isarchi3 - Mathbox for Thierry Arnoux

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Theorem isarchi3 33447

Description: This is the usual definition of the Archimedean property for an ordered group. (Contributed by Thierry Arnoux, 30-Jan-2018.)

**Hypotheses**
Ref	Expression
isarchi3.b	⊢ 𝐵 = (Base‘𝑊)
isarchi3.0	⊢ 0 = (0_g‘𝑊)
isarchi3.i	⊢ < = (lt‘𝑊)
isarchi3.x	⊢ · = (._g‘𝑊)

**Assertion**
Ref	Expression
isarchi3	⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))

Distinct variable groups: 𝑥,𝑛,𝑦,𝐵 𝑛,𝑊,𝑥,𝑦 < ,𝑛 · ,𝑛 0 ,𝑛 Allowed substitution hints: < (𝑥,𝑦) · (𝑥,𝑦) 0 (𝑥,𝑦)

Proof of Theorem isarchi3 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isogrp 20193	. . . . 5 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
2	1	simprbi 502	. . . 4 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
3		omndtos 20196	. . . 4 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
4	2, 3	syl 18	. . 3 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Toset)
5		grpmnd 19006	. . . . 5 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
6	5	adantr 485	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd) → 𝑊 ∈ Mnd)
7	1, 6	sylbi 220	. . 3 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Mnd)
8		isarchi3.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
9		isarchi3.0	. . . 4 ⊢ 0 = (0_g‘𝑊)
10		isarchi3.x	. . . 4 ⊢ · = (._g‘𝑊)
11		eqid 2769	. . . 4 ⊢ (le‘𝑊) = (le‘𝑊)
12		isarchi3.i	. . . 4 ⊢ < = (lt‘𝑊)
13	8, 9, 10, 11, 12	isarchi2 33445	. . 3 ⊢ ((𝑊 ∈ Toset ∧ 𝑊 ∈ Mnd) → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥))))
14	4, 7, 13	syl2anc 595	. 2 ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥))))
15		simpr 489	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
16	15	adantr 485	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑛 ∈ ℕ)
17	16	peano2nnd 12249	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 + 1) ∈ ℕ)
18		simp-4l 794	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ oGrp)
19	18	adantr 485	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑊 ∈ oGrp)
20		ogrpgrp 20194	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
21	8, 9	grpidcl 19031	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝐵)
22	19, 20, 21	3syl 19	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 0 ∈ 𝐵)
23		simp-4r 795	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝐵)
24	23	adantr 485	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑥 ∈ 𝐵)
25	20	ad4antr 744	. . . . . . . . . . . . . 14 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ Grp)
26	15	nnzd 12616	. . . . . . . . . . . . . 14 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
27	8, 10	mulgcl 19156	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → (𝑛 · 𝑥) ∈ 𝐵)
28	25, 26, 23, 27	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑛 · 𝑥) ∈ 𝐵)
29	28	adantr 485	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 · 𝑥) ∈ 𝐵)
30		simpllr 787	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 0 < 𝑥)
31		eqid 2769	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
32	8, 12, 31	ogrpaddlt 20207	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ oGrp ∧ ( 0 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ (𝑛 · 𝑥) ∈ 𝐵) ∧ 0 < 𝑥) → ( 0 (+_g‘𝑊)(𝑛 · 𝑥)) < (𝑥(+_g‘𝑊)(𝑛 · 𝑥)))
33	19, 22, 24, 29, 30, 32	syl131anc 1408	. . . . . . . . . . 11 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ( 0 (+_g‘𝑊)(𝑛 · 𝑥)) < (𝑥(+_g‘𝑊)(𝑛 · 𝑥)))
34	19, 20	syl 18	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑊 ∈ Grp)
35	8, 31, 9	grplid 19033	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ (𝑛 · 𝑥) ∈ 𝐵) → ( 0 (+_g‘𝑊)(𝑛 · 𝑥)) = (𝑛 · 𝑥))
36	34, 29, 35	syl2anc 595	. . . . . . . . . . 11 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ( 0 (+_g‘𝑊)(𝑛 · 𝑥)) = (𝑛 · 𝑥))
37		nncn 12240	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
38		ax-1cn 11157	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
39		addcom 11395	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 + 1) = (1 + 𝑛))
40	37, 38, 39	sylancl 597	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) = (1 + 𝑛))
41	40	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑛 + 1) · 𝑥) = ((1 + 𝑛) · 𝑥))
42	16, 41	syl 18	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ((𝑛 + 1) · 𝑥) = ((1 + 𝑛) · 𝑥))
43		grpsgrp 19026	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Smgrp)
44	19, 20, 43	3syl 19	. . . . . . . . . . . . 13 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑊 ∈ Smgrp)
45		1nn 12243	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
46	45	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 1 ∈ ℕ)
47	8, 10, 31	mulgnndir 19168	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Smgrp ∧ (1 ∈ ℕ ∧ 𝑛 ∈ ℕ ∧ 𝑥 ∈ 𝐵)) → ((1 + 𝑛) · 𝑥) = ((1 · 𝑥)(+_g‘𝑊)(𝑛 · 𝑥)))
48	44, 46, 16, 24, 47	syl13anc 1397	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ((1 + 𝑛) · 𝑥) = ((1 · 𝑥)(+_g‘𝑊)(𝑛 · 𝑥)))
49	8, 10	mulg1 19146	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐵 → (1 · 𝑥) = 𝑥)
50	24, 49	syl 18	. . . . . . . . . . . . 13 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (1 · 𝑥) = 𝑥)
51	50	oveq1d 7426	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ((1 · 𝑥)(+_g‘𝑊)(𝑛 · 𝑥)) = (𝑥(+_g‘𝑊)(𝑛 · 𝑥)))
52	42, 48, 51	3eqtrrd 2809	. . . . . . . . . . 11 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑥(+_g‘𝑊)(𝑛 · 𝑥)) = ((𝑛 + 1) · 𝑥))
53	33, 36, 52	3brtr3d 5146	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 · 𝑥) < ((𝑛 + 1) · 𝑥))
54		tospos 18473	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Toset → 𝑊 ∈ Poset)
55	18, 4, 54	3syl 19	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ Poset)
56		simpllr 787	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ 𝐵)
57	26	peano2zd 12702	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℤ)
58	8, 10	mulgcl 19156	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Grp ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑥 ∈ 𝐵) → ((𝑛 + 1) · 𝑥) ∈ 𝐵)
59	25, 57, 23, 58	syl3anc 1396	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) · 𝑥) ∈ 𝐵)
60	8, 11, 12	plelttr 18397	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Poset ∧ (𝑦 ∈ 𝐵 ∧ (𝑛 · 𝑥) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑥) ∈ 𝐵)) → ((𝑦(le‘𝑊)(𝑛 · 𝑥) ∧ (𝑛 · 𝑥) < ((𝑛 + 1) · 𝑥)) → 𝑦 < ((𝑛 + 1) · 𝑥)))
61	55, 56, 28, 59, 60	syl13anc 1397	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → ((𝑦(le‘𝑊)(𝑛 · 𝑥) ∧ (𝑛 · 𝑥) < ((𝑛 + 1) · 𝑥)) → 𝑦 < ((𝑛 + 1) · 𝑥)))
62	61	impl 460	. . . . . . . . . 10 ⊢ (((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) ∧ (𝑛 · 𝑥) < ((𝑛 + 1) · 𝑥)) → 𝑦 < ((𝑛 + 1) · 𝑥))
63	53, 62	mpdan 699	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑦 < ((𝑛 + 1) · 𝑥))
64		oveq1 7418	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 · 𝑥) = ((𝑛 + 1) · 𝑥))
65	64	breq2d 5125	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝑦 < (𝑚 · 𝑥) ↔ 𝑦 < ((𝑛 + 1) · 𝑥)))
66	65	rspcev 3590	. . . . . . . . 9 ⊢ (((𝑛 + 1) ∈ ℕ ∧ 𝑦 < ((𝑛 + 1) · 𝑥)) → ∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥))
67	17, 63, 66	syl2anc 595	. . . . . . . 8 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥))
68	67	r19.29an 3175	. . . . . . 7 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥))
69		oveq1 7418	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑥) = (𝑛 · 𝑥))
70	69	breq2d 5125	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑦 < (𝑚 · 𝑥) ↔ 𝑦 < (𝑛 · 𝑥)))
71	70	cbvrexvw 3250	. . . . . . 7 ⊢ (∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥) ↔ ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))
72	68, 71	sylib 221	. . . . . 6 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))
73	11, 12	pltle 18386	. . . . . . . . 9 ⊢ ((𝑊 ∈ oGrp ∧ 𝑦 ∈ 𝐵 ∧ (𝑛 · 𝑥) ∈ 𝐵) → (𝑦 < (𝑛 · 𝑥) → 𝑦(le‘𝑊)(𝑛 · 𝑥)))
74	18, 56, 28, 73	syl3anc 1396	. . . . . . . 8 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑦 < (𝑛 · 𝑥) → 𝑦(le‘𝑊)(𝑛 · 𝑥)))
75	74	reximdva 3184	. . . . . . 7 ⊢ ((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) → (∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥) → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)))
76	75	imp 411	. . . . . 6 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)) → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥))
77	72, 76	impbida 812	. . . . 5 ⊢ ((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) → (∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥) ↔ ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)))
78	77	pm5.74da 815	. . . 4 ⊢ (((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)) ↔ ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))
79	78	ralbidva 3192	. . 3 ⊢ ((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)) ↔ ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))
80	79	ralbidva 3192	. 2 ⊢ (𝑊 ∈ oGrp → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))
81	14, 80	bitrd 282	1 ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∃wrex 3095 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 ℂcc 11097 1c1 11100 + caddc 11102 ℕcn 12232 ℤcz 12590 Basecbs 17268 +_gcplusg 17309 lecple 17316 0_gc0g 17491 Posetcpo 18362 ltcplt 18363 Tosetctos 18469 Smgrpcsgrp 18775 Mndcmnd 18791 Grpcgrp 18999 ._gcmg 19132 oMndcomnd 20188 oGrpcogrp 20189 Archicarchi 33437

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 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 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-n0 12504 df-z 12591 df-uz 12862 df-fz 13535 df-seq 14037 df-0g 17493 df-proset 18349 df-poset 18368 df-plt 18383 df-toset 18470 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-grp 19002 df-minusg 19003 df-mulg 19133 df-omnd 20190 df-ogrp 20191 df-inftm 33438 df-archi 33439

This theorem is referenced by: archiexdiv 33450 isarchiofld 33459

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