Theorem suborng 32110

Description: Every subring of an ordered ring is also an ordered ring. (Contributed by Thierry Arnoux, 21-Jan-2018.)

Assertion

Ref	Expression
suborng	⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oRing)

Proof of Theorem suborng

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 485	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ Ring)
2		ringgrp 19969	. . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (𝑅 ↾s 𝐴) ∈ Grp)
3	2	adantl 482	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ Grp)
4		orngogrp 32096	. . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5		isogrp 31910	. . . . . 6 ⊢ (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
6	5	simprbi 497	. . . . 5 ⊢ (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
7	4, 6	syl 17	. . . 4 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
8		ringmnd 19974	. . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (𝑅 ↾s 𝐴) ∈ Mnd)
9		submomnd 31918	. . . 4 ⊢ ((𝑅 ∈ oMnd ∧ (𝑅 ↾s 𝐴) ∈ Mnd) → (𝑅 ↾s 𝐴) ∈ oMnd)
10	7, 8, 9	syl2an 596	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oMnd)
11		isogrp 31910	. . 3 ⊢ ((𝑅 ↾s 𝐴) ∈ oGrp ↔ ((𝑅 ↾s 𝐴) ∈ Grp ∧ (𝑅 ↾s 𝐴) ∈ oMnd))
12	3, 10, 11	sylanbrc 583	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oGrp)
13		simp-4l 781	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → 𝑅 ∈ oRing)
14		reldmress 17114	. . . . . . . . . . . . . . 15 ⊢ Rel dom ↾s
15	14	ovprc2 7397	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐴 ∈ V → (𝑅 ↾s 𝐴) = ∅)
16	15	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ V → (Base‘(𝑅 ↾s 𝐴)) = (Base‘∅))
17	16	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) = (Base‘∅))
18		base0 17088	. . . . . . . . . . . 12 ⊢ ∅ = (Base‘∅)
19	17, 18	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) = ∅)
20		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴))
21		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (1r‘(𝑅 ↾s 𝐴)) = (1r‘(𝑅 ↾s 𝐴))
22	20, 21	ringidcl 19989	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (1r‘(𝑅 ↾s 𝐴)) ∈ (Base‘(𝑅 ↾s 𝐴)))
23	22	ne0d 4295	. . . . . . . . . . . . 13 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (Base‘(𝑅 ↾s 𝐴)) ≠ ∅)
24	23	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) ≠ ∅)
25	24	neneqd 2948	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅 ↾s 𝐴)) = ∅)
26	19, 25	condan 816	. . . . . . . . . 10 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → 𝐴 ∈ V)
27		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
28		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	27, 28	ressbas 17118	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝐴)))
30		inss2 4189	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
31	29, 30	eqsstrrdi 3999	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (Base‘(𝑅 ↾s 𝐴)) ⊆ (Base‘𝑅))
32	26, 31	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (Base‘(𝑅 ↾s 𝐴)) ⊆ (Base‘𝑅))
33	32	ad3antrrr 728	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (Base‘(𝑅 ↾s 𝐴)) ⊆ (Base‘𝑅))
34		simpllr 774	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴)))
35	33, 34	sseldd 3945	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → 𝑎 ∈ (Base‘𝑅))
36		simprl 769	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎)
37		orngring 32095	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38		ringgrp 19969	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39	37, 38	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Grp)
40	39	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → 𝑅 ∈ Grp)
41	28	ressinbas 17126	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s 𝐴) = (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))))
42	29	oveq2d 7373	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))))
43	41, 42	eqtrd 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ V → (𝑅 ↾s 𝐴) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))))
44	26, 43	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))))
45	44, 3	eqeltrrd 2839	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) ∈ Grp)
46	28	issubg 18928	. . . . . . . . . . . . . 14 ⊢ ((Base‘(𝑅 ↾s 𝐴)) ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ (Base‘(𝑅 ↾s 𝐴)) ⊆ (Base‘𝑅) ∧ (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) ∈ Grp))
47	40, 32, 45, 46	syl3anbrc 1343	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (Base‘(𝑅 ↾s 𝐴)) ∈ (SubGrp‘𝑅))
48		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))
49		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	48, 49	subg0 18934	. . . . . . . . . . . . 13 ⊢ ((Base‘(𝑅 ↾s 𝐴)) ∈ (SubGrp‘𝑅) → (0g‘𝑅) = (0g‘(𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))))
51	47, 50	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘𝑅) = (0g‘(𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))))
52	44	fveq2d 6846	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))))
53	51, 52	eqtr4d 2779	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝐴)))
54	53	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝐴)))
55	26	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝐴 ∈ V)
56		eqid 2736	. . . . . . . . . . . 12 ⊢ (le‘𝑅) = (le‘𝑅)
57	27, 56	ressle 17261	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (le‘𝑅) = (le‘(𝑅 ↾s 𝐴)))
58	55, 57	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → (le‘𝑅) = (le‘(𝑅 ↾s 𝐴)))
59		eqidd 2737	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑎 = 𝑎)
60	54, 58, 59	breq123d 5119	. . . . . . . . 9 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → ((0g‘𝑅)(le‘𝑅)𝑎 ↔ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎))
61	60	adantr 481	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → ((0g‘𝑅)(le‘𝑅)𝑎 ↔ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎))
62	36, 61	mpbird 256	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (0g‘𝑅)(le‘𝑅)𝑎)
63		simplr 767	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴)))
64	33, 63	sseldd 3945	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → 𝑏 ∈ (Base‘𝑅))
65		simprr 771	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)
66		eqidd 2737	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑏 = 𝑏)
67	54, 58, 66	breq123d 5119	. . . . . . . . 9 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → ((0g‘𝑅)(le‘𝑅)𝑏 ↔ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏))
68	67	adantr 481	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → ((0g‘𝑅)(le‘𝑅)𝑏 ↔ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏))
69	65, 68	mpbird 256	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (0g‘𝑅)(le‘𝑅)𝑏)
70		eqid 2736	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
71	28, 56, 49, 70	orngmul 32098	. . . . . . 7 ⊢ ((𝑅 ∈ oRing ∧ (𝑎 ∈ (Base‘𝑅) ∧ (0g‘𝑅)(le‘𝑅)𝑎) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (0g‘𝑅)(le‘𝑅)𝑏)) → (0g‘𝑅)(le‘𝑅)(𝑎(.r‘𝑅)𝑏))
72	13, 35, 62, 64, 69, 71	syl122anc 1379	. . . . . 6 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (0g‘𝑅)(le‘𝑅)(𝑎(.r‘𝑅)𝑏))
73	54	adantr 481	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝐴)))
74	58	adantr 481	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (le‘𝑅) = (le‘(𝑅 ↾s 𝐴)))
75	55	adantr 481	. . . . . . . . 9 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → 𝐴 ∈ V)
76	27, 70	ressmulr 17188	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (.r‘𝑅) = (.r‘(𝑅 ↾s 𝐴)))
77	75, 76	syl 17	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (.r‘𝑅) = (.r‘(𝑅 ↾s 𝐴)))
78	77	oveqd 7374	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (𝑎(.r‘𝑅)𝑏) = (𝑎(.r‘(𝑅 ↾s 𝐴))𝑏))
79	73, 74, 78	breq123d 5119	. . . . . 6 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → ((0g‘𝑅)(le‘𝑅)(𝑎(.r‘𝑅)𝑏) ↔ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))(𝑎(.r‘(𝑅 ↾s 𝐴))𝑏)))
80	72, 79	mpbid 231	. . . . 5 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ ((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)) → (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))(𝑎(.r‘(𝑅 ↾s 𝐴))𝑏))
81	80	ex 413	. . . 4 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → (((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏) → (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))(𝑎(.r‘(𝑅 ↾s 𝐴))𝑏)))
82	81	anasss 467	. . 3 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝑎 ∈ (Base‘(𝑅 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴)))) → (((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏) → (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))(𝑎(.r‘(𝑅 ↾s 𝐴))𝑏)))
83	82	ralrimivva 3197	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → ∀𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))∀𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))(((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏) → (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))(𝑎(.r‘(𝑅 ↾s 𝐴))𝑏)))
84		eqid 2736	. . 3 ⊢ (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s 𝐴))
85		eqid 2736	. . 3 ⊢ (.r‘(𝑅 ↾s 𝐴)) = (.r‘(𝑅 ↾s 𝐴))
86		eqid 2736	. . 3 ⊢ (le‘(𝑅 ↾s 𝐴)) = (le‘(𝑅 ↾s 𝐴))
87	20, 84, 85, 86	isorng 32094	. 2 ⊢ ((𝑅 ↾s 𝐴) ∈ oRing ↔ ((𝑅 ↾s 𝐴) ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ oGrp ∧ ∀𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))∀𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))(((0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑎 ∧ (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏) → (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))(𝑎(.r‘(𝑅 ↾s 𝐴))𝑏))))
88	1, 12, 83, 87	syl3anbrc 1343	1 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oRing)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083   ↾_s cress 17112  ._rcmulr 17134  lecple 17140  0_gc0g 17321  Mndcmnd 18556  Grpcgrp 18748  SubGrpcsubg 18922  1_rcur 19913  Ringcrg 19964  oMndcomnd 31905  oGrpcogrp 31906  oRingcorng 32090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-dec 12619  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-ple 17153  df-0g 17323  df-poset 18202  df-toset 18306  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-subg 18925  df-mgp 19897  df-ur 19914  df-ring 19966  df-omnd 31907  df-ogrp 31908  df-orng 32092

This theorem is referenced by:  subofld  32111