Theorem isomnd 31909

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 6857	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑚))
3		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → (Base‘𝑚) = (Base‘𝑀))
5	2, 4	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑀))
6		isomnd.0	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
7	5, 6	eqtr4di 2794	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = 𝐵)
8		raleq 3309	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
9	8	raleqbi1dv 3307	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
10	9	raleqbi1dv 3307	. . . . . . . . 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 629	. . . . . . 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 3798	. . . . . 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 3798	. . . . 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 3784	. . . 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 6857	. . . . 5 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) ∈ V)
17		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = (+g‘𝑚))
18		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
19	18	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (+g‘𝑚) = (+g‘𝑀))
20	17, 19	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = (+g‘𝑀))
21		isomnd.1	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑀)
22	20, 21	eqtr4di 2794	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = + )
23	22	oveqd 7374	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (𝑎𝑝𝑐) = (𝑎 + 𝑐))
24	22	oveqd 7374	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (𝑏𝑝𝑐) = (𝑏 + 𝑐))
25	23, 24	breq12d 5118	. . . . . . . . . 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 3174	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
28	27	2ralbidv 3212	. . . . . . 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 629	. . . . . 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 3798	. . . . 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 3784	. . . 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 6857	. . . . . 6 ⊢ (𝑚 = 𝑀 → (le‘𝑚) ∈ V)
33		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑚))
34		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑚 = 𝑀)
35	34	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (le‘𝑚) = (le‘𝑀))
36	33, 35	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑀))
37		isomnd.2	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝑀)
38	36, 37	eqtr4di 2794	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = ≤ )
39	38	breqd 5116	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (𝑎𝑙𝑏 ↔ 𝑎 ≤ 𝑏))
40	38	breqd 5116	. . . . . . . . . 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 3174	. . . . . . . 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 3212	. . . . . . 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 629	. . . . . 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 3784	. . . . 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 2825	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 ∈ Toset ↔ 𝑀 ∈ Toset))
47	46	anbi1d 630	. . . . 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 278	. . . 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 304	. . 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 31907	. . 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 3648	. 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 277	1 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445  [wsbc 3739   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  lecple 17140  Tosetctos 18305  Mndcmnd 18556  oMndcomnd 31905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360  df-omnd 31907

This theorem is referenced by:  omndmnd  31912  omndtos  31913  omndadd  31914  submomnd  31918  xrge0omnd  31919  reofld  32136