Theorem isorng 20945

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 3922	. . 3 ⊢ (𝑅 ∈ (Ring ∩ oGrp) ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp))
2	1	anbi1i 635	. 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 6898	. . . . 5 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) ∈ V)
4		simpr 489	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → 𝑡 = (.r‘𝑟))
5		simpl 487	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → 𝑟 = 𝑅)
6	5	fveq2d 6887	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → (.r‘𝑟) = (.r‘𝑅))
7		isorng.2	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
8	6, 7	eqtr4di 2816	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → (.r‘𝑟) = · )
9	4, 8	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → 𝑡 = · )
10	9	oveqd 7429	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → (𝑎𝑡𝑏) = (𝑎 · 𝑏))
11	10	breq2d 5122	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → ( 0 𝑙(𝑎𝑡𝑏) ↔ 0 𝑙(𝑎 · 𝑏)))
12	11	imbi2d 343	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑡 = (.r‘𝑟)) → ((( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
13	12	2ralbidv 3229	. . . . . 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 3800	. . . . 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 3788	. . . 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 6898	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
17		simpr 489	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → 𝑣 = (Base‘𝑟))
18		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
19		isorng.0	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑅)
20	18, 19	eqtr4di 2816	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
21	20	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → (Base‘𝑟) = 𝐵)
22	17, 21	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑣 = (Base‘𝑟)) → 𝑣 = 𝐵)
23		raleq 3320	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
24	23	raleqbi1dv 3333	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
25	22, 24	syl 18	. . . . . . . . 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 3800	. . . . . . . 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 3800	. . . . . . 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 3800	. . . . . 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 3788	. . . . 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 6898	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) ∈ V)
31		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → 𝑧 = (0g‘𝑟))
32		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
33		isorng.1	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝑅)
34	32, 33	eqtr4di 2816	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
35	34	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (0g‘𝑟) = 0 )
36	31, 35	eqtrd 2798	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → 𝑧 = 0 )
37	36	breq1d 5120	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (𝑧𝑙𝑎 ↔ 0 𝑙𝑎))
38	36	breq1d 5120	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (𝑧𝑙𝑏 ↔ 0 𝑙𝑏))
39	37, 38	anbi12d 643	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) ↔ ( 0 𝑙𝑎 ∧ 0 𝑙𝑏)))
40	36	breq1d 5120	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (𝑧𝑙(𝑎𝑡𝑏) ↔ 0 𝑙(𝑎𝑡𝑏)))
41	39, 40	imbi12d 347	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑧 = (0g‘𝑟)) → (((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
42	41	2ralbidv 3229	. . . . . . . 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 3800	. . . . . . 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 3800	. . . . . 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 3788	. . . . 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 283	. . . 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 6898	. . . . 5 ⊢ (𝑟 = 𝑅 → (le‘𝑟) ∈ V)
48		simpr 489	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → 𝑙 = (le‘𝑟))
49		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → 𝑟 = 𝑅)
50	49	fveq2d 6887	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → (le‘𝑟) = (le‘𝑅))
51		isorng.3	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝑅)
52	50, 51	eqtr4di 2816	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → (le‘𝑟) = ≤ )
53	48, 52	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → 𝑙 = ≤ )
54	53	breqd 5121	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → ( 0 𝑙𝑎 ↔ 0 ≤ 𝑎))
55	53	breqd 5121	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → ( 0 𝑙𝑏 ↔ 0 ≤ 𝑏))
56	54, 55	anbi12d 643	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → (( 0 𝑙𝑎 ∧ 0 𝑙𝑏) ↔ ( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏)))
57	53	breqd 5121	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → ( 0 𝑙(𝑎 · 𝑏) ↔ 0 ≤ (𝑎 · 𝑏)))
58	56, 57	imbi12d 347	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑙 = (le‘𝑟)) → ((( 0 𝑙𝑎 ∧ 0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
59	58	2ralbidv 3229	. . . . 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 3788	. . . 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 312	. . 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 20943	. . 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 3655	. 2 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ (Ring ∩ oGrp) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
64		df-3an 1105	. 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 306	1 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455  [wsbc 3745   ∩ cin 3905   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  ._rcmulr 17312  lecple 17318  0_gc0g 17493  oGrpcogrp 20191  Ringcrg 20316  oRingcorng 20941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-orng 20943

This theorem is referenced by:  orngring  20946  orngogrp  20947  orngmul  20949  suborng  20960  zsoring  28583  reofld  33644