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Theorem suborng 32703

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 483	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (𝑅 ↾_s 𝐴) ∈ Ring)
2		ringgrp 20132	. . . 4 ⊢ ((𝑅 ↾_s 𝐴) ∈ Ring → (𝑅 ↾_s 𝐴) ∈ Grp)
3	2	adantl 480	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (𝑅 ↾_s 𝐴) ∈ Grp)
4		orngogrp 32689	. . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5		isogrp 32490	. . . . . 6 ⊢ (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
6	5	simprbi 495	. . . . 5 ⊢ (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
7	4, 6	syl 17	. . . 4 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
8		ringmnd 20137	. . . 4 ⊢ ((𝑅 ↾_s 𝐴) ∈ Ring → (𝑅 ↾_s 𝐴) ∈ Mnd)
9		submomnd 32498	. . . 4 ⊢ ((𝑅 ∈ oMnd ∧ (𝑅 ↾_s 𝐴) ∈ Mnd) → (𝑅 ↾_s 𝐴) ∈ oMnd)
10	7, 8, 9	syl2an 594	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (𝑅 ↾_s 𝐴) ∈ oMnd)
11		isogrp 32490	. . 3 ⊢ ((𝑅 ↾_s 𝐴) ∈ oGrp ↔ ((𝑅 ↾_s 𝐴) ∈ Grp ∧ (𝑅 ↾_s 𝐴) ∈ oMnd))
12	3, 10, 11	sylanbrc 581	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (𝑅 ↾_s 𝐴) ∈ oGrp)
13		simp-4l 779	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → 𝑅 ∈ oRing)
14		reldmress 17179	. . . . . . . . . . . . . . 15 ⊢ Rel dom ↾_s
15	14	ovprc2 7451	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐴 ∈ V → (𝑅 ↾_s 𝐴) = ∅)
16	15	fveq2d 6894	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ V → (Base‘(𝑅 ↾_s 𝐴)) = (Base‘∅))
17	16	adantl 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾_s 𝐴)) = (Base‘∅))
18		base0 17153	. . . . . . . . . . . 12 ⊢ ∅ = (Base‘∅)
19	17, 18	eqtr4di 2788	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾_s 𝐴)) = ∅)
20		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (Base‘(𝑅 ↾_s 𝐴)) = (Base‘(𝑅 ↾_s 𝐴))
21		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (1_r‘(𝑅 ↾_s 𝐴)) = (1_r‘(𝑅 ↾_s 𝐴))
22	20, 21	ringidcl 20154	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ↾_s 𝐴) ∈ Ring → (1_r‘(𝑅 ↾_s 𝐴)) ∈ (Base‘(𝑅 ↾_s 𝐴)))
23	22	ne0d 4334	. . . . . . . . . . . . 13 ⊢ ((𝑅 ↾_s 𝐴) ∈ Ring → (Base‘(𝑅 ↾_s 𝐴)) ≠ ∅)
24	23	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾_s 𝐴)) ≠ ∅)
25	24	neneqd 2943	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅 ↾_s 𝐴)) = ∅)
26	19, 25	condan 814	. . . . . . . . . 10 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → 𝐴 ∈ V)
27		eqid 2730	. . . . . . . . . . . 12 ⊢ (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴)
28		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	27, 28	ressbas 17183	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾_s 𝐴)))
30		inss2 4228	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
31	29, 30	eqsstrrdi 4036	. . . . . . . . . 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 726	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (Base‘(𝑅 ↾_s 𝐴)) ⊆ (Base‘𝑅))
34		simpllr 772	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴)))
35	33, 34	sseldd 3982	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → 𝑎 ∈ (Base‘𝑅))
36		simprl 767	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎)
37		orngring 32688	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38		ringgrp 20132	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39	37, 38	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Grp)
40	39	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → 𝑅 ∈ Grp)
41	28	ressinbas 17194	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾_s 𝐴) = (𝑅 ↾_s (𝐴 ∩ (Base‘𝑅))))
42	29	oveq2d 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾_s (𝐴 ∩ (Base‘𝑅))) = (𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴))))
43	41, 42	eqtrd 2770	. . . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴))) ∈ Grp)
46	28	issubg 19042	. . . . . . . . . . . . . 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 1341	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (Base‘(𝑅 ↾_s 𝐴)) ∈ (SubGrp‘𝑅))
48		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴))) = (𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴)))
49		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
50	48, 49	subg0 19048	. . . . . . . . . . . . 13 ⊢ ((Base‘(𝑅 ↾_s 𝐴)) ∈ (SubGrp‘𝑅) → (0_g‘𝑅) = (0_g‘(𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴)))))
51	47, 50	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (0_g‘𝑅) = (0_g‘(𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴)))))
52	44	fveq2d 6894	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (0_g‘(𝑅 ↾_s 𝐴)) = (0_g‘(𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴)))))
53	51, 52	eqtr4d 2773	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (0_g‘𝑅) = (0_g‘(𝑅 ↾_s 𝐴)))
54	53	ad2antrr 722	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) → (0_g‘𝑅) = (0_g‘(𝑅 ↾_s 𝐴)))
55	26	ad2antrr 722	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) → 𝐴 ∈ V)
56		eqid 2730	. . . . . . . . . . . 12 ⊢ (le‘𝑅) = (le‘𝑅)
57	27, 56	ressle 17329	. . . . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) → 𝑎 = 𝑎)
60	54, 58, 59	breq123d 5161	. . . . . . . . 9 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) → ((0_g‘𝑅)(le‘𝑅)𝑎 ↔ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎))
61	60	adantr 479	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → ((0_g‘𝑅)(le‘𝑅)𝑎 ↔ (0_g‘(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))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (0_g‘𝑅)(le‘𝑅)𝑎)
63		simplr 765	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴)))
64	33, 63	sseldd 3982	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → 𝑏 ∈ (Base‘𝑅))
65		simprr 769	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)
66		eqidd 2731	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) → 𝑏 = 𝑏)
67	54, 58, 66	breq123d 5161	. . . . . . . . 9 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) → ((0_g‘𝑅)(le‘𝑅)𝑏 ↔ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏))
68	67	adantr 479	. . . . . . . 8 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → ((0_g‘𝑅)(le‘𝑅)𝑏 ↔ (0_g‘(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))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (0_g‘𝑅)(le‘𝑅)𝑏)
70		eqid 2730	. . . . . . . 8 ⊢ (._r‘𝑅) = (._r‘𝑅)
71	28, 56, 49, 70	orngmul 32691	. . . . . . 7 ⊢ ((𝑅 ∈ oRing ∧ (𝑎 ∈ (Base‘𝑅) ∧ (0_g‘𝑅)(le‘𝑅)𝑎) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (0_g‘𝑅)(le‘𝑅)𝑏)) → (0_g‘𝑅)(le‘𝑅)(𝑎(._r‘𝑅)𝑏))
72	13, 35, 62, 64, 69, 71	syl122anc 1377	. . . . . 6 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (0_g‘𝑅)(le‘𝑅)(𝑎(._r‘𝑅)𝑏))
73	54	adantr 479	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (0_g‘𝑅) = (0_g‘(𝑅 ↾_s 𝐴)))
74	58	adantr 479	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (le‘𝑅) = (le‘(𝑅 ↾_s 𝐴)))
75	55	adantr 479	. . . . . . . . 9 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → 𝐴 ∈ V)
76	27, 70	ressmulr 17256	. . . . . . . . 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))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (._r‘𝑅) = (._r‘(𝑅 ↾_s 𝐴)))
78	77	oveqd 7428	. . . . . . 7 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (𝑎(._r‘𝑅)𝑏) = (𝑎(._r‘(𝑅 ↾_s 𝐴))𝑏))
79	73, 74, 78	breq123d 5161	. . . . . 6 ⊢ (((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → ((0_g‘𝑅)(le‘𝑅)(𝑎(._r‘𝑅)𝑏) ↔ (0_g‘(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))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))(𝑎(._r‘(𝑅 ↾_s 𝐴))𝑏))
81	80	ex 411	. . . 4 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) → (((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏) → (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))(𝑎(._r‘(𝑅 ↾_s 𝐴))𝑏)))
82	81	anasss 465	. . 3 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ (𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴)))) → (((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏) → (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))(𝑎(._r‘(𝑅 ↾_s 𝐴))𝑏)))
83	82	ralrimivva 3198	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → ∀𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))∀𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))(((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏) → (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))(𝑎(._r‘(𝑅 ↾_s 𝐴))𝑏)))
84		eqid 2730	. . 3 ⊢ (0_g‘(𝑅 ↾_s 𝐴)) = (0_g‘(𝑅 ↾_s 𝐴))
85		eqid 2730	. . 3 ⊢ (._r‘(𝑅 ↾_s 𝐴)) = (._r‘(𝑅 ↾_s 𝐴))
86		eqid 2730	. . 3 ⊢ (le‘(𝑅 ↾_s 𝐴)) = (le‘(𝑅 ↾_s 𝐴))
87	20, 84, 85, 86	isorng 32687	. 2 ⊢ ((𝑅 ↾_s 𝐴) ∈ oRing ↔ ((𝑅 ↾_s 𝐴) ∈ Ring ∧ (𝑅 ↾_s 𝐴) ∈ oGrp ∧ ∀𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))∀𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))(((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏) → (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))(𝑎(._r‘(𝑅 ↾_s 𝐴))𝑏))))
88	1, 12, 83, 87	syl3anbrc 1341	1 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (𝑅 ↾_s 𝐴) ∈ oRing)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 Vcvv 3472 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 Basecbs 17148 ↾_s cress 17177 ._rcmulr 17202 lecple 17208 0_gc0g 17389 Mndcmnd 18659 Grpcgrp 18855 SubGrpcsubg 19036 1_rcur 20075 Ringcrg 20127 oMndcomnd 32485 oGrpcogrp 32486 oRingcorng 32683

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-dec 12682 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-ple 17221 df-0g 17391 df-poset 18270 df-toset 18374 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-subg 19039 df-mgp 20029 df-ur 20076 df-ring 20129 df-omnd 32487 df-ogrp 32488 df-orng 32685

This theorem is referenced by: subofld 32704

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