Theorem suborng 20813

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 484	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ Ring)
2		ringgrp 20177	. . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (𝑅 ↾s 𝐴) ∈ Grp)
3	2	adantl 481	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ Grp)
4		orngogrp 20800	. . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5		isogrp 20057	. . . . . 6 ⊢ (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
6	5	simprbi 496	. . . . 5 ⊢ (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
7	4, 6	syl 17	. . . 4 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
8		ringmnd 20182	. . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (𝑅 ↾s 𝐴) ∈ Mnd)
9		submomnd 20065	. . . 4 ⊢ ((𝑅 ∈ oMnd ∧ (𝑅 ↾s 𝐴) ∈ Mnd) → (𝑅 ↾s 𝐴) ∈ oMnd)
10	7, 8, 9	syl2an 597	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oMnd)
11		isogrp 20057	. . 3 ⊢ ((𝑅 ↾s 𝐴) ∈ oGrp ↔ ((𝑅 ↾s 𝐴) ∈ Grp ∧ (𝑅 ↾s 𝐴) ∈ oMnd))
12	3, 10, 11	sylanbrc 584	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oGrp)
13		simp-4l 783	. . . . . . 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 17163	. . . . . . . . . . . . . . 15 ⊢ Rel dom ↾s
15	14	ovprc2 7400	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐴 ∈ V → (𝑅 ↾s 𝐴) = ∅)
16	15	fveq2d 6839	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ V → (Base‘(𝑅 ↾s 𝐴)) = (Base‘∅))
17	16	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) = (Base‘∅))
18		base0 17145	. . . . . . . . . . . 12 ⊢ ∅ = (Base‘∅)
19	17, 18	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) = ∅)
20		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴))
21		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (1r‘(𝑅 ↾s 𝐴)) = (1r‘(𝑅 ↾s 𝐴))
22	20, 21	ringidcl 20204	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) ≠ ∅)
25	24	neneqd 2938	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅 ↾s 𝐴)) = ∅)
26	19, 25	condan 818	. . . . . . . . . 10 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → 𝐴 ∈ V)
27		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
28		eqid 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	27, 28	ressbas 17167	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝐴)))
30		inss2 4191	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
31	29, 30	eqsstrrdi 3980	. . . . . . . . . 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 731	. . . . . . . 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 776	. . . . . . . 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 3935	. . . . . . 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 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))𝑎)
37		orngring 20799	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38		ringgrp 20177	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39	37, 38	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Grp)
40	39	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → 𝑅 ∈ Grp)
41	28	ressinbas 17176	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s 𝐴) = (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))))
42	29	oveq2d 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))))
43	41, 42	eqtrd 2772	. . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) ∈ Grp)
46	28	issubg 19060	. . . . . . . . . . . . . 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 1345	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (Base‘(𝑅 ↾s 𝐴)) ∈ (SubGrp‘𝑅))
48		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))
49		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	48, 49	subg0 19066	. . . . . . . . . . . . 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 6839	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))))
53	51, 52	eqtr4d 2775	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝐴)))
54	53	ad2antrr 727	. . . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝐴 ∈ V)
56		eqid 2737	. . . . . . . . . . . 12 ⊢ (le‘𝑅) = (le‘𝑅)
57	27, 56	ressle 17304	. . . . . . . . . . 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 2738	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑎 = 𝑎)
60	54, 58, 59	breq123d 5113	. . . . . . . . 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 480	. . . . . . . 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 257	. . . . . . 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 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)) → 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴)))
64	33, 63	sseldd 3935	. . . . . . 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 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)) → (0g‘(𝑅 ↾s 𝐴))(le‘(𝑅 ↾s 𝐴))𝑏)
66		eqidd 2738	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑏 = 𝑏)
67	54, 58, 66	breq123d 5113	. . . . . . . . 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 480	. . . . . . . 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 257	. . . . . . 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 2737	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
71	28, 56, 49, 70	orngmul 20802	. . . . . . 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 1382	. . . . . 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 480	. . . . . . 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 480	. . . . . . 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 480	. . . . . . . . 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 17231	. . . . . . . . 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 7377	. . . . . . 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 5113	. . . . . 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 232	. . . . 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 412	. . . 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 466	. . 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 3180	. 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 2737	. . 3 ⊢ (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s 𝐴))
85		eqid 2737	. . 3 ⊢ (.r‘(𝑅 ↾s 𝐴)) = (.r‘(𝑅 ↾s 𝐴))
86		eqid 2737	. . 3 ⊢ (le‘(𝑅 ↾s 𝐴)) = (le‘(𝑅 ↾s 𝐴))
87	20, 84, 85, 86	isorng 20798	. 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 1345	1 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oRing)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286   class class class wbr 5099  ‘cfv 6493  (class class class)co 7360  Basecbs 17140   ↾_s cress 17161  ._rcmulr 17182  lecple 17188  0_gc0g 17363  Mndcmnd 18663  Grpcgrp 18867  SubGrpcsubg 19054  oMndcomnd 20052  oGrpcogrp 20053  1_rcur 20120  Ringcrg 20172  oRingcorng 20794

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-dec 12612  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-ple 17201  df-0g 17365  df-poset 18240  df-toset 18342  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18870  df-subg 19057  df-omnd 20054  df-ogrp 20055  df-mgp 20080  df-ur 20121  df-ring 20174  df-orng 20796

This theorem is referenced by:  subofld  20814