Theorem submarchi 29740

Description: A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)

Assertion

Ref	Expression
submarchi	⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi)

Proof of Theorem submarchi

Dummy variables 𝑥 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		submrcl 17346	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2		eqid 2622	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		eqid 2622	. . . . . . 7 ⊢ (0g‘𝑊) = (0g‘𝑊)
4		eqid 2622	. . . . . . 7 ⊢ (.g‘𝑊) = (.g‘𝑊)
5		eqid 2622	. . . . . . 7 ⊢ (le‘𝑊) = (le‘𝑊)
6		eqid 2622	. . . . . . 7 ⊢ (lt‘𝑊) = (lt‘𝑊)
7	2, 3, 4, 5, 6	isarchi2 29739	. . . . . 6 ⊢ ((𝑊 ∈ Toset ∧ 𝑊 ∈ Mnd) → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥))))
8	1, 7	sylan2 491	. . . . 5 ⊢ ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥))))
9	8	biimpa 501	. . . 4 ⊢ (((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) ∧ 𝑊 ∈ Archi) → ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)))
10	9	an32s 846	. . 3 ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)))
11		eqid 2622	. . . . . . . 8 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
12	11	submbas 17355	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊 ↾s 𝐴)))
13	2	submss 17350	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
14	12, 13	eqsstr3d 3640	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊 ↾s 𝐴)) ⊆ (Base‘𝑊))
15		ssralv 3666	. . . . . . . 8 ⊢ ((Base‘(𝑊 ↾s 𝐴)) ⊆ (Base‘𝑊) → (∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) → ∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥))))
16	15	ralimdv 2963	. . . . . . 7 ⊢ ((Base‘(𝑊 ↾s 𝐴)) ⊆ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) → ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥))))
17		ssralv 3666	. . . . . . 7 ⊢ ((Base‘(𝑊 ↾s 𝐴)) ⊆ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥))))
18	16, 17	syld 47	. . . . . 6 ⊢ ((Base‘(𝑊 ↾s 𝐴)) ⊆ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥))))
19	14, 18	syl 17	. . . . 5 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥))))
20	19	adantl 482	. . . 4 ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥))))
21	11, 3	subm0 17356	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (0g‘𝑊) = (0g‘(𝑊 ↾s 𝐴)))
22	21	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → (0g‘𝑊) = (0g‘(𝑊 ↾s 𝐴)))
23	11, 5	ressle 16059	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊 ↾s 𝐴)))
24	23	difeq1d 3727	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊 ↾s 𝐴)) ∖ I ))
25	5, 6	pltfval 16959	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Mnd → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
27	11	submmnd 17354	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (𝑊 ↾s 𝐴) ∈ Mnd)
28		eqid 2622	. . . . . . . . . . . . 13 ⊢ (le‘(𝑊 ↾s 𝐴)) = (le‘(𝑊 ↾s 𝐴))
29		eqid 2622	. . . . . . . . . . . . 13 ⊢ (lt‘(𝑊 ↾s 𝐴)) = (lt‘(𝑊 ↾s 𝐴))
30	28, 29	pltfval 16959	. . . . . . . . . . . 12 ⊢ ((𝑊 ↾s 𝐴) ∈ Mnd → (lt‘(𝑊 ↾s 𝐴)) = ((le‘(𝑊 ↾s 𝐴)) ∖ I ))
31	27, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (lt‘(𝑊 ↾s 𝐴)) = ((le‘(𝑊 ↾s 𝐴)) ∖ I ))
32	24, 26, 31	3eqtr4d 2666	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = (lt‘(𝑊 ↾s 𝐴)))
33	32	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → (lt‘𝑊) = (lt‘(𝑊 ↾s 𝐴)))
34		eqidd 2623	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → 𝑥 = 𝑥)
35	22, 33, 34	breq123d 4667	. . . . . . . 8 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → ((0g‘𝑊)(lt‘𝑊)𝑥 ↔ (0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥))
36		eqidd 2623	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑦)
37	23	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → (le‘𝑊) = (le‘(𝑊 ↾s 𝐴)))
38		simplll 798	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (SubMnd‘𝑊))
39		simpr 477	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
40	39	nnnn0d 11351	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
41		simpllr 799	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴)))
42	38, 12	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 = (Base‘(𝑊 ↾s 𝐴)))
43	41, 42	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝐴)
44		eqid 2622	. . . . . . . . . . . 12 ⊢ (.g‘(𝑊 ↾s 𝐴)) = (.g‘(𝑊 ↾s 𝐴))
45	4, 11, 44	submmulg 17586	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑛 ∈ ℕ0 ∧ 𝑥 ∈ 𝐴) → (𝑛(.g‘𝑊)𝑥) = (𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))
46	38, 40, 43, 45	syl3anc 1326	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑛(.g‘𝑊)𝑥) = (𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))
47	36, 37, 46	breq123d 4667	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥) ↔ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥)))
48	47	rexbidva 3049	. . . . . . . 8 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → (∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥) ↔ ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥)))
49	35, 48	imbi12d 334	. . . . . . 7 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → (((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) ↔ ((0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))))
50	49	ralbidva 2985	. . . . . 6 ⊢ ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) → (∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) ↔ ∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))))
51	50	ralbidva 2985	. . . . 5 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) ↔ ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))))
52	51	adantl 482	. . . 4 ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) ↔ ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))))
53	20, 52	sylibd 229	. . 3 ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g‘𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))))
54	10, 53	mpd 15	. 2 ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥)))
55		resstos 29660	. . . 4 ⊢ ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Toset)
56	27	adantl 482	. . . 4 ⊢ ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Mnd)
57		eqid 2622	. . . . 5 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴))
58		eqid 2622	. . . . 5 ⊢ (0g‘(𝑊 ↾s 𝐴)) = (0g‘(𝑊 ↾s 𝐴))
59	57, 58, 44, 28, 29	isarchi2 29739	. . . 4 ⊢ (((𝑊 ↾s 𝐴) ∈ Toset ∧ (𝑊 ↾s 𝐴) ∈ Mnd) → ((𝑊 ↾s 𝐴) ∈ Archi ↔ ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))))
60	55, 56, 59	syl2anc 693	. . 3 ⊢ ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → ((𝑊 ↾s 𝐴) ∈ Archi ↔ ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))))
61	60	adantlr 751	. 2 ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → ((𝑊 ↾s 𝐴) ∈ Archi ↔ ∀𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))∀𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))((0g‘(𝑊 ↾s 𝐴))(lt‘(𝑊 ↾s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊 ↾s 𝐴))(𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))))
62	54, 61	mpbird 247	1 ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574   class class class wbr 4653   I cid 5023  ‘cfv 5888  (class class class)co 6650  ℕcn 11020  ℕ₀cn0 11292  Basecbs 15857   ↾_s cress 15858  lecple 15948  0_gc0g 16100  ltcplt 16941  Tosetctos 17033  Mndcmnd 17294  SubMndcsubmnd 17334  ._gcmg 17540  Archicarchi 29731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-seq 12802  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-ple 15961  df-0g 16102  df-preset 16928  df-poset 16946  df-plt 16958  df-toset 17034  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-inftm 29732  df-archi 29733

This theorem is referenced by:  nn0archi  29843