Theorem submarchi 33740

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 18990	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2		eqid 2761	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
3		eqid 2761	. . . . . . 7 ⊢ (0g‘𝑊) = (0g‘𝑊)
4		eqid 2761	. . . . . . 7 ⊢ (.g‘𝑊) = (.g‘𝑊)
5		eqid 2761	. . . . . . 7 ⊢ (le‘𝑊) = (le‘𝑊)
6		eqid 2761	. . . . . . 7 ⊢ (lt‘𝑊) = (lt‘𝑊)
7	2, 3, 4, 5, 6	isarchi2 33739	. . . . . 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 2761	. . . . . . . 8 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
12	11	submbas 19003	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊 ↾s 𝐴)))
13	2	submss 18997	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
14	12, 13	eqsstrrd 3966	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊 ↾s 𝐴)) ⊆ (Base‘𝑊))
15		ssralv 4000	. . . . . . . 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 3177	. . . . . . 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 4000	. . . . . . 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 19004	. . . . . . . . . 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 17544	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊 ↾s 𝐴)))
24	23	difeq1d 4073	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊 ↾s 𝐴)) ∖ I ))
25	5, 6	pltfval 18496	. . . . . . . . . . . 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 19002	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubMnd‘𝑊) → (𝑊 ↾s 𝐴) ∈ Mnd)
28		eqid 2761	. . . . . . . . . . . . 13 ⊢ (le‘(𝑊 ↾s 𝐴)) = (le‘(𝑊 ↾s 𝐴))
29		eqid 2761	. . . . . . . . . . . . 13 ⊢ (lt‘(𝑊 ↾s 𝐴)) = (lt‘(𝑊 ↾s 𝐴))
30	28, 29	pltfval 18496	. . . . . . . . . . . 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 2806	. . . . . . . . . 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 2762	. . . . . . . . 9 ⊢ (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) → 𝑥 = 𝑥)
35	22, 33, 34	breq123d 5117	. . . . . . . 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 2762	. . . . . . . . . 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 12660	. . . . . . . . . . 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 2864	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊 ↾s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝐴)
44		eqid 2761	. . . . . . . . . . . 12 ⊢ (.g‘(𝑊 ↾s 𝐴)) = (.g‘(𝑊 ↾s 𝐴))
45	4, 11, 44	submmulg 19321	. . . . . . . . . . 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 5117	. . . . . . . . 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 3185	. . . . . . . 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 3184	. . . . . 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 3184	. . . . 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 18597	. . . 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 2761	. . . . 5 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴))
58		eqid 2761	. . . . 5 ⊢ (0g‘(𝑊 ↾s 𝐴)) = (0g‘(𝑊 ↾s 𝐴))
59	57, 58, 44, 28, 29	isarchi2 33739	. . . 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 2145  ∀wral 3077  ∃wrex 3087   ∖ cdif 3896   ⊆ wss 3899   class class class wbr 5103   I cid 5545  ‘cfv 6537  (class class class)co 7418  ℕcn 12328  ℕ₀cn0 12599  Basecbs 17380   ↾_s cress 17401  lecple 17428  0_gc0g 17603  ltcplt 18475  Tosetctos 18581  Mndcmnd 18916  SubMndcsubmnd 18970  ._gcmg 19270  Archicarchi 33731

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-seq 14138  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-ple 17441  df-0g 17605  df-proset 18461  df-poset 18480  df-plt 18495  df-toset 18582  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-submnd 18972  df-mulg 19271  df-inftm 33732  df-archi 33733

This theorem is used by:  nn0archi  33901