Theorem submomnd 20011

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 20006	. . . 4 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
3	2	adantr 480	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → 𝑀 ∈ Toset)
4		reldmress 17143	. . . . . . . 8 ⊢ Rel dom ↾s
5	4	ovprc2 7389	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝑀 ↾s 𝐴) = ∅)
6	5	fveq2d 6826	. . . . . 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 17125	. . . . 5 ⊢ ∅ = (Base‘∅)
9	7, 8	eqtr4di 2782	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) = ∅)
10		eqid 2729	. . . . . . . 8 ⊢ (Base‘(𝑀 ↾s 𝐴)) = (Base‘(𝑀 ↾s 𝐴))
11		eqid 2729	. . . . . . . 8 ⊢ (0g‘(𝑀 ↾s 𝐴)) = (0g‘(𝑀 ↾s 𝐴))
12	10, 11	mndidcl 18623	. . . . . . 7 ⊢ ((𝑀 ↾s 𝐴) ∈ Mnd → (0g‘(𝑀 ↾s 𝐴)) ∈ (Base‘(𝑀 ↾s 𝐴)))
13	12	ne0d 4293	. . . . . 6 ⊢ ((𝑀 ↾s 𝐴) ∈ Mnd → (Base‘(𝑀 ↾s 𝐴)) ≠ ∅)
14	13	ad2antlr 727	. . . . 5 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) ≠ ∅)
15	14	neneqd 2930	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑀 ↾s 𝐴)) = ∅)
16	9, 15	condan 817	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → 𝐴 ∈ V)
17		resstos 18336	. . 3 ⊢ ((𝑀 ∈ Toset ∧ 𝐴 ∈ V) → (𝑀 ↾s 𝐴) ∈ Toset)
18	3, 16, 17	syl2anc 584	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ Toset)
19		simplll 774	. . . . . 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 2729	. . . . . . . . . . 11 ⊢ (𝑀 ↾s 𝐴) = (𝑀 ↾s 𝐴)
21		eqid 2729	. . . . . . . . . . 11 ⊢ (Base‘𝑀) = (Base‘𝑀)
22	20, 21	ressbas 17147	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑀)) = (Base‘(𝑀 ↾s 𝐴)))
23		inss2 4189	. . . . . . . . . 10 ⊢ (𝐴 ∩ (Base‘𝑀)) ⊆ (Base‘𝑀)
24	22, 23	eqsstrrdi 3981	. . . . . . . . 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 726	. . . . . . 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 1216	. . . . . . 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 3936	. . . . . 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 1217	. . . . . . 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 3936	. . . . . 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 1218	. . . . . . 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 3936	. . . . . 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 2729	. . . . . . . . . . 11 ⊢ (le‘𝑀) = (le‘𝑀)
34	20, 33	ressle 17284	. . . . . . . . . 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 5103	. . . . . . 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 2729	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
40	21, 33, 39	omndadd 20007	. . . . . 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 1385	. . . . 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 17195	. . . . . . . . 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 7366	. . . . . . 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 7366	. . . . . . 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 5106	. . . . . 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 232	. . . 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 3175	. 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 2729	. . 3 ⊢ (+g‘(𝑀 ↾s 𝐴)) = (+g‘(𝑀 ↾s 𝐴))
54		eqid 2729	. . 3 ⊢ (le‘(𝑀 ↾s 𝐴)) = (le‘(𝑀 ↾s 𝐴))
55	10, 53, 54	isomnd 20002	. 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 1344	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ oMnd)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  Basecbs 17120   ↾_s cress 17141  +_gcplusg 17161  lecple 17168  0_gc0g 17343  Tosetctos 18320  Mndcmnd 18608  oMndcomnd 19998

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-dec 12592  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-ple 17181  df-0g 17345  df-poset 18219  df-toset 18321  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-omnd 20000

This theorem is referenced by:  suborng  20761  nn0omnd  33283