Theorem fimgmcyc 43228

Description: Version of odcl2 19635 for finite magmas: the multiples of an element 𝐴 ∈ 𝐵 are eventually periodic. (Contributed by SN, 3-Jul-2025.)

Hypotheses

Ref	Expression
fimgmcyc.b	⊢ 𝐵 = (Base‘𝑀)
fimgmcyc.m	⊢ · = (.g‘𝑀)
fimgmcyc.s	⊢ (𝜑 → 𝑀 ∈ Mgm)
fimgmcyc.f	⊢ (𝜑 → 𝐵 ∈ Fin)
fimgmcyc.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)

Assertion

Ref	Expression
fimgmcyc	⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))

Distinct variable groups:   𝐴,𝑜,𝑝   · ,𝑜,𝑝   𝜑,𝑜,𝑝

Allowed substitution hints:   𝐵(𝑜,𝑝)   𝑀(𝑜,𝑝)

Proof of Theorem fimgmcyc

Dummy variables 𝑛 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		domnsym 9091	. . . . . . . . 9 ⊢ (ℕ ≼ 𝐵 → ¬ 𝐵 ≺ ℕ)
2		fimgmcyc.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		fisdomnn 42936	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → 𝐵 ≺ ℕ)
4	2, 3	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≺ ℕ)
5	1, 4	nsyl3 139	. . . . . . . 8 ⊢ (𝜑 → ¬ ℕ ≼ 𝐵)
6		fimgmcyc.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
7	6	fvexi 6896	. . . . . . . . 9 ⊢ 𝐵 ∈ V
8	7	f1dom 8970	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1→𝐵 → ℕ ≼ 𝐵)
9	5, 8	nsyl 141	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1→𝐵)
10		fimgmcyc.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Mgm)
11	10	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ Mgm)
12		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
13		fimgmcyc.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
14	13	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ 𝐵)
15		fimgmcyc.m	. . . . . . . . . . 11 ⊢ · = (.g‘𝑀)
16	6, 15	mulgnncl 19155	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mgm ∧ 𝑛 ∈ ℕ ∧ 𝐴 ∈ 𝐵) → (𝑛 · 𝐴) ∈ 𝐵)
17	11, 12, 14, 16	syl3anc 1396	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 · 𝐴) ∈ 𝐵)
18	17	fmpttd 7111	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ⟶𝐵)
19		dff13 7253	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1→𝐵 ↔ ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ⟶𝐵 ∧ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞)))
20	19	baib 544	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ⟶𝐵 → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1→𝐵 ↔ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞)))
21	18, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1→𝐵 ↔ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞)))
22	9, 21	mtbid 327	. . . . . 6 ⊢ (𝜑 → ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞))
23		oveq1 7418	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑜 → (𝑛 · 𝐴) = (𝑜 · 𝐴))
24		eqid 2769	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)) = (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))
25		ovex 7444	. . . . . . . . . . 11 ⊢ (𝑜 · 𝐴) ∈ V
26	23, 24, 25	fvmpt 6990	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = (𝑜 · 𝐴))
27		oveq1 7418	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑞 → (𝑛 · 𝐴) = (𝑞 · 𝐴))
28		ovex 7444	. . . . . . . . . . 11 ⊢ (𝑞 · 𝐴) ∈ V
29	27, 24, 28	fvmpt 6990	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) = (𝑞 · 𝐴))
30	26, 29	eqeqan12d 2783	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) ↔ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
31	30	imbi1d 344	. . . . . . . 8 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → ((((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞) ↔ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞)))
32	31	ralbidva 3192	. . . . . . 7 ⊢ (𝑜 ∈ ℕ → (∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞) ↔ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞)))
33	32	ralbiia 3115	. . . . . 6 ⊢ (∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞) ↔ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
34	22, 33	sylnib 331	. . . . 5 ⊢ (𝜑 → ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
35		df-ne 2965	. . . . . . . . 9 ⊢ (𝑜 ≠ 𝑞 ↔ ¬ 𝑜 = 𝑞)
36	35	anbi1i 635	. . . . . . . 8 ⊢ ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (¬ 𝑜 = 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
37		ancom 465	. . . . . . . 8 ⊢ ((¬ 𝑜 = 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑜 · 𝐴) = (𝑞 · 𝐴) ∧ ¬ 𝑜 = 𝑞))
38		annim 408	. . . . . . . 8 ⊢ (((𝑜 · 𝐴) = (𝑞 · 𝐴) ∧ ¬ 𝑜 = 𝑞) ↔ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
39	36, 37, 38	3bitri 300	. . . . . . 7 ⊢ ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
40	39	2rexbii 3147	. . . . . 6 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
41		rexnal2 3153	. . . . . 6 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞) ↔ ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
42	40, 41	bitri 278	. . . . 5 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
43	34, 42	sylibr 237	. . . 4 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
44	43	fimgmcyclem 43227	. . 3 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
45		nnz 12612	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℤ)
46		eluzp1 42992	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℤ → (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ↔ (𝑞 ∈ ℤ ∧ 𝑜 < 𝑞)))
47	45, 46	syl 18	. . . . . . . . 9 ⊢ (𝑜 ∈ ℕ → (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ↔ (𝑞 ∈ ℤ ∧ 𝑜 < 𝑞)))
48		idd 25	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ → 𝑞 ∈ ℤ))
49		nnz 12612	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℤ)
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℕ → 𝑞 ∈ ℤ))
51		0red 11211	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 ∈ ℝ)
52		nnre 12240	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℝ)
53	52	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑜 ∈ ℝ)
54		zre 12595	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ ℤ → 𝑞 ∈ ℝ)
55	54	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑞 ∈ ℝ)
56		nngt0 12267	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ ℕ → 0 < 𝑜)
57	56	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 < 𝑜)
58		simplr 780	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑜 < 𝑞)
59	51, 53, 55, 57, 58	lttrd 11371	. . . . . . . . . . . . . 14 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 < 𝑞)
60		elnnz 12601	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ ℕ ↔ (𝑞 ∈ ℤ ∧ 0 < 𝑞))
61	60	rbaibr 546	. . . . . . . . . . . . . 14 ⊢ (0 < 𝑞 → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
62	59, 61	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
63	62	ex 417	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ)))
64	48, 50, 63	pm5.21ndd 382	. . . . . . . . . . 11 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
65	64	ex 417	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → (𝑜 < 𝑞 → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ)))
66	65	pm5.32rd 588	. . . . . . . . 9 ⊢ (𝑜 ∈ ℕ → ((𝑞 ∈ ℤ ∧ 𝑜 < 𝑞) ↔ (𝑞 ∈ ℕ ∧ 𝑜 < 𝑞)))
67	47, 66	bitrd 282	. . . . . . . 8 ⊢ (𝑜 ∈ ℕ → (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ↔ (𝑞 ∈ ℕ ∧ 𝑜 < 𝑞)))
68	67	anbi1d 642	. . . . . . 7 ⊢ (𝑜 ∈ ℕ → ((𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑞 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
69		anass 473	. . . . . . 7 ⊢ (((𝑞 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
70	68, 69	bitrdi 290	. . . . . 6 ⊢ (𝑜 ∈ ℕ → ((𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
71	70	exbidv 1948	. . . . 5 ⊢ (𝑜 ∈ ℕ → (∃𝑞(𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑞(𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
72		df-rex 3096	. . . . 5 ⊢ (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑞(𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
73		df-rex 3096	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑞(𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
74	71, 72, 73	3bitr4g 317	. . . 4 ⊢ (𝑜 ∈ ℕ → (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
75	74	rexbiia 3116	. . 3 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
76	44, 75	sylibr 237	. 2 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴))
77		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑜 ∈ ℕ)
78	77	peano2nnd 12250	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
79	78	nnzd 12617	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℤ)
80		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℕ)
81	77, 80	nnaddcld 12288	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℕ)
82	81	nnzd 12617	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℤ)
83		1red 11209	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ∈ ℝ)
84	80	nnred 12248	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℝ)
85	77	nnred 12248	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑜 ∈ ℝ)
86	80	nnge1d 12284	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ≤ 𝑝)
87	83, 84, 85, 86	leadd2dd 11829	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ≤ (𝑜 + 𝑝))
88		eluz2 12868	. . . . 5 ⊢ ((𝑜 + 𝑝) ∈ (ℤ≥‘(𝑜 + 1)) ↔ ((𝑜 + 1) ∈ ℤ ∧ (𝑜 + 𝑝) ∈ ℤ ∧ (𝑜 + 1) ≤ (𝑜 + 𝑝)))
89	79, 82, 87, 88	syl3anbrc 1360	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ (ℤ≥‘(𝑜 + 1)))
90		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℕ)
91	90	nnzd 12617	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℤ)
92		eluzp1l 12889	. . . . . . 7 ⊢ ((𝑜 ∈ ℤ ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 < 𝑞)
93	91, 92	sylan 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 < 𝑞)
94		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 ∈ ℕ)
95		peano2nn 12245	. . . . . . . . 9 ⊢ (𝑜 ∈ ℕ → (𝑜 + 1) ∈ ℕ)
96	95	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
97		eluznn 12942	. . . . . . . 8 ⊢ (((𝑜 + 1) ∈ ℕ ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
98	96, 97	sylan 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
99		nnsub 12280	. . . . . . 7 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝑜 < 𝑞 ↔ (𝑞 − 𝑜) ∈ ℕ))
100	94, 98, 99	syl2anc 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → (𝑜 < 𝑞 ↔ (𝑞 − 𝑜) ∈ ℕ))
101	93, 100	mpbid 235	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → (𝑞 − 𝑜) ∈ ℕ)
102		eluzelcn 12874	. . . . . . 7 ⊢ (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) → 𝑞 ∈ ℂ)
103	102	ad2antlr 739	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑞 ∈ ℂ)
104		nncn 12241	. . . . . . . 8 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℂ)
105	104	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℂ)
106	105	ad2antrr 738	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑜 ∈ ℂ)
107		simpr 489	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑝 = (𝑞 − 𝑜))
108	103, 106, 107	rsubrotld 42963	. . . . 5 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑞 = (𝑜 + 𝑝))
109	101, 108	rspcedeq2vd 3598	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → ∃𝑝 ∈ ℕ 𝑞 = (𝑜 + 𝑝))
110		oveq1 7418	. . . . . 6 ⊢ (𝑞 = (𝑜 + 𝑝) → (𝑞 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))
111	110	eqeq2d 2780	. . . . 5 ⊢ (𝑞 = (𝑜 + 𝑝) → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
112	111	adantl 486	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 = (𝑜 + 𝑝)) → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
113	89, 109, 112	rexxfrd 5381	. . 3 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
114	113	rexbidva 3193	. 2 ⊢ (𝜑 → (∃𝑜 ∈ ℕ ∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
115	76, 114	mpbid 235	1 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   class class class wbr 5113   ↦ cmpt 5196  ⟶wf 6533  –1-1→wf1 6534  ‘cfv 6537  (class class class)co 7411   ≼ cdom 8941   ≺ csdm 8942  Fincfn 8943  ℂcc 11098  ℝcr 11099  0cc0 11100  1c1 11101   + caddc 11103   < clt 11243   ≤ cle 11244   − cmin 11441  ℕcn 12233  ℤcz 12591  ℤ_≥cuz 12862  Basecbs 17269  Mgmcmgm 18696  ._gcmg 19133

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-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

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-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-int 4917  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 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-seq 14038  df-hash 14367  df-mgm 18698  df-mulg 19134

This theorem is referenced by:  fidomncyc  43229