isarchi3 - Mathbox for Thierry Arnoux

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Theorem isarchi3 33194

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 33079	. . . . 5 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
2	1	simprbi 496	. . . 4 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
3		omndtos 33082	. . . 4 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
4	2, 3	syl 17	. . 3 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Toset)
5		grpmnd 18958	. . . . 5 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
6	5	adantr 480	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd) → 𝑊 ∈ Mnd)
7	1, 6	sylbi 217	. . 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 2737	. . . 4 ⊢ (le‘𝑊) = (le‘𝑊)
12		isarchi3.i	. . . 4 ⊢ < = (lt‘𝑊)
13	8, 9, 10, 11, 12	isarchi2 33192	. . 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 584	. 2 ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥))))
15		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
16	15	adantr 480	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑛 ∈ ℕ)
17	16	peano2nnd 12283	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 + 1) ∈ ℕ)
18		simp-4l 783	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ oGrp)
19	18	adantr 480	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑊 ∈ oGrp)
20		ogrpgrp 33080	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
21	8, 9	grpidcl 18983	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝐵)
22	19, 20, 21	3syl 18	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 0 ∈ 𝐵)
23		simp-4r 784	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝐵)
24	23	adantr 480	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑥 ∈ 𝐵)
25	20	ad4antr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ Grp)
26	15	nnzd 12640	. . . . . . . . . . . . . 14 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
27	8, 10	mulgcl 19109	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → (𝑛 · 𝑥) ∈ 𝐵)
28	25, 26, 23, 27	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑛 · 𝑥) ∈ 𝐵)
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 · 𝑥) ∈ 𝐵)
30		simpllr 776	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 0 < 𝑥)
31		eqid 2737	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
32	8, 12, 31	ogrpaddlt 33094	. . . . . . . . . . . 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 1385	. . . . . . . . . . 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 17	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑊 ∈ Grp)
35	8, 31, 9	grplid 18985	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ (𝑛 · 𝑥) ∈ 𝐵) → ( 0 (+_g‘𝑊)(𝑛 · 𝑥)) = (𝑛 · 𝑥))
36	34, 29, 35	syl2anc 584	. . . . . . . . . . 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 12274	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
38		ax-1cn 11213	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
39		addcom 11447	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 + 1) = (1 + 𝑛))
40	37, 38, 39	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) = (1 + 𝑛))
41	40	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑛 + 1) · 𝑥) = ((1 + 𝑛) · 𝑥))
42	16, 41	syl 17	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ((𝑛 + 1) · 𝑥) = ((1 + 𝑛) · 𝑥))
43		grpsgrp 18978	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Smgrp)
44	19, 20, 43	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑊 ∈ Smgrp)
45		1nn 12277	. . . . . . . . . . . . . 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 19121	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 19099	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐵 → (1 · 𝑥) = 𝑥)
50	24, 49	syl 17	. . . . . . . . . . . . 13 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (1 · 𝑥) = 𝑥)
51	50	oveq1d 7446	. . . . . . . . . . . 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 2782	. . . . . . . . . . 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 5174	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 · 𝑥) < ((𝑛 + 1) · 𝑥))
54		tospos 18465	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Toset → 𝑊 ∈ Poset)
55	18, 4, 54	3syl 18	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ Poset)
56		simpllr 776	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ 𝐵)
57	26	peano2zd 12725	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℤ)
58	8, 10	mulgcl 19109	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Grp ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑥 ∈ 𝐵) → ((𝑛 + 1) · 𝑥) ∈ 𝐵)
59	25, 57, 23, 58	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) · 𝑥) ∈ 𝐵)
60	8, 11, 12	plelttr 18389	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 455	. . . . . . . . . 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 687	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑦 < ((𝑛 + 1) · 𝑥))
64		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 · 𝑥) = ((𝑛 + 1) · 𝑥))
65	64	breq2d 5155	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝑦 < (𝑚 · 𝑥) ↔ 𝑦 < ((𝑛 + 1) · 𝑥)))
66	65	rspcev 3622	. . . . . . . . 9 ⊢ (((𝑛 + 1) ∈ ℕ ∧ 𝑦 < ((𝑛 + 1) · 𝑥)) → ∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥))
67	17, 63, 66	syl2anc 584	. . . . . . . 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 3158	. . . . . . 7 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥))
69		oveq1 7438	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑥) = (𝑛 · 𝑥))
70	69	breq2d 5155	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑦 < (𝑚 · 𝑥) ↔ 𝑦 < (𝑛 · 𝑥)))
71	70	cbvrexvw 3238	. . . . . . 7 ⊢ (∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥) ↔ ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))
72	68, 71	sylib 218	. . . . . 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 18378	. . . . . . . . 9 ⊢ ((𝑊 ∈ oGrp ∧ 𝑦 ∈ 𝐵 ∧ (𝑛 · 𝑥) ∈ 𝐵) → (𝑦 < (𝑛 · 𝑥) → 𝑦(le‘𝑊)(𝑛 · 𝑥)))
74	18, 56, 28, 73	syl3anc 1373	. . . . . . . 8 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑦 < (𝑛 · 𝑥) → 𝑦(le‘𝑊)(𝑛 · 𝑥)))
75	74	reximdva 3168	. . . . . . 7 ⊢ ((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) → (∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥) → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)))
76	75	imp 406	. . . . . 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 801	. . . . 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 804	. . . 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 3176	. . 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 3176	. 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 279	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 ℕcn 12266 ℤcz 12613 Basecbs 17247 +_gcplusg 17297 lecple 17304 0_gc0g 17484 Posetcpo 18353 ltcplt 18354 Tosetctos 18461 Smgrpcsgrp 18731 Mndcmnd 18747 Grpcgrp 18951 ._gcmg 19085 oMndcomnd 33074 oGrpcogrp 33075 Archicarchi 33184

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-0g 17486 df-proset 18340 df-poset 18359 df-plt 18375 df-toset 18462 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-mulg 19086 df-omnd 33076 df-ogrp 33077 df-inftm 33185 df-archi 33186

This theorem is referenced by: archiexdiv 33197 isarchiofld 33347

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