Theorem isorng 33326

Description: An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)

Hypotheses

Ref	Expression
isorng.0	⊢ 𝐵 = (Base‘𝑅)
isorng.1	⊢ 0 = (0g‘𝑅)
isorng.2	⊢ · = (.r‘𝑅)
isorng.3	⊢ ≤ = (le‘𝑅)

Assertion

Ref	Expression
isorng	⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))

Distinct variable groups:   𝑎,𝑏,𝐵   𝑅,𝑎,𝑏

Allowed substitution hints:   · (𝑎,𝑏)   ≤ (𝑎,𝑏)   0 (𝑎,𝑏)

Proof of Theorem isorng

Dummy variables 𝑙 𝑟 𝑡 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3947	. . 3 ⊢ (𝑅 ∈ (Ring ∩ oGrp) ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp))
2	1	anbi1i 624	. 2 ⊢ ((𝑅 ∈ (Ring ∩ oGrp) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))) ↔ ((𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
3		fvexd 6896	. . . . 5 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) ∈ V)
4		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → 𝑡 = (.r‘𝑟))
5		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → 𝑟 = 𝑅)
6	5	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → (.r‘𝑟) = (.r‘𝑅))
7		isorng.2	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
8	6, 7	eqtr4di 2789	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → (.r‘𝑟) = · )
9	4, 8	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → 𝑡 = · )
10	9	oveqd 7427	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → (𝑎𝑡𝑏) = (𝑎 · 𝑏))
11	10	breq2d 5136	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → ( 0 𝑙(𝑎𝑡𝑏) ↔ 0 𝑙(𝑎 · 𝑏)))
12	11	imbi2d 340	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → ((( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
13	12	2ralbidv 3209	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
14	13	sbcbidv 3826	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → ([(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
15	3, 14	sbcied 3814	. . . 4 ⊢ (𝑟 = 𝑅 → ([(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
16		fvexd 6896	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
17		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → 𝑣 = (Base‘𝑟))
18		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
19		isorng.0	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑅)
20	18, 19	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
21	20	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → (Base‘𝑟) = 𝐵)
22	17, 21	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → 𝑣 = 𝐵)
23		raleq 3306	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
24	23	raleqbi1dv 3321	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
25	22, 24	syl 17	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → (∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
26	25	sbcbidv 3826	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → ([(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
27	26	sbcbidv 3826	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → ([(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
28	27	sbcbidv 3826	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → ([(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
29	16, 28	sbcied 3814	. . . . 5 ⊢ (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑣][(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
30		fvexd 6896	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) ∈ V)
31		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → 𝑧 = (0g‘𝑟))
32		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
33		isorng.1	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝑅)
34	32, 33	eqtr4di 2789	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
35	34	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (0g‘𝑟) = 0 )
36	31, 35	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → 𝑧 = 0 )
37	36	breq1d 5134	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (𝑧𝑙𝑎 ↔ 0 𝑙𝑎))
38	36	breq1d 5134	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (𝑧𝑙𝑏 ↔ 0 𝑙𝑏))
39	37, 38	anbi12d 632	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) ↔ ( 0 𝑙𝑎 ∧ 0 𝑙𝑏)))
40	36	breq1d 5134	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (𝑧𝑙(𝑎𝑡𝑏) ↔ 0 𝑙(𝑎𝑡𝑏)))
41	39, 40	imbi12d 344	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
42	41	2ralbidv 3209	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
43	42	sbcbidv 3826	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → ([(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
44	43	sbcbidv 3826	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → ([(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
45	30, 44	sbcied 3814	. . . . 5 ⊢ (𝑟 = 𝑅 → ([(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
46	29, 45	bitr2d 280	. . . 4 ⊢ (𝑟 = 𝑅 → ([(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ [(Base‘𝑟) / 𝑣][(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
47		fvexd 6896	. . . . 5 ⊢ (𝑟 = 𝑅 → (le‘𝑟) ∈ V)
48		simpr 484	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → 𝑙 = (le‘𝑟))
49		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → 𝑟 = 𝑅)
50	49	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → (le‘𝑟) = (le‘𝑅))
51		isorng.3	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝑅)
52	50, 51	eqtr4di 2789	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → (le‘𝑟) = ≤ )
53	48, 52	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → 𝑙 = ≤ )
54	53	breqd 5135	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → ( 0 𝑙𝑎 ↔ 0 ≤ 𝑎))
55	53	breqd 5135	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → ( 0 𝑙𝑏 ↔ 0 ≤ 𝑏))
56	54, 55	anbi12d 632	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) ↔ ( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏)))
57	53	breqd 5135	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → ( 0 𝑙(𝑎 · 𝑏) ↔ 0 ≤ (𝑎 · 𝑏)))
58	56, 57	imbi12d 344	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → ((( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
59	58	2ralbidv 3209	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
60	47, 59	sbcied 3814	. . . 4 ⊢ (𝑟 = 𝑅 → ([(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
61	15, 46, 60	3bitr3d 309	. . 3 ⊢ (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑣][(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
62		df-orng 33324	. . 3 ⊢ oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))}
63	61, 62	elrab2 3679	. 2 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ (Ring ∩ oGrp) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
64		df-3an 1088	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))) ↔ ((𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
65	2, 63, 64	3bitr4i 303	1 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464  [wsbc 3770   ∩ cin 3930   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  ._rcmulr 17277  lecple 17283  0_gc0g 17458  Ringcrg 20198  oGrpcogrp 33071  oRingcorng 33322

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-orng 33324

This theorem is referenced by:  orngring  33327  orngogrp  33328  orngmul  33330  suborng  33342  reofld  33364