)))
54	44, 24, 53	sylancr 414	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (.r‘𝐿) = (.r‘(𝐿 ↾s 𝑠)))
55	54	oveqdr 6035	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐿)𝑦) = (𝑥(.r‘(𝐿 ↾s 𝑠))𝑦))
56	43, 50, 55	3eqtr3d 2270	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘(𝐾 ↾s 𝑠))𝑦) = (𝑥(.r‘(𝐿 ↾s 𝑠))𝑦))
57	31, 56	sylan2 286	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ (𝑠 ∩ 𝐵) ∧ 𝑦 ∈ (𝑠 ∩ 𝐵))) → (𝑥(.r‘(𝐾 ↾s 𝑠))𝑦) = (𝑥(.r‘(𝐿 ↾s 𝑠))𝑦))
58	19, 28, 42, 57	ringpropd 14009	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((𝐾 ↾s 𝑠) ∈ Ring ↔ (𝐿 ↾s 𝑠) ∈ Ring))
59	10, 58	anbi12d 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((𝐾 ∈ Ring ∧ (𝐾 ↾s 𝑠) ∈ Ring) ↔ (𝐿 ∈ Ring ∧ (𝐿 ↾s 𝑠) ∈ Ring)))
60	4, 5	eqtr3d 2264	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
61	60	sseq2d 3254	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ⊆ (Base‘𝐾) ↔ 𝑠 ⊆ (Base‘𝐿)))
62	61	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (𝑠 ⊆ (Base‘𝐾) ↔ 𝑠 ⊆ (Base‘𝐿)))
63	4	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘𝐾))
64	5	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘𝐿))
65		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐾 ∈ Ring)
66	63, 64, 43, 65, 24	rngidpropdg 14118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (1r‘𝐾) = (1r‘𝐿))
67	66	eleq1d 2298	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((1r‘𝐾) ∈ 𝑠 ↔ (1r‘𝐿) ∈ 𝑠))
68	62, 67	anbi12d 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((𝑠 ⊆ (Base‘𝐾) ∧ (1r‘𝐾) ∈ 𝑠) ↔ (𝑠 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝑠)))
69	59, 68	anbi12d 473	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (((𝐾 ∈ Ring ∧ (𝐾 ↾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r‘𝐾) ∈ 𝑠)) ↔ ((𝐿 ∈ Ring ∧ (𝐿 ↾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝑠))))
70		eqid 2229	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
71		eqid 2229	. . . . . 6 ⊢ (1r‘𝐾) = (1r‘𝐾)
72	70, 71	issubrg 14193	. . . . 5 ⊢ (𝑠 ∈ (SubRing‘𝐾) ↔ ((𝐾 ∈ Ring ∧ (𝐾 ↾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐾) ∧ (1r‘𝐾) ∈ 𝑠)))
73		eqid 2229	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
74		eqid 2229	. . . . . 6 ⊢ (1r‘𝐿) = (1r‘𝐿)
75	73, 74	issubrg 14193	. . . . 5 ⊢ (𝑠 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿 ↾s 𝑠) ∈ Ring) ∧ (𝑠 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝑠)))
76	69, 72, 75	3bitr4g 223	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (𝑠 ∈ (SubRing‘𝐾) ↔ 𝑠 ∈ (SubRing‘𝐿)))
77	76	ex 115	. . 3 ⊢ (𝜑 → (𝐾 ∈ Ring → (𝑠 ∈ (SubRing‘𝐾) ↔ 𝑠 ∈ (SubRing‘𝐿))))
78	2, 9, 77	pm5.21ndd 710	. 2 ⊢ (𝜑 → (𝑠 ∈ (SubRing‘𝐾) ↔ 𝑠 ∈ (SubRing‘𝐿)))
79	78	eqrdv 2227	1 ⊢ (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∩ cin 3196   ⊆ wss 3197  ‘cfv 5318  (class class class)co 6007  Basecbs 13040   ↾_s cress 13041  +_gcplusg 13118  ._rcmulr 13119  1_rcur 13930  Ringcrg 13967  SubRingcsubrg 14189

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-mgp 13892  df-ur 13931  df-ring 13969  df-subrg 14191

This theorem is referenced by: (None)