Theorem fimgmcyc 42489

Description: Version of odcl2 19607 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 9165	. . . . . . . . 9 ⊢ (ℕ ≼ 𝐵 → ¬ 𝐵 ≺ ℕ)
2		fimgmcyc.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		fisdomnn 42239	. . . . . . . . . 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 6934	. . . . . . . . 9 ⊢ 𝐵 ∈ V
8	7	f1dom 9034	. . . . . . . 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 19129	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mgm ∧ 𝑛 ∈ ℕ ∧ 𝐴 ∈ 𝐵) → (𝑛 · 𝐴) ∈ 𝐵)
17	11, 12, 14, 16	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 · 𝐴) ∈ 𝐵)
18	17	fmpttd 7149	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ⟶𝐵)
19		dff13 7292	. . . . . . . . 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 7455	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑜 → (𝑛 · 𝐴) = (𝑜 · 𝐴))
24		eqid 2740	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)) = (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))
25		ovex 7481	. . . . . . . . . . 11 ⊢ (𝑜 · 𝐴) ∈ V
26	23, 24, 25	fvmpt 7029	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = (𝑜 · 𝐴))
27		oveq1 7455	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑞 → (𝑛 · 𝐴) = (𝑞 · 𝐴))
28		ovex 7481	. . . . . . . . . . 11 ⊢ (𝑞 · 𝐴) ∈ V
29	27, 24, 28	fvmpt 7029	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) = (𝑞 · 𝐴))
30	26, 29	eqeqan12d 2754	. . . . . . . . 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 3182	. . . . . . 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 3097	. . . . . 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 2947	. . . . . . . . 9 ⊢ (𝑜 ≠ 𝑞 ↔ ¬ 𝑜 = 𝑞)
36	35	anbi1i 623	. . . . . . . 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 3135	. . . . . 6 ⊢ (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
41		rexnal2 3141	. . . . . 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 42488	. . 3 ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
45		nnz 12660	. . . . . . . . . 10 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℤ)
46		eluzp1 42295	. . . . . . . . . 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 12660	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℤ)
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℕ → 𝑞 ∈ ℤ))
51		0red 11293	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 ∈ ℝ)
52		nnre 12300	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℝ)
53	52	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑜 ∈ ℝ)
54		zre 12643	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ ℤ → 𝑞 ∈ ℝ)
55	54	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑞 ∈ ℝ)
56		nngt0 12324	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ ℕ → 0 < 𝑜)
57	56	ad2antrr 725	. . . . . . . . . . . . . . 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 11451	. . . . . . . . . . . . . 14 ⊢ (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 < 𝑞)
60		elnnz 12649	. . . . . . . . . . . . . . 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 577	. . . . . . . . 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 630	. . . . . . 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 1920	. . . . 5 ⊢ (𝑜 ∈ ℕ → (∃𝑞(𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑞(𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
72		df-rex 3077	. . . . 5 ⊢ (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑞(𝑞 ∈ (ℤ≥‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
73		df-rex 3077	. . . . 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 3098	. . 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 12310	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
79	78	nnzd 12666	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℤ)
80		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℕ)
81	77, 80	nnaddcld 12345	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℕ)
82	81	nnzd 12666	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℤ)
83		1red 11291	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ∈ ℝ)
84	80	nnred 12308	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℝ)
85	77	nnred 12308	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑜 ∈ ℝ)
86	80	nnge1d 12341	. . . . . 6 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ≤ 𝑝)
87	83, 84, 85, 86	leadd2dd 11905	. . . . 5 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ≤ (𝑜 + 𝑝))
88		eluz2 12909	. . . . 5 ⊢ ((𝑜 + 𝑝) ∈ (ℤ≥‘(𝑜 + 1)) ↔ ((𝑜 + 1) ∈ ℤ ∧ (𝑜 + 𝑝) ∈ ℤ ∧ (𝑜 + 1) ≤ (𝑜 + 𝑝)))
89	79, 82, 87, 88	syl3anbrc 1343	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ (ℤ≥‘(𝑜 + 1)))
90		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℕ)
91	90	nnzd 12666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℤ)
92		eluzp1l 12930	. . . . . . 7 ⊢ ((𝑜 ∈ ℤ ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑜 < 𝑞)
93	91, 92	sylan 579	. . . . . 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 12305	. . . . . . . . 9 ⊢ (𝑜 ∈ ℕ → (𝑜 + 1) ∈ ℕ)
96	95	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
97		eluznn 12983	. . . . . . . 8 ⊢ (((𝑜 + 1) ∈ ℕ ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
98	96, 97	sylan 579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
99		nnsub 12337	. . . . . . 7 ⊢ ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝑜 < 𝑞 ↔ (𝑞 − 𝑜) ∈ ℕ))
100	94, 98, 99	syl2anc 583	. . . . . 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 12915	. . . . . . 7 ⊢ (𝑞 ∈ (ℤ≥‘(𝑜 + 1)) → 𝑞 ∈ ℂ)
103	102	ad2antlr 726	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑞 ∈ ℂ)
104		nncn 12301	. . . . . . . 8 ⊢ (𝑜 ∈ ℕ → 𝑜 ∈ ℂ)
105	104	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → 𝑜 ∈ ℂ)
106	105	ad2antrr 725	. . . . . 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 42267	. . . . 5 ⊢ ((((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) ∧ 𝑝 = (𝑞 − 𝑜)) → 𝑞 = (𝑜 + 𝑝))
109	101, 108	rspcedeq2vd 3643	. . . 4 ⊢ (((𝜑 ∧ 𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ≥‘(𝑜 + 1))) → ∃𝑝 ∈ ℕ 𝑞 = (𝑜 + 𝑝))
110		oveq1 7455	. . . . . 6 ⊢ (𝑞 = (𝑜 + 𝑝) → (𝑞 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))
111	110	eqeq2d 2751	. . . . 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 5427	. . 3 ⊢ ((𝜑 ∧ 𝑜 ∈ ℕ) → (∃𝑞 ∈ (ℤ≥‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
114	113	rexbidva 3183	. 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 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  –1-1→wf1 6570  ‘cfv 6573  (class class class)co 7448   ≼ cdom 9001   ≺ csdm 9002  Fincfn 9003  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325   − cmin 11520  ℕcn 12293  ℤcz 12639  ℤ_≥cuz 12903  Basecbs 17258  Mgmcmgm 18676  ._gcmg 19107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-seq 14053  df-hash 14380  df-mgm 18678  df-mulg 19108

This theorem is referenced by:  fidomncyc  42490