Theorem fimgmcyc 43020

Description: Version of odcl2 19531 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 9031	. . . . . . . . 9 ⊢ (ℕ ≼ 𝐵 → ¬ 𝐵 ≺ ℕ)
2		fimgmcyc.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		fisdomnn 42728	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → 𝐵 ≺ ℕ)
4	2, 3	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≺ ℕ)
5	1, 4	nsyl3 138	. . . . . . . 8 ⊢ (𝜑 → ¬ ℕ ≼ 𝐵)
6		fimgmcyc.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
7	6	fvexi 6841	. . . . . . . . 9 ⊢ 𝐵 ∈ V
8	7	f1dom 8910	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1→𝐵 → ℕ ≼ 𝐵)
9	5, 8	nsyl 140	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1→𝐵)
10		fimgmcyc.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Mgm)
11	10	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ Mgm)
12		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
13		fimgmcyc.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
14	13	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ 𝐵)
15		fimgmcyc.m	. . . . . . . . . . 11 ⊢ · = (.g‘𝑀)
16	6, 15	mulgnncl 19056	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mgm ∧ 𝑛 ∈ ℕ ∧ 𝐴 ∈ 𝐵) → (𝑛 · 𝐴) ∈ 𝐵)
17	11, 12, 14, 16	syl3anc 1379	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 · 𝐴) ∈ 𝐵)
18	17	fmpttd 7056	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ⟶𝐵)
19		dff13 7198	. . . . . . . . 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 540	. . . . . . . 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 17	. . . . . . 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 325	. . . . . 6 ⊢ (𝜑 → ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞))
23		oveq1 7363	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑜 → (𝑛 · 𝐴) = (𝑜 · 𝐴))
24		eqid 2739	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)) = (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))
25		ovex 7389	. . . . . . . . . . 11 ⊢ (𝑜 · 𝐴) ∈ V
26	23, 24, 25	fvmpt 6935	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = (𝑜 · 𝐴))
27		oveq1 7363	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑞 → (𝑛 · 𝐴) = (𝑞 · 𝐴))
28		ovex 7389	. . . . . . . . . . 11 ⊢ (𝑞 · 𝐴) ∈ V
29	27, 24, 28	fvmpt 6935	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) = (𝑞 · 𝐴))
30	26, 29	eqeqan12d 2753	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) ↔ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
31	30	imbi1d 342	. . . . . . . 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 3160	. . . . . . 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 3083	. . . . . 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 329	. . . . 5 ⊢ (𝜑 → ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
35		df-ne 2935	. . . . . . . . 9 ⊢ (𝑜 ≠ 𝑞 ↔ ¬ 𝑜 = 𝑞)
36	35	anbi1i 630	. . . . . . . 8 ⊢ ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (¬ 𝑜 = 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
37		ancom 461	. . . . . . . 8 ⊢ ((¬ 𝑜 = 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑜 · 𝐴) = (𝑞 · 𝐴) ∧ ¬ 𝑜 = 𝑞))
38		annim 404	. . . . . . . 8 ⊢ (((𝑜 · 𝐴) = (𝑞 · 𝐴) ∧ ¬ 𝑜 = 𝑞) ↔ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
39	36, 37, 38	3bitri 298	. . . . . . 7 ⊢ ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
40	39	2rexbii 3115	. . . . . 6 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
41		rexnal2 3121	. . . . . 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 276	. . . . 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 235	. . . 4 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
44	43	fimgmcyclem 43019	. . 3 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
45		nnz 12536	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℤ)
46		eluzp1 42784	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℤ → (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ↔ (𝑞 ∈ ℤ ∧ 𝑜 < 𝑞)))
47	45, 46	syl 17	. . . . . . . . 9 ⊢ (𝑜 ∈ ℕ → (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ↔ (𝑞 ∈ ℤ ∧ 𝑜 < 𝑞)))
48		idd 24	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ → 𝑞 ∈ ℤ))
49		nnz 12536	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℤ)
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℕ → 𝑞 ∈ ℤ))
51		0red 11138	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 ∈ ℝ)
52		nnre 12172	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℝ)
53	52	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑜 ∈ ℝ)
54		zre 12519	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ ℤ → 𝑞 ∈ ℝ)
55	54	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑞 ∈ ℝ)
56		nngt0 12199	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ ℕ → 0 < 𝑜)
57	56	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 < 𝑜)
58		simplr 774	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑜 < 𝑞)
59	51, 53, 55, 57, 58	lttrd 11298	. . . . . . . . . . . . . 14 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 < 𝑞)
60		elnnz 12525	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ ℕ ↔ (𝑞 ∈ ℤ ∧ 0 < 𝑞))
61	60	rbaibr 542	. . . . . . . . . . . . . 14 ⊢ (0 < 𝑞 → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
62	59, 61	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
63	62	ex 413	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ)))
64	48, 50, 63	pm5.21ndd 380	. . . . . . . . . . 11 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
65	64	ex 413	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → (𝑜 < 𝑞 → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ)))
66	65	pm5.32rd 583	. . . . . . . . 9 ⊢ (𝑜 ∈ ℕ → ((𝑞 ∈ ℤ ∧ 𝑜 < 𝑞) ↔ (𝑞 ∈ ℕ ∧ 𝑜 < 𝑞)))
67	47, 66	bitrd 280	. . . . . . . 8 ⊢ (𝑜 ∈ ℕ → (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ↔ (𝑞 ∈ ℕ ∧ 𝑜 < 𝑞)))
68	67	anbi1d 637	. . . . . . 7 ⊢ (𝑜 ∈ ℕ → ((𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑞 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
69		anass 469	. . . . . . 7 ⊢ (((𝑞 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
70	68, 69	bitrdi 288	. . . . . 6 ⊢ (𝑜 ∈ ℕ → ((𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
71	70	exbidv 1928	. . . . 5 ⊢ (𝑜 ∈ ℕ → (∃𝑞(𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑞(𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
72		df-rex 3064	. . . . 5 ⊢ (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑞(𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
73		df-rex 3064	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑞(𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
74	71, 72, 73	3bitr4g 315	. . . 4 ⊢ (𝑜 ∈ ℕ → (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
75	74	rexbiia 3084	. . 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 235	. 2 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴))
77		simplr 774	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑜 ∈ ℕ)
78	77	peano2nnd 12182	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
79	78	nnzd 12541	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℤ)
80		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℕ)
81	77, 80	nnaddcld 12220	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℕ)
82	81	nnzd 12541	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℤ)
83		1red 11136	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ∈ ℝ)
84	80	nnred 12180	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℝ)
85	77	nnred 12180	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑜 ∈ ℝ)
86	80	nnge1d 12216	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ≤ 𝑝)
87	83, 84, 85, 86	leadd2dd 11756	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ≤ (𝑜 + 𝑝))
88		eluz2 12785	. . . . 5 ⊢ ((𝑜 + 𝑝) ∈ (ℤ≥‘(𝑜 + 1)) ↔ ((𝑜 + 1) ∈ ℤ ∧ (𝑜 + 𝑝) ∈ ℤ ∧ (𝑜 + 1) ≤ (𝑜 + 𝑝)))
89	79, 82, 87, 88	syl3anbrc 1350	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ (ℤ≥‘(𝑜 + 1)))
90		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℕ)
91	90	nnzd 12541	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℤ)
92		eluzp1l 12806	. . . . . . 7 ⊢ ((𝑜 ∈ ℤ ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 < 𝑞)
93	91, 92	sylan 586	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 < 𝑞)
94		simplr 774	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 ∈ ℕ)
95		peano2nn 12177	. . . . . . . . 9 ⊢ (𝑜 ∈ ℕ → (𝑜 + 1) ∈ ℕ)
96	95	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
97		eluznn 12859	. . . . . . . 8 ⊢ (((𝑜 + 1) ∈ ℕ ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
98	96, 97	sylan 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
99		nnsub 12212	. . . . . . 7 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝑜 < 𝑞 ↔ (𝑞 − 𝑜) ∈ ℕ))
100	94, 98, 99	syl2anc 590	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → (𝑜 < 𝑞 ↔ (𝑞 − 𝑜) ∈ ℕ))
101	93, 100	mpbid 233	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → (𝑞 − 𝑜) ∈ ℕ)
102		eluzelcn 12791	. . . . . . 7 ⊢ (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) → 𝑞 ∈ ℂ)
103	102	ad2antlr 733	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑞 ∈ ℂ)
104		nncn 12173	. . . . . . . 8 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℂ)
105	104	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℂ)
106	105	ad2antrr 732	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑜 ∈ ℂ)
107		simpr 485	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑝 = (𝑞 − 𝑜))
108	103, 106, 107	rsubrotld 42755	. . . . 5 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑞 = (𝑜 + 𝑝))
109	101, 108	rspcedeq2vd 3568	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → ∃𝑝 ∈ ℕ 𝑞 = (𝑜 + 𝑝))
110		oveq1 7363	. . . . . 6 ⊢ (𝑞 = (𝑜 + 𝑝) → (𝑞 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))
111	110	eqeq2d 2750	. . . . 5 ⊢ (𝑞 = (𝑜 + 𝑝) → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
112	111	adantl 482	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 = (𝑜 + 𝑝)) → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
113	89, 109, 112	rexxfrd 5338	. . 3 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
114	113	rexbidva 3161	. 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 233	1 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063   class class class wbr 5072   ↦ cmpt 5153  ⟶wf 6481  –1-1→wf1 6482  ‘cfv 6485  (class class class)co 7356   ≼ cdom 8881   ≺ csdm 8882  Fincfn 8883  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   < clt 11170   ≤ cle 11171   − cmin 11368  ℕcn 12165  ℤcz 12515  ℤ_≥cuz 12779  Basecbs 17170  Mgmcmgm 18597  ._gcmg 19034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-seq 13955  df-hash 14284  df-mgm 18599  df-mulg 19035

This theorem is referenced by:  fidomncyc  43021