Theorem submarchi 33140

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 18729	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2		eqid 2729	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		eqid 2729	. . . . . . 7 ⊢ (0g‘𝑊) = (0g‘𝑊)
4		eqid 2729	. . . . . . 7 ⊢ (.g‘𝑊) = (.g‘𝑊)
5		eqid 2729	. . . . . . 7 ⊢ (le‘𝑊) = (le‘𝑊)
6		eqid 2729	. . . . . . 7 ⊢ (lt‘𝑊) = (lt‘𝑊)
7	2, 3, 4, 5, 6	isarchi2 33139	. . . . . 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 593	. . . . 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 476	. . . 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 652	. . 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 2729	. . . . . . . 8 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
12	11	submbas 18741	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊 ↾s 𝐴)))
13	2	submss 18736	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
14	12, 13	eqsstrrd 3982	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊 ↾s 𝐴)) ⊆ (Base‘𝑊))
15		ssralv 4015	. . . . . . . 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 3147	. . . . . . 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 4015	. . . . . . 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 481	. . . 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 18742	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (0g‘𝑊) = (0g‘(𝑊 ↾s 𝐴)))
22	21	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → (0g‘𝑊) = (0g‘(𝑊 ↾s 𝐴)))
23	11, 5	ressle 17343	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊 ↾s 𝐴)))
24	23	difeq1d 4088	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊 ↾s 𝐴)) ∖ I ))
25	5, 6	pltfval 18290	. . . . . . . . . . . 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 18740	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (𝑊 ↾s 𝐴) ∈ Mnd)
28		eqid 2729	. . . . . . . . . . . . 13 ⊢ (le‘(𝑊 ↾s 𝐴)) = (le‘(𝑊 ↾s 𝐴))
29		eqid 2729	. . . . . . . . . . . . 13 ⊢ (lt‘(𝑊 ↾s 𝐴)) = (lt‘(𝑊 ↾s 𝐴))
30	28, 29	pltfval 18290	. . . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = (lt‘(𝑊 ↾s 𝐴)))
33	32	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → (lt‘𝑊) = (lt‘(𝑊 ↾s 𝐴)))
34		eqidd 2730	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → 𝑥 = 𝑥)
35	22, 33, 34	breq123d 5121	. . . . . . . 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 2730	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑦)
37	23	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → (le‘𝑊) = (le‘(𝑊 ↾s 𝐴)))
38		simplll 774	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (SubMnd‘𝑊))
39		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
40	39	nnnn0d 12503	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
41		simpllr 775	. . . . . . . . . . . 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 2831	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝐴)
44		eqid 2729	. . . . . . . . . . . 12 ⊢ (.g‘(𝑊 ↾s 𝐴)) = (.g‘(𝑊 ↾s 𝐴))
45	4, 11, 44	submmulg 19050	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑛 ∈ ℕ0 ∧ 𝑥 ∈ 𝐴) → (𝑛(.g‘𝑊)𝑥) = (𝑛(.g‘(𝑊 ↾s 𝐴))𝑥))
46	38, 40, 43, 45	syl3anc 1373	. . . . . . . . . 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 5121	. . . . . . . . 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 3155	. . . . . . . 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 344	. . . . . . 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 3154	. . . . . 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 3154	. . . . 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 481	. . . 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 239	. . 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 32893	. . . 4 ⊢ ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Toset)
56	27	adantl 481	. . . 4 ⊢ ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Mnd)
57		eqid 2729	. . . . 5 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴))
58		eqid 2729	. . . . 5 ⊢ (0g‘(𝑊 ↾s 𝐴)) = (0g‘(𝑊 ↾s 𝐴))
59	57, 58, 44, 28, 29	isarchi2 33139	. . . 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 584	. . 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 715	. 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 257	1 ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∖ cdif 3911   ⊆ wss 3914   class class class wbr 5107   I cid 5532  ‘cfv 6511  (class class class)co 7387  ℕcn 12186  ℕ₀cn0 12442  Basecbs 17179   ↾_s cress 17200  lecple 17227  0_gc0g 17402  ltcplt 18269  Tosetctos 18375  Mndcmnd 18661  SubMndcsubmnd 18709  ._gcmg 18999  Archicarchi 33131

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 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-seq 13967  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-ple 17240  df-0g 17404  df-proset 18255  df-poset 18274  df-plt 18289  df-toset 18376  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-mulg 19000  df-inftm 33132  df-archi 33133

This theorem is referenced by:  nn0archi  33318