Theorem submarchi 33147

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 18705	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2		eqid 2731	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		eqid 2731	. . . . . . 7 ⊢ (0g‘𝑊) = (0g‘𝑊)
4		eqid 2731	. . . . . . 7 ⊢ (.g‘𝑊) = (.g‘𝑊)
5		eqid 2731	. . . . . . 7 ⊢ (le‘𝑊) = (le‘𝑊)
6		eqid 2731	. . . . . . 7 ⊢ (lt‘𝑊) = (lt‘𝑊)
7	2, 3, 4, 5, 6	isarchi2 33146	. . . . . 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 2731	. . . . . . . 8 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
12	11	submbas 18717	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊 ↾s 𝐴)))
13	2	submss 18712	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
14	12, 13	eqsstrrd 3965	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊 ↾s 𝐴)) ⊆ (Base‘𝑊))
15		ssralv 3998	. . . . . . . 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 3146	. . . . . . 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 3998	. . . . . . 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 18718	. . . . . . . . . 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 17279	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊 ↾s 𝐴)))
24	23	difeq1d 4070	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊 ↾s 𝐴)) ∖ I ))
25	5, 6	pltfval 18230	. . . . . . . . . . . 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 18716	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (𝑊 ↾s 𝐴) ∈ Mnd)
28		eqid 2731	. . . . . . . . . . . . 13 ⊢ (le‘(𝑊 ↾s 𝐴)) = (le‘(𝑊 ↾s 𝐴))
29		eqid 2731	. . . . . . . . . . . . 13 ⊢ (lt‘(𝑊 ↾s 𝐴)) = (lt‘(𝑊 ↾s 𝐴))
30	28, 29	pltfval 18230	. . . . . . . . . . . 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 2776	. . . . . . . . . 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 2732	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → 𝑥 = 𝑥)
35	22, 33, 34	breq123d 5100	. . . . . . . 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 2732	. . . . . . . . . 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 12437	. . . . . . . . . . 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 2834	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝐴)
44		eqid 2731	. . . . . . . . . . . 12 ⊢ (.g‘(𝑊 ↾s 𝐴)) = (.g‘(𝑊 ↾s 𝐴))
45	4, 11, 44	submmulg 19026	. . . . . . . . . . 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 5100	. . . . . . . . 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 3154	. . . . . . . 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 3153	. . . . . 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 3153	. . . . 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 18331	. . . 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 2731	. . . . 5 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴))
58		eqid 2731	. . . . 5 ⊢ (0g‘(𝑊 ↾s 𝐴)) = (0g‘(𝑊 ↾s 𝐴))
59	57, 58, 44, 28, 29	isarchi2 33146	. . . 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ∖ cdif 3894   ⊆ wss 3897   class class class wbr 5086   I cid 5505  ‘cfv 6476  (class class class)co 7341  ℕcn 12120  ℕ₀cn0 12376  Basecbs 17115   ↾_s cress 17136  lecple 17163  0_gc0g 17338  ltcplt 18209  Tosetctos 18315  Mndcmnd 18637  SubMndcsubmnd 18685  ._gcmg 18975  Archicarchi 33138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-z 12464  df-dec 12584  df-uz 12728  df-fz 13403  df-seq 13904  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-ple 17176  df-0g 17340  df-proset 18195  df-poset 18214  df-plt 18229  df-toset 18316  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-submnd 18687  df-mulg 18976  df-inftm 33139  df-archi 33140

This theorem is referenced by:  nn0archi  33304