Theorem submomnd 20251

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 490	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ Mnd)
2		omndtos 20246	. . . 4 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
3	2	adantr 486	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → 𝑀 ∈ Toset)
4		reldmress 17319	. . . . . . . 8 ⊢ Rel dom ↾s
5	4	ovprc2 7460	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝑀 ↾s 𝐴) = ∅)
6	5	fveq2d 6890	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (Base‘(𝑀 ↾s 𝐴)) = (Base‘∅))
7	6	adantl 487	. . . . 5 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) = (Base‘∅))
8		base0 17301	. . . . 5 ⊢ ∅ = (Base‘∅)
9	7, 8	eqtr4di 2818	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) = ∅)
10		eqid 2765	. . . . . . . 8 ⊢ (Base‘(𝑀 ↾s 𝐴)) = (Base‘(𝑀 ↾s 𝐴))
11		eqid 2765	. . . . . . . 8 ⊢ (0g‘(𝑀 ↾s 𝐴)) = (0g‘(𝑀 ↾s 𝐴))
12	10, 11	mndidcl 18845	. . . . . . 7 ⊢ ((𝑀 ↾s 𝐴) ∈ Mnd → (0g‘(𝑀 ↾s 𝐴)) ∈ (Base‘(𝑀 ↾s 𝐴)))
13	12	ne0d 4295	. . . . . 6 ⊢ ((𝑀 ↾s 𝐴) ∈ Mnd → (Base‘(𝑀 ↾s 𝐴)) ≠ ∅)
14	13	ad2antlr 740	. . . . 5 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀 ↾s 𝐴)) ≠ ∅)
15	14	neneqd 2965	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑀 ↾s 𝐴)) = ∅)
16	9, 15	condan 830	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → 𝐴 ∈ V)
17		resstos 18513	. . 3 ⊢ ((𝑀 ∈ Toset ∧ 𝐴 ∈ V) → (𝑀 ↾s 𝐴) ∈ Toset)
18	3, 16, 17	syl2anc 596	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ Toset)
19		simplll 787	. . . . . 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 2765	. . . . . . . . . . 11 ⊢ (𝑀 ↾s 𝐴) = (𝑀 ↾s 𝐴)
21		eqid 2765	. . . . . . . . . . 11 ⊢ (Base‘𝑀) = (Base‘𝑀)
22	20, 21	ressbas 17323	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑀)) = (Base‘(𝑀 ↾s 𝐴)))
23		inss2 4190	. . . . . . . . . 10 ⊢ (𝐴 ∩ (Base‘𝑀)) ⊆ (Base‘𝑀)
24	22, 23	eqsstrrdi 3983	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (Base‘(𝑀 ↾s 𝐴)) ⊆ (Base‘𝑀))
25	16, 24	syl 18	. . . . . . . 8 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (Base‘(𝑀 ↾s 𝐴)) ⊆ (Base‘𝑀))
26	25	ad2antrr 739	. . . . . . 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 1234	. . . . . . 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 3939	. . . . . 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 1235	. . . . . . 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 3939	. . . . . 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 1236	. . . . . . 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 3939	. . . . . 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 2765	. . . . . . . . . . 11 ⊢ (le‘𝑀) = (le‘𝑀)
34	20, 33	ressle 17460	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (le‘𝑀) = (le‘(𝑀 ↾s 𝐴)))
35	16, 34	syl 18	. . . . . . . . 9 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (le‘𝑀) = (le‘(𝑀 ↾s 𝐴)))
36	35	adantr 486	. . . . . . . 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 5122	. . . . . . 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 483	. . . . . 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 2765	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
40	21, 33, 39	omndadd 20247	. . . . . 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 1410	. . . . 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 486	. . . . . . . . 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 17371	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (+g‘𝑀) = (+g‘(𝑀 ↾s 𝐴)))
44	42, 43	syl 18	. . . . . . . 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 7437	. . . . . . 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 18	. . . . . . 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 7437	. . . . . . 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 5125	. . . . . 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 486	. . . . 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 235	. . . 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 418	. . 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 3213	. 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 2765	. . 3 ⊢ (+g‘(𝑀 ↾s 𝐴)) = (+g‘(𝑀 ↾s 𝐴))
54		eqid 2765	. . 3 ⊢ (le‘(𝑀 ↾s 𝐴)) = (le‘(𝑀 ↾s 𝐴))
55	10, 53, 54	isomnd 20242	. 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 1362	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ oMnd)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286   class class class wbr 5111  ‘cfv 6541  (class class class)co 7420  Basecbs 17296   ↾_s cress 17317  +_gcplusg 17337  lecple 17344  0_gc0g 17519  Tosetctos 18497  Mndcmnd 18829  oMndcomnd 20238

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7743  ax-cnex 11176  ax-resscn 11177  ax-1cn 11178  ax-icn 11179  ax-addcl 11180  ax-addrcl 11181  ax-mulcl 11182  ax-mulrcl 11183  ax-mulcom 11184  ax-addass 11185  ax-mulass 11186  ax-distr 11187  ax-i2m1 11188  ax-1ne0 11189  ax-1rid 11190  ax-rnegex 11191  ax-rrecex 11192  ax-cnre 11193  ax-pre-lttri 11194  ax-pre-lttrn 11195  ax-pre-ltadd 11196  ax-pre-mulgt0 11197

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7870  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-er 8701  df-en 8951  df-dom 8952  df-sdom 8953  df-pnf 11265  df-mnf 11266  df-xr 11267  df-ltxr 11268  df-le 11269  df-sub 11463  df-neg 11464  df-nn 12254  df-2 12323  df-3 12324  df-4 12325  df-5 12326  df-6 12327  df-7 12328  df-8 12329  df-9 12330  df-n0 12525  df-z 12612  df-dec 12733  df-sets 17251  df-slot 17269  df-ndx 17281  df-base 17297  df-ress 17318  df-plusg 17350  df-ple 17357  df-0g 17521  df-poset 18396  df-toset 18498  df-mgm 18725  df-sgrp 18814  df-mnd 18830  df-omnd 20240

This theorem is used by:  suborng  21034  nn0omnd  33733