Theorem suborng 31523

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 19799	. . . 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 31509	. . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5		isogrp 31337	. . . . . 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 19804	. . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (𝑅 ↾s 𝐴) ∈ Mnd)
9		submomnd 31345	. . . 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 31337	. . 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 780	. . . . . . 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 16954	. . . . . . . . . . . . . . 15 ⊢ Rel dom ↾s
15	14	ovprc2 7312	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐴 ∈ V → (𝑅 ↾s 𝐴) = ∅)
16	15	fveq2d 6775	. . . . . . . . . . . . 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 16928	. . . . . . . . . . . 12 ⊢ ∅ = (Base‘∅)
19	17, 18	eqtr4di 2798	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) = ∅)
20		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴))
21		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (1r‘(𝑅 ↾s 𝐴)) = (1r‘(𝑅 ↾s 𝐴))
22	20, 21	ringidcl 19818	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (1r‘(𝑅 ↾s 𝐴)) ∈ (Base‘(𝑅 ↾s 𝐴)))
23	22	ne0d 4275	. . . . . . . . . . . . 13 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (Base‘(𝑅 ↾s 𝐴)) ≠ ∅)
24	23	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) ≠ ∅)
25	24	neneqd 2950	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅 ↾s 𝐴)) = ∅)
26	19, 25	condan 815	. . . . . . . . . 10 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → 𝐴 ∈ V)
27		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
28		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	27, 28	ressbas 16958	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝐴)))
30		inss2 4169	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
31	29, 30	eqsstrrdi 3981	. . . . . . . . . 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 727	. . . . . . . 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 773	. . . . . . . 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 3927	. . . . . . 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 768	. . . . . . . 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 31508	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38		ringgrp 19799	. . . . . . . . . . . . . . . 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 16966	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s 𝐴) = (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))))
42	29	oveq2d 7288	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))))
43	41, 42	eqtrd 2780	. . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) ∈ Grp)
46	28	issubg 18766	. . . . . . . . . . . . . 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 1342	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (Base‘(𝑅 ↾s 𝐴)) ∈ (SubGrp‘𝑅))
48		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))
49		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	48, 49	subg0 18772	. . . . . . . . . . . . 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 6775	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))))
53	51, 52	eqtr4d 2783	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝐴)))
54	53	ad2antrr 723	. . . . . . . . . 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 723	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝐴 ∈ V)
56		eqid 2740	. . . . . . . . . . . 12 ⊢ (le‘𝑅) = (le‘𝑅)
57	27, 56	ressle 17101	. . . . . . . . . . 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 2741	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑎 = 𝑎)
60	54, 58, 59	breq123d 5093	. . . . . . . . 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 766	. . . . . . . 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 3927	. . . . . . 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 770	. . . . . . . 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 2741	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑏 = 𝑏)
67	54, 58, 66	breq123d 5093	. . . . . . . . 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 2740	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
71	28, 56, 49, 70	orngmul 31511	. . . . . . 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 1378	. . . . . 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 17028	. . . . . . . . 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 7289	. . . . . . 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 5093	. . . . . 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 3117	. 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 2740	. . 3 ⊢ (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s 𝐴))
85		eqid 2740	. . 3 ⊢ (.r‘(𝑅 ↾s 𝐴)) = (.r‘(𝑅 ↾s 𝐴))
86		eqid 2740	. . 3 ⊢ (le‘(𝑅 ↾s 𝐴)) = (le‘(𝑅 ↾s 𝐴))
87	20, 84, 85, 86	isorng 31507	. 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 1342	1 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oRing)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ≠ wne 2945  ∀wral 3066  Vcvv 3431   ∩ cin 3891   ⊆ wss 3892  ∅c0 4262   class class class wbr 5079  ‘cfv 6432  (class class class)co 7272  Basecbs 16923   ↾_s cress 16952  ._rcmulr 16974  lecple 16980  0_gc0g 17161  Mndcmnd 18396  Grpcgrp 18588  SubGrpcsubg 18760  1_rcur 19748  Ringcrg 19794  oMndcomnd 31332  oGrpcogrp 31333  oRingcorng 31503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583  ax-cnex 10938  ax-resscn 10939  ax-1cn 10940  ax-icn 10941  ax-addcl 10942  ax-addrcl 10943  ax-mulcl 10944  ax-mulrcl 10945  ax-mulcom 10946  ax-addass 10947  ax-mulass 10948  ax-distr 10949  ax-i2m1 10950  ax-1ne0 10951  ax-1rid 10952  ax-rnegex 10953  ax-rrecex 10954  ax-cnre 10955  ax-pre-lttri 10956  ax-pre-lttrn 10957  ax-pre-ltadd 10958  ax-pre-mulgt0 10959

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7229  df-ov 7275  df-oprab 7276  df-mpo 7277  df-om 7708  df-2nd 7826  df-frecs 8089  df-wrecs 8120  df-recs 8194  df-rdg 8233  df-er 8490  df-en 8726  df-dom 8727  df-sdom 8728  df-pnf 11022  df-mnf 11023  df-xr 11024  df-ltxr 11025  df-le 11026  df-sub 11218  df-neg 11219  df-nn 11985  df-2 12047  df-3 12048  df-4 12049  df-5 12050  df-6 12051  df-7 12052  df-8 12053  df-9 12054  df-dec 12449  df-sets 16876  df-slot 16894  df-ndx 16906  df-base 16924  df-ress 16953  df-plusg 16986  df-mulr 16987  df-ple 16993  df-0g 17163  df-poset 18042  df-toset 18146  df-mgm 18337  df-sgrp 18386  df-mnd 18397  df-grp 18591  df-subg 18763  df-mgp 19732  df-ur 19749  df-ring 19796  df-omnd 31334  df-ogrp 31335  df-orng 31505

This theorem is referenced by:  subofld  31524