Theorem isomnd 20146

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 6878	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑚))
3		fveq2 6863	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4	3	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → (Base‘𝑚) = (Base‘𝑀))
5	2, 4	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑀))
6		isomnd.0	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
7	5, 6	eqtr4di 2814	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = 𝐵)
8		raleq 3316	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
9	8	raleqbi1dv 3329	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
10	9	raleqbi1dv 3329	. . . . . . . . 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 639	. . . . . . 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 3799	. . . . . 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 3799	. . . . 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 3787	. . . 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 6878	. . . . 5 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) ∈ V)
17		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = (+g‘𝑚))
18		fveq2 6863	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
19	18	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (+g‘𝑚) = (+g‘𝑀))
20	17, 19	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = (+g‘𝑀))
21		isomnd.1	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑀)
22	20, 21	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = + )
23	22	oveqd 7409	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (𝑎𝑝𝑐) = (𝑎 + 𝑐))
24	22	oveqd 7409	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (𝑏𝑝𝑐) = (𝑏 + 𝑐))
25	23, 24	breq12d 5112	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → ((𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐) ↔ (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))
26	25	imbi2d 342	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → ((𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
27	26	ralbidv 3184	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
28	27	2ralbidv 3225	. . . . . . 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 639	. . . . . 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 3799	. . . . 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 3787	. . . 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 6878	. . . . . 6 ⊢ (𝑚 = 𝑀 → (le‘𝑚) ∈ V)
33		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑚))
34		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑚 = 𝑀)
35	34	fveq2d 6867	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (le‘𝑚) = (le‘𝑀))
36	33, 35	eqtrd 2796	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑀))
37		isomnd.2	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝑀)
38	36, 37	eqtr4di 2814	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = ≤ )
39	38	breqd 5110	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (𝑎𝑙𝑏 ↔ 𝑎 ≤ 𝑏))
40	38	breqd 5110	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → ((𝑎 + 𝑐)𝑙(𝑏 + 𝑐) ↔ (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))
41	39, 40	imbi12d 346	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → ((𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
42	41	ralbidv 3184	. . . . . . . 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 3225	. . . . . . 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 639	. . . . . 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 3787	. . . . 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 2849	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 ∈ Toset ↔ 𝑀 ∈ Toset))
47	46	anbi1d 640	. . . . 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 281	. . . 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 307	. . 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 20144	. . 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 3653	. 2 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ (𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))))
52		3anass 1105	. 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 280	1 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453  [wsbc 3744   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  lecple 17276  Tosetctos 18429  Mndcmnd 18751  oMndcomnd 20142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-ov 7395  df-omnd 20144

This theorem is referenced by:  omndmnd  20149  omndtos  20150  omndadd  20151  submomnd  20155  xrge0omnd  21477  zsoring  28479  reofld  33490