isarchi3 - Mathbox for Thierry Arnoux

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Theorem isarchi3 30816

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 30703	. . . . 5 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
2	1	simprbi 499	. . . 4 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
3		omndtos 30706	. . . 4 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
4	2, 3	syl 17	. . 3 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Toset)
5		grpmnd 18110	. . . . 5 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
6	5	adantr 483	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd) → 𝑊 ∈ Mnd)
7	1, 6	sylbi 219	. . 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 2821	. . . 4 ⊢ (le‘𝑊) = (le‘𝑊)
12		isarchi3.i	. . . 4 ⊢ < = (lt‘𝑊)
13	8, 9, 10, 11, 12	isarchi2 30814	. . 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 586	. 2 ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥))))
15		simpr 487	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
16	15	adantr 483	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑛 ∈ ℕ)
17	16	peano2nnd 11655	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 + 1) ∈ ℕ)
18		simp-4l 781	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ oGrp)
19	18	adantr 483	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑊 ∈ oGrp)
20		ogrpgrp 30704	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Grp)
21	8, 9	grpidcl 18131	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝐵)
24	23	adantr 483	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑥 ∈ 𝐵)
25	20	ad4antr 730	. . . . . . . . . . . . . 14 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ Grp)
26	15	nnzd 12087	. . . . . . . . . . . . . 14 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
27	8, 10	mulgcl 18245	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Grp ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ 𝐵) → (𝑛 · 𝑥) ∈ 𝐵)
28	25, 26, 23, 27	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑛 · 𝑥) ∈ 𝐵)
29	28	adantr 483	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 · 𝑥) ∈ 𝐵)
30		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 0 < 𝑥)
31		eqid 2821	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
32	8, 12, 31	ogrpaddlt 30718	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 18133	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Grp ∧ (𝑛 · 𝑥) ∈ 𝐵) → ( 0 (+_g‘𝑊)(𝑛 · 𝑥)) = (𝑛 · 𝑥))
36	34, 29, 35	syl2anc 586	. . . . . . . . . . 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 11646	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
38		ax-1cn 10595	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
39		addcom 10826	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 + 1) = (1 + 𝑛))
40	37, 38, 39	sylancl 588	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) = (1 + 𝑛))
41	40	oveq1d 7171	. . . . . . . . . . . . 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 18127	. . . . . . . . . . . . . 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 11649	. . . . . . . . . . . . . 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 18256	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 18235	. . . . . . . . . . . . . 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 7171	. . . . . . . . . . . 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 2861	. . . . . . . . . . 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 5097	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → (𝑛 · 𝑥) < ((𝑛 + 1) · 𝑥))
54		tospos 30645	. . . . . . . . . . . . 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 774	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ 𝐵)
57	26	peano2zd 12091	. . . . . . . . . . . . 13 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℤ)
58	8, 10	mulgcl 18245	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Grp ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑥 ∈ 𝐵) → ((𝑛 + 1) · 𝑥) ∈ 𝐵)
59	25, 57, 23, 58	syl3anc 1367	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) · 𝑥) ∈ 𝐵)
60	8, 11, 12	plelttr 17582	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 458	. . . . . . . . . 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 685	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) ∧ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → 𝑦 < ((𝑛 + 1) · 𝑥))
64		oveq1 7163	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 · 𝑥) = ((𝑛 + 1) · 𝑥))
65	64	breq2d 5078	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝑦 < (𝑚 · 𝑥) ↔ 𝑦 < ((𝑛 + 1) · 𝑥)))
66	65	rspcev 3623	. . . . . . . . 9 ⊢ (((𝑛 + 1) ∈ ℕ ∧ 𝑦 < ((𝑛 + 1) · 𝑥)) → ∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥))
67	17, 63, 66	syl2anc 586	. . . . . . . 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 3288	. . . . . . 7 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)) → ∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥))
69		oveq1 7163	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑥) = (𝑛 · 𝑥))
70	69	breq2d 5078	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑦 < (𝑚 · 𝑥) ↔ 𝑦 < (𝑛 · 𝑥)))
71	70	cbvrexvw 3450	. . . . . . 7 ⊢ (∃𝑚 ∈ ℕ 𝑦 < (𝑚 · 𝑥) ↔ ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))
72	68, 71	sylib 220	. . . . . 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 17571	. . . . . . . . 9 ⊢ ((𝑊 ∈ oGrp ∧ 𝑦 ∈ 𝐵 ∧ (𝑛 · 𝑥) ∈ 𝐵) → (𝑦 < (𝑛 · 𝑥) → 𝑦(le‘𝑊)(𝑛 · 𝑥)))
74	18, 56, 28, 73	syl3anc 1367	. . . . . . . 8 ⊢ (((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) ∧ 𝑛 ∈ ℕ) → (𝑦 < (𝑛 · 𝑥) → 𝑦(le‘𝑊)(𝑛 · 𝑥)))
75	74	reximdva 3274	. . . . . . 7 ⊢ ((((𝑊 ∈ oGrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 0 < 𝑥) → (∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥) → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛 · 𝑥)))
76	75	imp 409	. . . . . 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 799	. . . . 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 802	. . . 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 3196	. . 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 3196	. 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 281	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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 ℕcn 11638 ℤcz 11982 Basecbs 16483 +_gcplusg 16565 lecple 16572 0_gc0g 16713 Posetcpo 17550 ltcplt 17551 Tosetctos 17643 Smgrpcsgrp 17900 Mndcmnd 17911 Grpcgrp 18103 ._gcmg 18224 oMndcomnd 30698 oGrpcogrp 30699 Archicarchi 30806

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-seq 13371 df-0g 16715 df-proset 17538 df-poset 17556 df-plt 17568 df-toset 17644 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mulg 18225 df-omnd 30700 df-ogrp 30701 df-inftm 30807 df-archi 30808

This theorem is referenced by: archiexdiv 30819 isarchiofld 30890

W3C validator