Theorem isomnd 33078

Description: A (left) ordered monoid is a monoid with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypotheses

Ref	Expression
isomnd.0	⊢ 𝐵 = (Base‘𝑀)
isomnd.1	⊢ + = (+g‘𝑀)
isomnd.2	⊢ ≤ = (le‘𝑀)

Assertion

Ref	Expression
isomnd	⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))

Distinct variable groups:   𝑎,𝑏,𝑐,𝐵   𝑀,𝑎,𝑏,𝑐

Allowed substitution hints:   + (𝑎,𝑏,𝑐)   ≤ (𝑎,𝑏,𝑐)

Proof of Theorem isomnd

Dummy variables 𝑙 𝑚 𝑝 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6921	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑚))
3		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → (Base‘𝑚) = (Base‘𝑀))
5	2, 4	eqtrd 2777	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑀))
6		isomnd.0	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
7	5, 6	eqtr4di 2795	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = 𝐵)
8		raleq 3323	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
9	8	raleqbi1dv 3338	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
10	9	raleqbi1dv 3338	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
11	7, 10	syl 17	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → (∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
12	11	anbi2d 630	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → ((𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ (𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))))
13	12	sbcbidv 3845	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → ([(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))))
14	13	sbcbidv 3845	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → ([(+g‘𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(+g‘𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))))
15	1, 14	sbcied 3832	. . . 4 ⊢ (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑣][(+g‘𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(+g‘𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))))
16		fvexd 6921	. . . . 5 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) ∈ V)
17		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = (+g‘𝑚))
18		fveq2 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
19	18	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (+g‘𝑚) = (+g‘𝑀))
20	17, 19	eqtrd 2777	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = (+g‘𝑀))
21		isomnd.1	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑀)
22	20, 21	eqtr4di 2795	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = + )
23	22	oveqd 7448	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (𝑎𝑝𝑐) = (𝑎 + 𝑐))
24	22	oveqd 7448	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (𝑏𝑝𝑐) = (𝑏 + 𝑐))
25	23, 24	breq12d 5156	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → ((𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐) ↔ (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))
26	25	imbi2d 340	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → ((𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
27	26	ralbidv 3178	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
28	27	2ralbidv 3221	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
29	28	anbi2d 630	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → ((𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ (𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))))
30	29	sbcbidv 3845	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → ([(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))))
31	16, 30	sbcied 3832	. . . 4 ⊢ (𝑚 = 𝑀 → ([(+g‘𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))))
32		fvexd 6921	. . . . . 6 ⊢ (𝑚 = 𝑀 → (le‘𝑚) ∈ V)
33		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑚))
34		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑚 = 𝑀)
35	34	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (le‘𝑚) = (le‘𝑀))
36	33, 35	eqtrd 2777	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑀))
37		isomnd.2	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝑀)
38	36, 37	eqtr4di 2795	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = ≤ )
39	38	breqd 5154	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (𝑎𝑙𝑏 ↔ 𝑎 ≤ 𝑏))
40	38	breqd 5154	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → ((𝑎 + 𝑐)𝑙(𝑏 + 𝑐) ↔ (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))
41	39, 40	imbi12d 344	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → ((𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
42	41	ralbidv 3178	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
43	42	2ralbidv 3221	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
44	43	anbi2d 630	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → ((𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))) ↔ (𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))))
45	32, 44	sbcied 3832	. . . . 5 ⊢ (𝑚 = 𝑀 → ([(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))) ↔ (𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))))
46		eleq1 2829	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 ∈ Toset ↔ 𝑀 ∈ Toset))
47	46	anbi1d 631	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) ↔ (𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))))
48	45, 47	bitrd 279	. . . 4 ⊢ (𝑚 = 𝑀 → ([(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))) ↔ (𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))))
49	15, 31, 48	3bitrd 305	. . 3 ⊢ (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑣][(+g‘𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ (𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))))
50		df-omnd 33076	. . 3 ⊢ oMnd = {𝑚 ∈ Mnd ∣ [(Base‘𝑚) / 𝑣][(+g‘𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))}
51	49, 50	elrab2 3695	. 2 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ (𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))))
52		3anass 1095	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))) ↔ (𝑀 ∈ Mnd ∧ (𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))))
53	51, 52	bitr4i 278	1 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480  [wsbc 3788   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  lecple 17304  Tosetctos 18461  Mndcmnd 18747  oMndcomnd 33074

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-omnd 33076

This theorem is referenced by:  omndmnd  33081  omndtos  33082  omndadd  33083  submomnd  33087  xrge0omnd  33088  reofld  33372