Theorem suborng 21134

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 490	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ Ring)
2		ringgrp 20465	. . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (𝑅 ↾s 𝐴) ∈ Grp)
3	2	adantl 487	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ Grp)
4		orngogrp 21121	. . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5		isogrp 20339	. . . . . 6 ⊢ (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd))
6	5	simprbi 503	. . . . 5 ⊢ (𝑅 ∈ oGrp → 𝑅 ∈ oMnd)
7	4, 6	syl 18	. . . 4 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oMnd)
8		ringmnd 20471	. . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (𝑅 ↾s 𝐴) ∈ Mnd)
9		submomnd 20347	. . . 4 ⊢ ((𝑅 ∈ oMnd ∧ (𝑅 ↾s 𝐴) ∈ Mnd) → (𝑅 ↾s 𝐴) ∈ oMnd)
10	7, 8, 9	syl2an 608	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oMnd)
11		isogrp 20339	. . 3 ⊢ ((𝑅 ↾s 𝐴) ∈ oGrp ↔ ((𝑅 ↾s 𝐴) ∈ Grp ∧ (𝑅 ↾s 𝐴) ∈ oMnd))
12	3, 10, 11	sylanbrc 595	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oGrp)
13		simp-4l 795	. . . . . . 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 17411	. . . . . . . . . . . . . . 15 ⊢ Rel dom ↾s
15	14	ovprc2 7460	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐴 ∈ V → (𝑅 ↾s 𝐴) = ∅)
16	15	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ V → (Base‘(𝑅 ↾s 𝐴)) = (Base‘∅))
17	16	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) = (Base‘∅))
18		base0 17393	. . . . . . . . . . . 12 ⊢ ∅ = (Base‘∅)
19	17, 18	eqtr4di 2813	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) = ∅)
20		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴))
21		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (1r‘(𝑅 ↾s 𝐴)) = (1r‘(𝑅 ↾s 𝐴))
22	20, 21	ringidcl 20495	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (1r‘(𝑅 ↾s 𝐴)) ∈ (Base‘(𝑅 ↾s 𝐴)))
23	22	ne0d 4287	. . . . . . . . . . . . 13 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (Base‘(𝑅 ↾s 𝐴)) ≠ ∅)
24	23	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) ≠ ∅)
25	24	neneqd 2960	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑅 ↾s 𝐴)) = ∅)
26	19, 25	condan 830	. . . . . . . . . 10 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → 𝐴 ∈ V)
27		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
28		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	27, 28	ressbas 17415	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝐴)))
30		inss2 4134	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
31	29, 30	eqsstrrdi 3953	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (Base‘(𝑅 ↾s 𝐴)) ⊆ (Base‘𝑅))
32	26, 31	syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (Base‘(𝑅 ↾s 𝐴)) ⊆ (Base‘𝑅))
33	32	ad3antrrr 743	. . . . . . . 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 788	. . . . . . . 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 3909	. . . . . . 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 783	. . . . . . . 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 21120	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38		ringgrp 20465	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39	37, 38	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Grp)
40	39	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → 𝑅 ∈ Grp)
41	28	ressinbas 17424	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s 𝐴) = (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))))
42	29	oveq2d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))))
43	41, 42	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ V → (𝑅 ↾s 𝐴) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))))
44	26, 43	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))))
45	44, 3	eqeltrrd 2861	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) ∈ Grp)
46	28	issubg 19337	. . . . . . . . . . . . . 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 1362	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (Base‘(𝑅 ↾s 𝐴)) ∈ (SubGrp‘𝑅))
48		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))
49		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	48, 49	subg0 19343	. . . . . . . . . . . . 13 ⊢ ((Base‘(𝑅 ↾s 𝐴)) ∈ (SubGrp‘𝑅) → (0g‘𝑅) = (0g‘(𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))))
51	47, 50	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘𝑅) = (0g‘(𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))))
52	44	fveq2d 6889	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))))
53	51, 52	eqtr4d 2798	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝐴)))
54	53	ad2antrr 739	. . . . . . . . . 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 739	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝐴 ∈ V)
56		eqid 2760	. . . . . . . . . . . 12 ⊢ (le‘𝑅) = (le‘𝑅)
57	27, 56	ressle 17552	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (le‘𝑅) = (le‘(𝑅 ↾s 𝐴)))
58	55, 57	syl 18	. . . . . . . . . 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 2761	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑎 = 𝑎)
60	54, 58, 59	breq123d 5116	. . . . . . . . 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 486	. . . . . . . 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 260	. . . . . . 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 781	. . . . . . . 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 3909	. . . . . . 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 785	. . . . . . . 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 2761	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑏 = 𝑏)
67	54, 58, 66	breq123d 5116	. . . . . . . . 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 486	. . . . . . . 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 260	. . . . . . 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 2760	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
71	28, 56, 49, 70	orngmul 21123	. . . . . . 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 1406	. . . . . 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 486	. . . . . . 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 486	. . . . . . 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 486	. . . . . . . . 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 17479	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (.r‘𝑅) = (.r‘(𝑅 ↾s 𝐴)))
77	75, 76	syl 18	. . . . . . . 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 7437	. . . . . . 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 5116	. . . . . 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 235	. . . . 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 418	. . . 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 472	. . 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 3205	. 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 2760	. . 3 ⊢ (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s 𝐴))
85		eqid 2760	. . 3 ⊢ (.r‘(𝑅 ↾s 𝐴)) = (.r‘(𝑅 ↾s 𝐴))
86		eqid 2760	. . 3 ⊢ (le‘(𝑅 ↾s 𝐴)) = (le‘(𝑅 ↾s 𝐴))
87	20, 84, 85, 86	isorng 21119	. 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 1362	1 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oRing)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17388   ↾_s cress 17409  ._rcmulr 17430  lecple 17436  0_gc0g 17611  Mndcmnd 18924  Grpcgrp 19145  SubGrpcsubg 19331  oMndcomnd 20334  oGrpcogrp 20335  1_rcur 20408  Ringcrg 20460  oRingcorng 21115

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-er 8718  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-nn 12337  df-2 12406  df-3 12407  df-4 12408  df-5 12409  df-6 12410  df-7 12411  df-8 12412  df-9 12413  df-n0 12608  df-z 12695  df-dec 12816  df-sets 17343  df-slot 17361  df-ndx 17373  df-base 17389  df-ress 17410  df-plusg 17442  df-mulr 17443  df-ple 17449  df-0g 17613  df-poset 18488  df-toset 18590  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-grp 19148  df-subg 19334  df-omnd 20336  df-ogrp 20337  df-mgp 20362  df-ur 20409  df-ring 20462  df-orng 21117

This theorem is used by:  subofld  21135