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Theorem suborng 32421

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 20054	. . . 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 32407	. . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5		isogrp 32207	. . . . . 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 20059	. . . 4 ⊢ ((𝑅 ↾_s 𝐴) ∈ Ring → (𝑅 ↾_s 𝐴) ∈ Mnd)
9		submomnd 32215	. . . 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 32207	. . 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))) ∧ ((0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑎 ∧ (0_g‘(𝑅 ↾_s 𝐴))(le‘(𝑅 ↾_s 𝐴))𝑏)) → 𝑅 ∈ oRing)
14		reldmress 17171	. . . . . . . . . . . . . . 15 ⊢ Rel dom ↾_s
15	14	ovprc2 7445	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐴 ∈ V → (𝑅 ↾_s 𝐴) = ∅)
16	15	fveq2d 6892	. . . . . . . . . . . . 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 17145	. . . . . . . . . . . 12 ⊢ ∅ = (Base‘∅)
19	17, 18	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾_s 𝐴)) = ∅)
20		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (Base‘(𝑅 ↾_s 𝐴)) = (Base‘(𝑅 ↾_s 𝐴))
21		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (1_r‘(𝑅 ↾_s 𝐴)) = (1_r‘(𝑅 ↾_s 𝐴))
22	20, 21	ringidcl 20076	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾_s 𝐴)) ≠ ∅)
25	24	neneqd 2945	. . . . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴)
28		eqid 2732	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	27, 28	ressbas 17175	. . . . . . . . . . 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 728	. . . . . . . 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 774	. . . . . . . 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 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))𝑎)
37		orngring 32406	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38		ringgrp 20054	. . . . . . . . . . . . . . . 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 17186	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾_s 𝐴) = (𝑅 ↾_s (𝐴 ∩ (Base‘𝑅))))
42	29	oveq2d 7421	. . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴))) ∈ Grp)
46	28	issubg 19000	. . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . 14 ⊢ (𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴))) = (𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴)))
49		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
50	48, 49	subg0 19006	. . . . . . . . . . . . 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 6892	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (0_g‘(𝑅 ↾_s 𝐴)) = (0_g‘(𝑅 ↾_s (Base‘(𝑅 ↾_s 𝐴)))))
53	51, 52	eqtr4d 2775	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) → (0_g‘𝑅) = (0_g‘(𝑅 ↾_s 𝐴)))
54	53	ad2antrr 724	. . . . . . . . . 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 724	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾_s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾_s 𝐴))) → 𝐴 ∈ V)
56		eqid 2732	. . . . . . . . . . . 12 ⊢ (le‘𝑅) = (le‘𝑅)
57	27, 56	ressle 17321	. . . . . . . . . . 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 2733	. . . . . . . . . 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 481	. . . . . . . 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 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)) → 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 771	. . . . . . . 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 2733	. . . . . . . . . 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 481	. . . . . . . 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 2732	. . . . . . . 8 ⊢ (._r‘𝑅) = (._r‘𝑅)
71	28, 56, 49, 70	orngmul 32409	. . . . . . 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 1379	. . . . . 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 481	. . . . . . 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 481	. . . . . . 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 481	. . . . . . . . 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 17248	. . . . . . . . 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 7422	. . . . . . 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 413	. . . 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 467	. . 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 3200	. 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 2732	. . 3 ⊢ (0_g‘(𝑅 ↾_s 𝐴)) = (0_g‘(𝑅 ↾_s 𝐴))
85		eqid 2732	. . 3 ⊢ (._r‘(𝑅 ↾_s 𝐴)) = (._r‘(𝑅 ↾_s 𝐴))
86		eqid 2732	. . 3 ⊢ (le‘(𝑅 ↾_s 𝐴)) = (le‘(𝑅 ↾_s 𝐴))
87	20, 84, 85, 86	isorng 32405	. 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 1343	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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 ↾_s cress 17169 ._rcmulr 17194 lecple 17200 0_gc0g 17381 Mndcmnd 18621 Grpcgrp 18815 SubGrpcsubg 18994 1_rcur 19998 Ringcrg 20049 oMndcomnd 32202 oGrpcogrp 32203 oRingcorng 32401

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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-dec 12674 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-ple 17213 df-0g 17383 df-poset 18262 df-toset 18366 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-subg 18997 df-mgp 19982 df-ur 19999 df-ring 20051 df-omnd 32204 df-ogrp 32205 df-orng 32403

This theorem is referenced by: subofld 32422

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