Theorem submomnd 20102

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 20097	. . . 4 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
3	2	adantr 480	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → 𝑀 ∈ Toset)
4		reldmress 17197	. . . . . . . 8 ⊢ Rel dom ↾s
5	4	ovprc2 7402	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝑀 ↾s 𝐴) = ∅)
6	5	fveq2d 6840	. . . . . 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 17179	. . . . 5 ⊢ ∅ = (Base‘∅)
9	7, 8	eqtr4di 2790	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) = ∅)
10		eqid 2737	. . . . . . . 8 ⊢ (Base‘(𝑀 ↾s 𝐴)) = (Base‘(𝑀 ↾s 𝐴))
11		eqid 2737	. . . . . . . 8 ⊢ (0g‘(𝑀 ↾s 𝐴)) = (0g‘(𝑀 ↾s 𝐴))
12	10, 11	mndidcl 18712	. . . . . . 7 ⊢ ((𝑀 ↾s 𝐴) ∈ Mnd → (0g‘(𝑀 ↾s 𝐴)) ∈ (Base‘(𝑀 ↾s 𝐴)))
13	12	ne0d 4283	. . . . . 6 ⊢ ((𝑀 ↾s 𝐴) ∈ Mnd → (Base‘(𝑀 ↾s 𝐴)) ≠ ∅)
14	13	ad2antlr 728	. . . . 5 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) ≠ ∅)
15	14	neneqd 2938	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑀 ↾s 𝐴)) = ∅)
16	9, 15	condan 818	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → 𝐴 ∈ V)
17		resstos 18391	. . 3 ⊢ ((𝑀 ∈ Toset ∧ 𝐴 ∈ V) → (𝑀 ↾s 𝐴) ∈ Toset)
18	3, 16, 17	syl2anc 585	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ Toset)
19		simplll 775	. . . . . 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 2737	. . . . . . . . . . 11 ⊢ (𝑀 ↾s 𝐴) = (𝑀 ↾s 𝐴)
21		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘𝑀) = (Base‘𝑀)
22	20, 21	ressbas 17201	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑀)) = (Base‘(𝑀 ↾s 𝐴)))
23		inss2 4179	. . . . . . . . . 10 ⊢ (𝐴 ∩ (Base‘𝑀)) ⊆ (Base‘𝑀)
24	22, 23	eqsstrrdi 3968	. . . . . . . . 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 727	. . . . . . 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 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) → 𝑎 ∈ (Base‘(𝑀 ↾s 𝐴)))
28	26, 27	sseldd 3923	. . . . . 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 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) → 𝑏 ∈ (Base‘(𝑀 ↾s 𝐴)))
30	26, 29	sseldd 3923	. . . . . 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 1219	. . . . . . 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 3923	. . . . . 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 2737	. . . . . . . . . . 11 ⊢ (le‘𝑀) = (le‘𝑀)
34	20, 33	ressle 17338	. . . . . . . . . 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 5097	. . . . . . 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 2737	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
40	21, 33, 39	omndadd 20098	. . . . . 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 1386	. . . . 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 17249	. . . . . . . . 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 7379	. . . . . . 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 7379	. . . . . . 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 5100	. . . . . 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 3184	. 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 2737	. . 3 ⊢ (+g‘(𝑀 ↾s 𝐴)) = (+g‘(𝑀 ↾s 𝐴))
54		eqid 2737	. . 3 ⊢ (le‘(𝑀 ↾s 𝐴)) = (le‘(𝑀 ↾s 𝐴))
55	10, 53, 54	isomnd 20093	. 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 1345	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ oMnd)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274   class class class wbr 5086  ‘cfv 6494  (class class class)co 7362  Basecbs 17174   ↾_s cress 17195  +_gcplusg 17215  lecple 17222  0_gc0g 17397  Tosetctos 18375  Mndcmnd 18697  oMndcomnd 20089

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-dec 12640  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-ple 17235  df-0g 17399  df-poset 18274  df-toset 18376  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-omnd 20091

This theorem is referenced by:  suborng  20848  nn0omnd  33423