Theorem isomnd 29032

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		fvex 6113	. . . . . 6 ⊢ (Base‘𝑚) ∈ V
2	1	a1i 11	. . . . 5 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
3		simpr 476	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑚))
4		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → (Base‘𝑚) = (Base‘𝑀))
6	3, 5	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑀))
7		isomnd.0	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
8	6, 7	syl6eqr 2662	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑣 = (Base‘𝑚)) → 𝑣 = 𝐵)
9		raleq 3115	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
10	9	raleqbi1dv 3123	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
11	10	raleqbi1dv 3123	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → (∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
12	8, 11	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))))
13	12	anbi2d 736	. . . . . . 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)))))
14	13	sbcbidv 3457	. . . . . 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)))))
15	14	sbcbidv 3457	. . . . 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)))))
16	2, 15	sbcied 3439	. . . 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)))))
17		fvex 6113	. . . . . 6 ⊢ (+g‘𝑚) ∈ V
18	17	a1i 11	. . . . 5 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) ∈ V)
19		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = (+g‘𝑚))
20		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
21	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (+g‘𝑚) = (+g‘𝑀))
22	19, 21	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = (+g‘𝑀))
23		isomnd.1	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑀)
24	22, 23	syl6eqr 2662	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → 𝑝 = + )
25	24	oveqd 6566	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (𝑎𝑝𝑐) = (𝑎 + 𝑐))
26	24	oveqd 6566	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (𝑏𝑝𝑐) = (𝑏 + 𝑐))
27	25, 26	breq12d 4596	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → ((𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐) ↔ (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))
28	27	imbi2d 329	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → ((𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
29	28	ralbidv 2969	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑝 = (+g‘𝑚)) → (∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
30	29	2ralbidv 2972	. . . . . . 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))))
31	30	anbi2d 736	. . . . . 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)))))
32	31	sbcbidv 3457	. . . . 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)))))
33	18, 32	sbcied 3439	. . . 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)))))
34		fvex 6113	. . . . . . 7 ⊢ (le‘𝑚) ∈ V
35	34	a1i 11	. . . . . 6 ⊢ (𝑚 = 𝑀 → (le‘𝑚) ∈ V)
36		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑚))
37		simpl 472	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑚 = 𝑀)
38	37	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (le‘𝑚) = (le‘𝑀))
39	36, 38	eqtrd 2644	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑀))
40		isomnd.2	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝑀)
41	39, 40	syl6eqr 2662	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → 𝑙 = ≤ )
42	41	breqd 4594	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (𝑎𝑙𝑏 ↔ 𝑎 ≤ 𝑏))
43	41	breqd 4594	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → ((𝑎 + 𝑐)𝑙(𝑏 + 𝑐) ↔ (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))
44	42, 43	imbi12d 333	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → ((𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
45	44	ralbidv 2969	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑙 = (le‘𝑚)) → (∀𝑐 ∈ 𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
46	45	2ralbidv 2972	. . . . . . 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))))
47	46	anbi2d 736	. . . . . 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)))))
48	35, 47	sbcied 3439	. . . . 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)))))
49		eleq1 2676	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 ∈ Toset ↔ 𝑀 ∈ Toset))
50	49	anbi1d 737	. . . . 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)))))
51	48, 50	bitrd 267	. . . 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)))))
52	16, 33, 51	3bitrd 293	. . 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)))))
53		df-omnd 29030	. . 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)))}
54	52, 53	elrab2 3333	. 2 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ (𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))))
55		3anass 1035	. 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)))))
56	54, 55	bitr4i 266	1 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  lecple 15775  Tosetctos 16856  Mndcmnd 17117  oMndcomnd 29028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-omnd 29030

This theorem is referenced by:  omndmnd  29035  omndtos  29036  omndadd  29037  submomnd  29041  xrge0omnd  29042  reofld  29171