Theorem submomnd 31238

Description: A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)

Assertion

Ref	Expression
submomnd	⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ oMnd)

Proof of Theorem submomnd

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ Mnd)
2		omndtos 31233	. . . 4 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
3	2	adantr 480	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → 𝑀 ∈ Toset)
4		reldmress 16869	. . . . . . . 8 ⊢ Rel dom ↾s
5	4	ovprc2 7295	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝑀 ↾s 𝐴) = ∅)
6	5	fveq2d 6760	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (Base‘(𝑀 ↾s 𝐴)) = (Base‘∅))
7	6	adantl 481	. . . . 5 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) = (Base‘∅))
8		base0 16845	. . . . 5 ⊢ ∅ = (Base‘∅)
9	7, 8	eqtr4di 2797	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) = ∅)
10		eqid 2738	. . . . . . . 8 ⊢ (Base‘(𝑀 ↾s 𝐴)) = (Base‘(𝑀 ↾s 𝐴))
11		eqid 2738	. . . . . . . 8 ⊢ (0g‘(𝑀 ↾s 𝐴)) = (0g‘(𝑀 ↾s 𝐴))
12	10, 11	mndidcl 18315	. . . . . . 7 ⊢ ((𝑀 ↾s 𝐴) ∈ Mnd → (0g‘(𝑀 ↾s 𝐴)) ∈ (Base‘(𝑀 ↾s 𝐴)))
13	12	ne0d 4266	. . . . . 6 ⊢ ((𝑀 ↾s 𝐴) ∈ Mnd → (Base‘(𝑀 ↾s 𝐴)) ≠ ∅)
14	13	ad2antlr 723	. . . . 5 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) ≠ ∅)
15	14	neneqd 2947	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑀 ↾s 𝐴)) = ∅)
16	9, 15	condan 814	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → 𝐴 ∈ V)
17		resstos 31147	. . 3 ⊢ ((𝑀 ∈ Toset ∧ 𝐴 ∈ V) → (𝑀 ↾s 𝐴) ∈ Toset)
18	3, 16, 17	syl2anc 583	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ Toset)
19		simplll 771	. . . . . 6 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → 𝑀 ∈ oMnd)
20		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑀 ↾s 𝐴) = (𝑀 ↾s 𝐴)
21		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘𝑀) = (Base‘𝑀)
22	20, 21	ressbas 16873	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑀)) = (Base‘(𝑀 ↾s 𝐴)))
23		inss2 4160	. . . . . . . . . 10 ⊢ (𝐴 ∩ (Base‘𝑀)) ⊆ (Base‘𝑀)
24	22, 23	eqsstrrdi 3972	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (Base‘(𝑀 ↾s 𝐴)) ⊆ (Base‘𝑀))
25	16, 24	syl 17	. . . . . . . 8 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (Base‘(𝑀 ↾s 𝐴)) ⊆ (Base‘𝑀))
26	25	ad2antrr 722	. . . . . . 7 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → (Base‘(𝑀 ↾s 𝐴)) ⊆ (Base‘𝑀))
27		simplr1 1213	. . . . . . 7 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → 𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)))
28	26, 27	sseldd 3918	. . . . . 6 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → 𝑎 ∈ (Base‘𝑀))
29		simplr2 1214	. . . . . . 7 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)))
30	26, 29	sseldd 3918	. . . . . 6 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → 𝑏 ∈ (Base‘𝑀))
31		simplr3 1215	. . . . . . 7 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))
32	26, 31	sseldd 3918	. . . . . 6 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → 𝑐 ∈ (Base‘𝑀))
33		eqid 2738	. . . . . . . . . . 11 ⊢ (le‘𝑀) = (le‘𝑀)
34	20, 33	ressle 17013	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (le‘𝑀) = (le‘(𝑀 ↾s 𝐴)))
35	16, 34	syl 17	. . . . . . . . 9 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (le‘𝑀) = (le‘(𝑀 ↾s 𝐴)))
36	35	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) → (le‘𝑀) = (le‘(𝑀 ↾s 𝐴)))
37	36	breqd 5081	. . . . . . 7 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) → (𝑎(le‘𝑀)𝑏 ↔ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏))
38	37	biimpar 477	. . . . . 6 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → 𝑎(le‘𝑀)𝑏)
39		eqid 2738	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
40	21, 33, 39	omndadd 31234	. . . . . 6 ⊢ ((𝑀 ∈ oMnd ∧ (𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀) ∧ 𝑐 ∈ (Base‘𝑀)) ∧ 𝑎(le‘𝑀)𝑏) → (𝑎(+g‘𝑀)𝑐)(le‘𝑀)(𝑏(+g‘𝑀)𝑐))
41	19, 28, 30, 32, 38, 40	syl131anc 1381	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → (𝑎(+g‘𝑀)𝑐)(le‘𝑀)(𝑏(+g‘𝑀)𝑐))
42	16	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) → 𝐴 ∈ V)
43	20, 39	ressplusg 16926	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (+g‘𝑀) = (+g‘(𝑀 ↾s 𝐴)))
44	42, 43	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) → (+g‘𝑀) = (+g‘(𝑀 ↾s 𝐴)))
45	44	oveqd 7272	. . . . . . 7 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) → (𝑎(+g‘𝑀)𝑐) = (𝑎(+g‘(𝑀 ↾s 𝐴))𝑐))
46	42, 34	syl 17	. . . . . . 7 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) → (le‘𝑀) = (le‘(𝑀 ↾s 𝐴)))
47	44	oveqd 7272	. . . . . . 7 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) → (𝑏(+g‘𝑀)𝑐) = (𝑏(+g‘(𝑀 ↾s 𝐴))𝑐))
48	45, 46, 47	breq123d 5084	. . . . . 6 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) → ((𝑎(+g‘𝑀)𝑐)(le‘𝑀)(𝑏(+g‘𝑀)𝑐) ↔ (𝑎(+g‘(𝑀 ↾s 𝐴))𝑐)(le‘(𝑀 ↾s 𝐴))(𝑏(+g‘(𝑀 ↾s 𝐴))𝑐)))
49	48	adantr 480	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → ((𝑎(+g‘𝑀)𝑐)(le‘𝑀)(𝑏(+g‘𝑀)𝑐) ↔ (𝑎(+g‘(𝑀 ↾s 𝐴))𝑐)(le‘(𝑀 ↾s 𝐴))(𝑏(+g‘(𝑀 ↾s 𝐴))𝑐)))
50	41, 49	mpbid 231	. . . 4 ⊢ ((((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) ∧ 𝑎(le‘(𝑀 ↾s 𝐴))𝑏) → (𝑎(+g‘(𝑀 ↾s 𝐴))𝑐)(le‘(𝑀 ↾s 𝐴))(𝑏(+g‘(𝑀 ↾s 𝐴))𝑐))
51	50	ex 412	. . 3 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀 ↾s 𝐴)))) → (𝑎(le‘(𝑀 ↾s 𝐴))𝑏 → (𝑎(+g‘(𝑀 ↾s 𝐴))𝑐)(le‘(𝑀 ↾s 𝐴))(𝑏(+g‘(𝑀 ↾s 𝐴))𝑐)))
52	51	ralrimivvva 3115	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → ∀𝑎 ∈ (Base‘(𝑀 ↾s 𝐴))∀𝑏 ∈ (Base‘(𝑀 ↾s 𝐴))∀𝑐 ∈ (Base‘(𝑀 ↾s 𝐴))(𝑎(le‘(𝑀 ↾s 𝐴))𝑏 → (𝑎(+g‘(𝑀 ↾s 𝐴))𝑐)(le‘(𝑀 ↾s 𝐴))(𝑏(+g‘(𝑀 ↾s 𝐴))𝑐)))
53		eqid 2738	. . 3 ⊢ (+g‘(𝑀 ↾s 𝐴)) = (+g‘(𝑀 ↾s 𝐴))
54		eqid 2738	. . 3 ⊢ (le‘(𝑀 ↾s 𝐴)) = (le‘(𝑀 ↾s 𝐴))
55	10, 53, 54	isomnd 31229	. 2 ⊢ ((𝑀 ↾s 𝐴) ∈ oMnd ↔ ((𝑀 ↾s 𝐴) ∈ Mnd ∧ (𝑀 ↾s 𝐴) ∈ Toset ∧ ∀𝑎 ∈ (Base‘(𝑀 ↾s 𝐴))∀𝑏 ∈ (Base‘(𝑀 ↾s 𝐴))∀𝑐 ∈ (Base‘(𝑀 ↾s 𝐴))(𝑎(le‘(𝑀 ↾s 𝐴))𝑏 → (𝑎(+g‘(𝑀 ↾s 𝐴))𝑐)(le‘(𝑀 ↾s 𝐴))(𝑏(+g‘(𝑀 ↾s 𝐴))𝑐))))
56	1, 18, 52, 55	syl3anbrc 1341	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ oMnd)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840   ↾_s cress 16867  +_gcplusg 16888  lecple 16895  0_gc0g 17067  Tosetctos 18049  Mndcmnd 18300  oMndcomnd 31225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-dec 12367  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-ple 16908  df-0g 17069  df-poset 17946  df-toset 18050  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-omnd 31227

This theorem is referenced by:  suborng  31416  nn0omnd  31447