isarchi3 - Mathbox for Thierry Arnoux

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Theorem isarchi3 33748

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 20338	. . . . 5 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
2	1	simprbi 503	. . . 4 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
3		omndtos 20341	. . . 4 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
4	2, 3	syl 18	. . 3 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Toset)
5		grpmnd 19151	. . . . 5 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
6	5	adantr 486	. . . 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 2761	. . . 4 ⊢ (le‘𝑊) = (le‘𝑊)
12		isarchi3.i	. . . 4 ⊢ < = (lt‘𝑊)
13	8, 9, 10, 11, 12	isarchi2 33746	. . 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 596	. 2 ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥))))
15		simpr 490	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
16	15	adantr 486	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑛 ∈ ℕ)
17	16	peano2nnd 12352	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 + 1) ∈ ℕ)
18		simp-4l 795	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ oGrp)
19	18	adantr 486	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑊 ∈ oGrp)
20		ogrpgrp 20339	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
21	8, 9	grpidcl 19176	. . . . . . . . . . . . 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 796	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝐵)
24	23	adantr 486	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑥 ∈ 𝐵)
25	20	ad4antr 745	. . . . . . . . . . . . . 14 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ Grp)
26	15	nnzd 12719	. . . . . . . . . . . . . 14 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
27	8, 10	mulgcl 19301	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → (𝑛 · 𝑥) ∈ 𝐵)
28	25, 26, 23, 27	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑛 · 𝑥) ∈ 𝐵)
29	28	adantr 486	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 · 𝑥) ∈ 𝐵)
30		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 0 < 𝑥)
31		eqid 2761	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
32	8, 12, 31	ogrpaddlt 20352	. . . . . . . . . . . 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 1410	. . . . . . . . . . 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 19178	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ (𝑛 · 𝑥) ∈ 𝐵) → ( 0 (+_g‘𝑊)(𝑛 · 𝑥)) = (𝑛 · 𝑥))
36	34, 29, 35	syl2anc 596	. . . . . . . . . . 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 12343	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
38		ax-1cn 11258	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
39		addcom 11496	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 + 1) = (1 + 𝑛))
40	37, 38, 39	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) = (1 + 𝑛))
41	40	oveq1d 7435	. . . . . . . . . . . . 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 19171	. . . . . . . . . . . . . 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 12346	. . . . . . . . . . . . . 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 19313	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 19291	. . . . . . . . . . . . . 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 7435	. . . . . . . . . . . 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 2801	. . . . . . . . . . 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 5136	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 · 𝑥) < ((𝑛 + 1) · 𝑥))
54		tospos 18592	. . . . . . . . . . . . 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 788	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ 𝐵)
57	26	peano2zd 12806	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℤ)
58	8, 10	mulgcl 19301	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Grp ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑥 ∈ 𝐵) → ((𝑛 + 1) · 𝑥) ∈ 𝐵)
59	25, 57, 23, 58	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) · 𝑥) ∈ 𝐵)
60	8, 11, 12	plelttr 18516	. . . . . . . . . . . 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 1399	. . . . . . . . . . 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 461	. . . . . . . . . 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 700	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑦 < ((𝑛 + 1) · 𝑥))
64		oveq1 7427	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 · 𝑥) = ((𝑛 + 1) · 𝑥))
65	64	breq2d 5115	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝑦 < (𝑚 · 𝑥) ↔ 𝑦 < ((𝑛 + 1) · 𝑥)))
66	65	rspcev 3577	. . . . . . . . 9 ⊢ (((𝑛 + 1) ∈ ℕ ∧ 𝑦 < ((𝑛 + 1) · 𝑥)) → ∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥))
67	17, 63, 66	syl2anc 596	. . . . . . . 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 3167	. . . . . . 7 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥))
69		oveq1 7427	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑥) = (𝑛 · 𝑥))
70	69	breq2d 5115	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑦 < (𝑚 · 𝑥) ↔ 𝑦 < (𝑛 · 𝑥)))
71	70	cbvrexvw 3242	. . . . . . 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 18505	. . . . . . . . 9 ⊢ ((𝑊 ∈ oGrp ∧ 𝑦 ∈ 𝐵 ∧ (𝑛 · 𝑥) ∈ 𝐵) → (𝑦 < (𝑛 · 𝑥) → 𝑦(le‘𝑊)(𝑛 · 𝑥)))
74	18, 56, 28, 73	syl3anc 1398	. . . . . . . 8 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑦 < (𝑛 · 𝑥) → 𝑦(le‘𝑊)(𝑛 · 𝑥)))
75	74	reximdva 3176	. . . . . . 7 ⊢ ((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) → (∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥) → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)))
76	75	imp 412	. . . . . 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 813	. . . . 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 816	. . . 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 3184	. . 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 3184	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∃wrex 3087 class class class wbr 5103 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 1c1 11201 + caddc 11203 ℕcn 12335 ℤcz 12693 Basecbs 17387 +_gcplusg 17428 lecple 17435 0_gc0g 17610 Posetcpo 18481 ltcplt 18482 Tosetctos 18588 Smgrpcsgrp 18907 Mndcmnd 18923 Grpcgrp 19144 ._gcmg 19277 oMndcomnd 20333 oGrpcogrp 20334 Archicarchi 33738

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 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-seq 14145 df-0g 17612 df-proset 18468 df-poset 18487 df-plt 18502 df-toset 18589 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-grp 19147 df-minusg 19148 df-mulg 19278 df-omnd 20335 df-ogrp 20336 df-inftm 33739 df-archi 33740

This theorem is used by: archiexdiv 33751 isarchiofld 33760

W3C validator