Theorem suborng 33325

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 20256	. . . 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 33311	. . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
5		isogrp 33062	. . . . . 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 20261	. . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (𝑅 ↾s 𝐴) ∈ Mnd)
9		submomnd 33070	. . . 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 33062	. . 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 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 17276	. . . . . . . . . . . . . . 15 ⊢ Rel dom ↾s
15	14	ovprc2 7471	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐴 ∈ V → (𝑅 ↾s 𝐴) = ∅)
16	15	fveq2d 6911	. . . . . . . . . . . . 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 17250	. . . . . . . . . . . 12 ⊢ ∅ = (Base‘∅)
19	17, 18	eqtr4di 2793	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑅 ↾s 𝐴)) = ∅)
20		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴))
21		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (1r‘(𝑅 ↾s 𝐴)) = (1r‘(𝑅 ↾s 𝐴))
22	20, 21	ringidcl 20280	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (1r‘(𝑅 ↾s 𝐴)) ∈ (Base‘(𝑅 ↾s 𝐴)))
23	22	ne0d 4348	. . . . . . . . . . . . 13 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (Base‘(𝑅 ↾s 𝐴)) ≠ ∅)
24	23	ad2antlr 727	. . . . . . . . . . . 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 818	. . . . . . . . . 10 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → 𝐴 ∈ V)
27		eqid 2735	. . . . . . . . . . . 12 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
28		eqid 2735	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	27, 28	ressbas 17280	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝐴)))
30		inss2 4246	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (Base‘𝑅)) ⊆ (Base‘𝑅)
31	29, 30	eqsstrrdi 4051	. . . . . . . . . 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 730	. . . . . . . 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 3996	. . . . . . 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 33310	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)
38		ringgrp 20256	. . . . . . . . . . . . . . . 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 17291	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s 𝐴) = (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))))
42	29	oveq2d 7447	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ V → (𝑅 ↾s (𝐴 ∩ (Base‘𝑅))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))))
43	41, 42	eqtrd 2775	. . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) ∈ Grp)
46	28	issubg 19157	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴))) = (𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))
49		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	48, 49	subg0 19163	. . . . . . . . . . . . 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 6911	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s (Base‘(𝑅 ↾s 𝐴)))))
53	51, 52	eqtr4d 2778	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝐴)))
54	53	ad2antrr 726	. . . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝐴 ∈ V)
56		eqid 2735	. . . . . . . . . . . 12 ⊢ (le‘𝑅) = (le‘𝑅)
57	27, 56	ressle 17426	. . . . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑎 = 𝑎)
60	54, 58, 59	breq123d 5162	. . . . . . . . 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 3996	. . . . . . 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 2736	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ 𝑎 ∈ (Base‘(𝑅 ↾s 𝐴))) ∧ 𝑏 ∈ (Base‘(𝑅 ↾s 𝐴))) → 𝑏 = 𝑏)
67	54, 58, 66	breq123d 5162	. . . . . . . . 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 2735	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
71	28, 56, 49, 70	orngmul 33313	. . . . . . 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 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 17353	. . . . . . . . 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 7448	. . . . . . 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 5162	. . . . . 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 3200	. 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 2735	. . 3 ⊢ (0g‘(𝑅 ↾s 𝐴)) = (0g‘(𝑅 ↾s 𝐴))
85		eqid 2735	. . 3 ⊢ (.r‘(𝑅 ↾s 𝐴)) = (.r‘(𝑅 ↾s 𝐴))
86		eqid 2735	. . 3 ⊢ (le‘(𝑅 ↾s 𝐴)) = (le‘(𝑅 ↾s 𝐴))
87	20, 84, 85, 86	isorng 33309	. 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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245   ↾_s cress 17274  ._rcmulr 17299  lecple 17305  0_gc0g 17486  Mndcmnd 18760  Grpcgrp 18964  SubGrpcsubg 19151  1_rcur 20199  Ringcrg 20251  oMndcomnd 33057  oGrpcogrp 33058  oRingcorng 33305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-dec 12732  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-ple 17318  df-0g 17488  df-poset 18371  df-toset 18475  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-subg 19154  df-mgp 20153  df-ur 20200  df-ring 20253  df-omnd 33059  df-ogrp 33060  df-orng 33307

This theorem is referenced by:  subofld  33326