Theorem fimgmcyc 42731

Description: Version of odcl2 19492 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 9029	. . . . . . . . 9 ⊢ (ℕ ≼ 𝐵 → ¬ 𝐵 ≺ ℕ)
2		fimgmcyc.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		fisdomnn 42441	. . . . . . . . . 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 6846	. . . . . . . . 9 ⊢ 𝐵 ∈ V
8	7	f1dom 8908	. . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ Mgm)
12		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
13		fimgmcyc.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
14	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ 𝐵)
15		fimgmcyc.m	. . . . . . . . . . 11 ⊢ · = (.g‘𝑀)
16	6, 15	mulgnncl 19017	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mgm ∧ 𝑛 ∈ ℕ ∧ 𝐴 ∈ 𝐵) → (𝑛 · 𝐴) ∈ 𝐵)
17	11, 12, 14, 16	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 · 𝐴) ∈ 𝐵)
18	17	fmpttd 7058	. . . . . . . 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 535	. . . . . . . 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 324	. . . . . 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 2734	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)) = (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))
25		ovex 7389	. . . . . . . . . . 11 ⊢ (𝑜 · 𝐴) ∈ V
26	23, 24, 25	fvmpt 6939	. . . . . . . . . 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 6939	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) = (𝑞 · 𝐴))
30	26, 29	eqeqan12d 2748	. . . . . . . . 9 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) ↔ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
31	30	imbi1d 341	. . . . . . . 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 3155	. . . . . . 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 3078	. . . . . 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 328	. . . . 5 ⊢ (𝜑 → ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
35		df-ne 2931	. . . . . . . . 9 ⊢ (𝑜 ≠ 𝑞 ↔ ¬ 𝑜 = 𝑞)
36	35	anbi1i 624	. . . . . . . 8 ⊢ ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (¬ 𝑜 = 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
37		ancom 460	. . . . . . . 8 ⊢ ((¬ 𝑜 = 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑜 · 𝐴) = (𝑞 · 𝐴) ∧ ¬ 𝑜 = 𝑞))
38		annim 403	. . . . . . . 8 ⊢ (((𝑜 · 𝐴) = (𝑞 · 𝐴) ∧ ¬ 𝑜 = 𝑞) ↔ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
39	36, 37, 38	3bitri 297	. . . . . . 7 ⊢ ((𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
40	39	2rexbii 3110	. . . . . 6 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
41		rexnal2 3116	. . . . . 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 275	. . . . 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 234	. . . 4 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
44	43	fimgmcyclem 42730	. . 3 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
45		nnz 12507	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℤ)
46		eluzp1 42504	. . . . . . . . . 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 12507	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℤ)
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℕ → 𝑞 ∈ ℤ))
51		0red 11133	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 ∈ ℝ)
52		nnre 12150	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℝ)
53	52	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑜 ∈ ℝ)
54		zre 12490	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ ℤ → 𝑞 ∈ ℝ)
55	54	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑞 ∈ ℝ)
56		nngt0 12174	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ ℕ → 0 < 𝑜)
57	56	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 < 𝑜)
58		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑜 < 𝑞)
59	51, 53, 55, 57, 58	lttrd 11292	. . . . . . . . . . . . . 14 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 < 𝑞)
60		elnnz 12496	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ ℕ ↔ (𝑞 ∈ ℤ ∧ 0 < 𝑞))
61	60	rbaibr 537	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ)))
64	48, 50, 63	pm5.21ndd 379	. . . . . . . . . . 11 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
65	64	ex 412	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → (𝑜 < 𝑞 → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ)))
66	65	pm5.32rd 578	. . . . . . . . 9 ⊢ (𝑜 ∈ ℕ → ((𝑞 ∈ ℤ ∧ 𝑜 < 𝑞) ↔ (𝑞 ∈ ℕ ∧ 𝑜 < 𝑞)))
67	47, 66	bitrd 279	. . . . . . . 8 ⊢ (𝑜 ∈ ℕ → (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ↔ (𝑞 ∈ ℕ ∧ 𝑜 < 𝑞)))
68	67	anbi1d 631	. . . . . . 7 ⊢ (𝑜 ∈ ℕ → ((𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ((𝑞 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
69		anass 468	. . . . . . 7 ⊢ (((𝑞 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
70	68, 69	bitrdi 287	. . . . . 6 ⊢ (𝑜 ∈ ℕ → ((𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
71	70	exbidv 1922	. . . . 5 ⊢ (𝑜 ∈ ℕ → (∃𝑞(𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑞(𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
72		df-rex 3059	. . . . 5 ⊢ (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑞(𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
73		df-rex 3059	. . . . 5 ⊢ (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑞(𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
74	71, 72, 73	3bitr4g 314	. . . 4 ⊢ (𝑜 ∈ ℕ → (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
75	74	rexbiia 3079	. . 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 234	. 2 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴))
77		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑜 ∈ ℕ)
78	77	peano2nnd 12160	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
79	78	nnzd 12512	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℤ)
80		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℕ)
81	77, 80	nnaddcld 12195	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℕ)
82	81	nnzd 12512	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℤ)
83		1red 11131	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ∈ ℝ)
84	80	nnred 12158	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℝ)
85	77	nnred 12158	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑜 ∈ ℝ)
86	80	nnge1d 12191	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ≤ 𝑝)
87	83, 84, 85, 86	leadd2dd 11750	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ≤ (𝑜 + 𝑝))
88		eluz2 12755	. . . . 5 ⊢ ((𝑜 + 𝑝) ∈ (ℤ≥‘(𝑜 + 1)) ↔ ((𝑜 + 1) ∈ ℤ ∧ (𝑜 + 𝑝) ∈ ℤ ∧ (𝑜 + 1) ≤ (𝑜 + 𝑝)))
89	79, 82, 87, 88	syl3anbrc 1344	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ (ℤ≥‘(𝑜 + 1)))
90		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℕ)
91	90	nnzd 12512	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℤ)
92		eluzp1l 12776	. . . . . . 7 ⊢ ((𝑜 ∈ ℤ ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 < 𝑞)
93	91, 92	sylan 580	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 < 𝑞)
94		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 ∈ ℕ)
95		peano2nn 12155	. . . . . . . . 9 ⊢ (𝑜 ∈ ℕ → (𝑜 + 1) ∈ ℕ)
96	95	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
97		eluznn 12829	. . . . . . . 8 ⊢ (((𝑜 + 1) ∈ ℕ ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
98	96, 97	sylan 580	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
99		nnsub 12187	. . . . . . 7 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝑜 < 𝑞 ↔ (𝑞 − 𝑜) ∈ ℕ))
100	94, 98, 99	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → (𝑜 < 𝑞 ↔ (𝑞 − 𝑜) ∈ ℕ))
101	93, 100	mpbid 232	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → (𝑞 − 𝑜) ∈ ℕ)
102		eluzelcn 12761	. . . . . . 7 ⊢ (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) → 𝑞 ∈ ℂ)
103	102	ad2antlr 727	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑞 ∈ ℂ)
104		nncn 12151	. . . . . . . 8 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℂ)
105	104	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℂ)
106	105	ad2antrr 726	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑜 ∈ ℂ)
107		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑝 = (𝑞 − 𝑜))
108	103, 106, 107	rsubrotld 42475	. . . . 5 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑞 = (𝑜 + 𝑝))
109	101, 108	rspcedeq2vd 3582	. . . 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 2745	. . . . 5 ⊢ (𝑞 = (𝑜 + 𝑝) → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
112	111	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 = (𝑜 + 𝑝)) → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
113	89, 109, 112	rexxfrd 5352	. . 3 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
114	113	rexbidva 3156	. 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 232	1 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058   class class class wbr 5096   ↦ cmpt 5177  ⟶wf 6486  –1-1→wf1 6487  ‘cfv 6490  (class class class)co 7356   ≼ cdom 8879   ≺ csdm 8880  Fincfn 8881  ℂcc 11022  ℝcr 11023  0cc0 11024  1c1 11025   + caddc 11027   < clt 11164   ≤ cle 11165   − cmin 11362  ℕcn 12143  ℤcz 12486  ℤ_≥cuz 12749  Basecbs 17134  Mgmcmgm 18561  ._gcmg 18995

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  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 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-seq 13923  df-hash 14252  df-mgm 18563  df-mulg 18996

This theorem is referenced by:  fidomncyc  42732