Theorem submarchi 33546

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 18884	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2		eqid 2765	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		eqid 2765	. . . . . . 7 ⊢ (0g‘𝑊) = (0g‘𝑊)
4		eqid 2765	. . . . . . 7 ⊢ (.g‘𝑊) = (.g‘𝑊)
5		eqid 2765	. . . . . . 7 ⊢ (le‘𝑊) = (le‘𝑊)
6		eqid 2765	. . . . . . 7 ⊢ (lt‘𝑊) = (lt‘𝑊)
7	2, 3, 4, 5, 6	isarchi2 33545	. . . . . 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 605	. . . . 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 482	. . . 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 665	. . 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 2765	. . . . . . . 8 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
12	11	submbas 18897	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊 ↾s 𝐴)))
13	2	submss 18891	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
14	12, 13	eqsstrrd 3973	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊 ↾s 𝐴)) ⊆ (Base‘𝑊))
15		ssralv 4007	. . . . . . . 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 3181	. . . . . . 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 4007	. . . . . . 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 48	. . . . . 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 18	. . . . 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 487	. . . 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 18898	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (0g‘𝑊) = (0g‘(𝑊 ↾s 𝐴)))
22	21	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → (0g‘𝑊) = (0g‘(𝑊 ↾s 𝐴)))
23	11, 5	ressle 17451	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊 ↾s 𝐴)))
24	23	difeq1d 4080	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊 ↾s 𝐴)) ∖ I ))
25	5, 6	pltfval 18403	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Mnd → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
26	1, 25	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
27	11	submmnd 18896	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (𝑊 ↾s 𝐴) ∈ Mnd)
28		eqid 2765	. . . . . . . . . . . . 13 ⊢ (le‘(𝑊 ↾s 𝐴)) = (le‘(𝑊 ↾s 𝐴))
29		eqid 2765	. . . . . . . . . . . . 13 ⊢ (lt‘(𝑊 ↾s 𝐴)) = (lt‘(𝑊 ↾s 𝐴))
30	28, 29	pltfval 18403	. . . . . . . . . . . 12 ⊢ ((𝑊 ↾s 𝐴) ∈ Mnd → (lt‘(𝑊 ↾s 𝐴)) = ((le‘(𝑊 ↾s 𝐴)) ∖ I ))
31	27, 30	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (lt‘(𝑊 ↾s 𝐴)) = ((le‘(𝑊 ↾s 𝐴)) ∖ I ))
32	24, 26, 31	3eqtr4d 2810	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = (lt‘(𝑊 ↾s 𝐴)))
33	32	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → (lt‘𝑊) = (lt‘(𝑊 ↾s 𝐴)))
34		eqidd 2766	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → 𝑥 = 𝑥)
35	22, 33, 34	breq123d 5125	. . . . . . . 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 2766	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑦)
37	23	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → (le‘𝑊) = (le‘(𝑊 ↾s 𝐴)))
38		simplll 787	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (SubMnd‘𝑊))
39		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
40	39	nnnn0d 12576	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
41		simpllr 788	. . . . . . . . . . . 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 18	. . . . . . . . . . . 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 2868	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝐴)
44		eqid 2765	. . . . . . . . . . . 12 ⊢ (.g‘(𝑊 ↾s 𝐴)) = (.g‘(𝑊 ↾s 𝐴))
45	4, 11, 44	submmulg 19208	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑛 ∈ ℕ0 ∧ 𝑥 ∈ 𝐴) → (𝑛(.g‘𝑊)𝑥) = (𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))
46	38, 40, 43, 45	syl3anc 1398	. . . . . . . . . 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 5125	. . . . . . . . 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 3189	. . . . . . . 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 347	. . . . . . 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 3188	. . . . . 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 3188	. . . . 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 487	. . . 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 242	. . 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 16	. 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 18504	. . . 4 ⊢ ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Toset)
56	27	adantl 487	. . . 4 ⊢ ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Mnd)
57		eqid 2765	. . . . 5 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴))
58		eqid 2765	. . . . 5 ⊢ (0g‘(𝑊 ↾s 𝐴)) = (0g‘(𝑊 ↾s 𝐴))
59	57, 58, 44, 28, 29	isarchi2 33545	. . . 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 596	. . 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 728	. 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 260	1 ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ∖ cdif 3903   ⊆ wss 3906   class class class wbr 5111   I cid 5557  ‘cfv 6540  (class class class)co 7416  ℕcn 12244  ℕ₀cn0 12515  Basecbs 17287   ↾_s cress 17308  lecple 17335  0_gc0g 17510  ltcplt 18382  Tosetctos 18488  Mndcmnd 18814  SubMndcsubmnd 18864  ._gcmg 19157  Archicarchi 33537

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 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

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 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-z 12603  df-dec 12724  df-uz 12875  df-fz 13548  df-seq 14052  df-sets 17242  df-slot 17260  df-ndx 17272  df-base 17288  df-ress 17309  df-plusg 17341  df-ple 17348  df-0g 17512  df-proset 18368  df-poset 18387  df-plt 18402  df-toset 18489  df-mgm 18716  df-sgrp 18799  df-mnd 18815  df-submnd 18866  df-mulg 19158  df-inftm 33538  df-archi 33539

This theorem is used by:  nn0archi  33707