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Theorem fimgmcyc 42520
Description: Version of odcl2 19597 for finite magmas: the multiples of an element 𝐴𝐵 are eventually periodic. (Contributed by SN, 3-Jul-2025.)
Hypotheses
Ref Expression
fimgmcyc.b 𝐵 = (Base‘𝑀)
fimgmcyc.m · = (.g𝑀)
fimgmcyc.s (𝜑𝑀 ∈ Mgm)
fimgmcyc.f (𝜑𝐵 ∈ Fin)
fimgmcyc.a (𝜑𝐴𝐵)
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.
StepHypRef Expression
1 domnsym 9137 . . . . . . . . 9 (ℕ ≼ 𝐵 → ¬ 𝐵 ≺ ℕ)
2 fimgmcyc.f . . . . . . . . . 10 (𝜑𝐵 ∈ Fin)
3 fisdomnn 42263 . . . . . . . . . 10 (𝐵 ∈ Fin → 𝐵 ≺ ℕ)
42, 3syl 17 . . . . . . . . 9 (𝜑𝐵 ≺ ℕ)
51, 4nsyl3 138 . . . . . . . 8 (𝜑 → ¬ ℕ ≼ 𝐵)
6 fimgmcyc.b . . . . . . . . . 10 𝐵 = (Base‘𝑀)
76fvexi 6920 . . . . . . . . 9 𝐵 ∈ V
87f1dom 9012 . . . . . . . 8 ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1𝐵 → ℕ ≼ 𝐵)
95, 8nsyl 140 . . . . . . 7 (𝜑 → ¬ (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1𝐵)
10 fimgmcyc.s . . . . . . . . . . 11 (𝜑𝑀 ∈ Mgm)
1110adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ Mgm)
12 simpr 484 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
13 fimgmcyc.a . . . . . . . . . . 11 (𝜑𝐴𝐵)
1413adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐴𝐵)
15 fimgmcyc.m . . . . . . . . . . 11 · = (.g𝑀)
166, 15mulgnncl 19119 . . . . . . . . . 10 ((𝑀 ∈ Mgm ∧ 𝑛 ∈ ℕ ∧ 𝐴𝐵) → (𝑛 · 𝐴) ∈ 𝐵)
1711, 12, 14, 16syl3anc 1370 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑛 · 𝐴) ∈ 𝐵)
1817fmpttd 7134 . . . . . . . 8 (𝜑 → (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ⟶𝐵)
19 dff13 7274 . . . . . . . . 9 ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1𝐵 ↔ ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ⟶𝐵 ∧ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞)))
2019baib 535 . . . . . . . 8 ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ⟶𝐵 → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1𝐵 ↔ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞)))
2118, 20syl 17 . . . . . . 7 (𝜑 → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)):ℕ–1-1𝐵 ↔ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞)))
229, 21mtbid 324 . . . . . 6 (𝜑 → ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞))
23 oveq1 7437 . . . . . . . . . . 11 (𝑛 = 𝑜 → (𝑛 · 𝐴) = (𝑜 · 𝐴))
24 eqid 2734 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴)) = (𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))
25 ovex 7463 . . . . . . . . . . 11 (𝑜 · 𝐴) ∈ V
2623, 24, 25fvmpt 7015 . . . . . . . . . 10 (𝑜 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = (𝑜 · 𝐴))
27 oveq1 7437 . . . . . . . . . . 11 (𝑛 = 𝑞 → (𝑛 · 𝐴) = (𝑞 · 𝐴))
28 ovex 7463 . . . . . . . . . . 11 (𝑞 · 𝐴) ∈ V
2927, 24, 28fvmpt 7015 . . . . . . . . . 10 (𝑞 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) = (𝑞 · 𝐴))
3026, 29eqeqan12d 2748 . . . . . . . . 9 ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) ↔ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
3130imbi1d 341 . . . . . . . 8 ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → ((((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑜) = ((𝑛 ∈ ℕ ↦ (𝑛 · 𝐴))‘𝑞) → 𝑜 = 𝑞) ↔ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞)))
3231ralbidva 3173 . . . . . . 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)))
3332ralbiia 3088 . . . . . 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))
3422, 33sylnib 328 . . . . 5 (𝜑 → ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
35 df-ne 2938 . . . . . . . . 9 (𝑜𝑞 ↔ ¬ 𝑜 = 𝑞)
3635anbi1i 624 . . . . . . . 8 ((𝑜𝑞 ∧ (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))
3936, 37, 383bitri 297 . . . . . . 7 ((𝑜𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
40392rexbii 3126 . . . . . 6 (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
41 rexnal2 3132 . . . . . 6 (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ ¬ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞) ↔ ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
4240, 41bitri 275 . . . . 5 (∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ¬ ∀𝑜 ∈ ℕ ∀𝑞 ∈ ℕ ((𝑜 · 𝐴) = (𝑞 · 𝐴) → 𝑜 = 𝑞))
4334, 42sylibr 234 . . . 4 (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
4443fimgmcyclem 42519 . . 3 (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
45 nnz 12631 . . . . . . . . . 10 (𝑜 ∈ ℕ → 𝑜 ∈ ℤ)
46 eluzp1 42319 . . . . . . . . . 10 (𝑜 ∈ ℤ → (𝑞 ∈ (ℤ‘(𝑜 + 1)) ↔ (𝑞 ∈ ℤ ∧ 𝑜 < 𝑞)))
4745, 46syl 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 12631 . . . . . . . . . . . . 13 (𝑞 ∈ ℕ → 𝑞 ∈ ℤ)
5049a1i 11 . . . . . . . . . . . 12 ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℕ → 𝑞 ∈ ℤ))
51 0red 11261 . . . . . . . . . . . . . . 15 (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 ∈ ℝ)
52 nnre 12270 . . . . . . . . . . . . . . . 16 (𝑜 ∈ ℕ → 𝑜 ∈ ℝ)
5352ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑜 ∈ ℝ)
54 zre 12614 . . . . . . . . . . . . . . . 16 (𝑞 ∈ ℤ → 𝑞 ∈ ℝ)
5554adantl 481 . . . . . . . . . . . . . . 15 (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑞 ∈ ℝ)
56 nngt0 12294 . . . . . . . . . . . . . . . 16 (𝑜 ∈ ℕ → 0 < 𝑜)
5756ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 < 𝑜)
58 simplr 769 . . . . . . . . . . . . . . 15 (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 𝑜 < 𝑞)
5951, 53, 55, 57, 58lttrd 11419 . . . . . . . . . . . . . 14 (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → 0 < 𝑞)
60 elnnz 12620 . . . . . . . . . . . . . . 15 (𝑞 ∈ ℕ ↔ (𝑞 ∈ ℤ ∧ 0 < 𝑞))
6160rbaibr 537 . . . . . . . . . . . . . 14 (0 < 𝑞 → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
6259, 61syl 17 . . . . . . . . . . . . 13 (((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) ∧ 𝑞 ∈ ℤ) → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
6362ex 412 . . . . . . . . . . . 12 ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ)))
6448, 50, 63pm5.21ndd 379 . . . . . . . . . . 11 ((𝑜 ∈ ℕ ∧ 𝑜 < 𝑞) → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ))
6564ex 412 . . . . . . . . . 10 (𝑜 ∈ ℕ → (𝑜 < 𝑞 → (𝑞 ∈ ℤ ↔ 𝑞 ∈ ℕ)))
6665pm5.32rd 578 . . . . . . . . 9 (𝑜 ∈ ℕ → ((𝑞 ∈ ℤ ∧ 𝑜 < 𝑞) ↔ (𝑞 ∈ ℕ ∧ 𝑜 < 𝑞)))
6747, 66bitrd 279 . . . . . . . 8 (𝑜 ∈ ℕ → (𝑞 ∈ (ℤ‘(𝑜 + 1)) ↔ (𝑞 ∈ ℕ ∧ 𝑜 < 𝑞)))
6867anbi1d 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))))
7068, 69bitrdi 287 . . . . . 6 (𝑜 ∈ ℕ → ((𝑞 ∈ (ℤ‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ (𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
7170exbidv 1918 . . . . 5 (𝑜 ∈ ℕ → (∃𝑞(𝑞 ∈ (ℤ‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑞(𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))))
72 df-rex 3068 . . . . 5 (∃𝑞 ∈ (ℤ‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑞(𝑞 ∈ (ℤ‘(𝑜 + 1)) ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
73 df-rex 3068 . . . . 5 (∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)) ↔ ∃𝑞(𝑞 ∈ ℕ ∧ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
7471, 72, 733bitr4g 314 . . . 4 (𝑜 ∈ ℕ → (∃𝑞 ∈ (ℤ‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴))))
7574rexbiia 3089 . . 3 (∃𝑜 ∈ ℕ ∃𝑞 ∈ (ℤ‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
7644, 75sylibr 234 . 2 (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ (ℤ‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴))
77 simplr 769 . . . . . . 7 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑜 ∈ ℕ)
7877peano2nnd 12280 . . . . . 6 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
7978nnzd 12637 . . . . 5 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ∈ ℤ)
80 simpr 484 . . . . . . 7 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℕ)
8177, 80nnaddcld 12315 . . . . . 6 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℕ)
8281nnzd 12637 . . . . 5 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ ℤ)
83 1red 11259 . . . . . 6 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ∈ ℝ)
8480nnred 12278 . . . . . 6 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑝 ∈ ℝ)
8577nnred 12278 . . . . . 6 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 𝑜 ∈ ℝ)
8680nnge1d 12311 . . . . . 6 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → 1 ≤ 𝑝)
8783, 84, 85, 86leadd2dd 11875 . . . . 5 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 1) ≤ (𝑜 + 𝑝))
88 eluz2 12881 . . . . 5 ((𝑜 + 𝑝) ∈ (ℤ‘(𝑜 + 1)) ↔ ((𝑜 + 1) ∈ ℤ ∧ (𝑜 + 𝑝) ∈ ℤ ∧ (𝑜 + 1) ≤ (𝑜 + 𝑝)))
8979, 82, 87, 88syl3anbrc 1342 . . . 4 (((𝜑𝑜 ∈ ℕ) ∧ 𝑝 ∈ ℕ) → (𝑜 + 𝑝) ∈ (ℤ‘(𝑜 + 1)))
90 simpr 484 . . . . . . . 8 ((𝜑𝑜 ∈ ℕ) → 𝑜 ∈ ℕ)
9190nnzd 12637 . . . . . . 7 ((𝜑𝑜 ∈ ℕ) → 𝑜 ∈ ℤ)
92 eluzp1l 12902 . . . . . . 7 ((𝑜 ∈ ℤ ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) → 𝑜 < 𝑞)
9391, 92sylan 580 . . . . . 6 (((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) → 𝑜 < 𝑞)
94 simplr 769 . . . . . . 7 (((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) → 𝑜 ∈ ℕ)
95 peano2nn 12275 . . . . . . . . 9 (𝑜 ∈ ℕ → (𝑜 + 1) ∈ ℕ)
9695adantl 481 . . . . . . . 8 ((𝜑𝑜 ∈ ℕ) → (𝑜 + 1) ∈ ℕ)
97 eluznn 12957 . . . . . . . 8 (((𝑜 + 1) ∈ ℕ ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
9896, 97sylan 580 . . . . . . 7 (((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) → 𝑞 ∈ ℕ)
99 nnsub 12307 . . . . . . 7 ((𝑜 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝑜 < 𝑞 ↔ (𝑞𝑜) ∈ ℕ))
10094, 98, 99syl2anc 584 . . . . . 6 (((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) → (𝑜 < 𝑞 ↔ (𝑞𝑜) ∈ ℕ))
10193, 100mpbid 232 . . . . 5 (((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) → (𝑞𝑜) ∈ ℕ)
102 eluzelcn 12887 . . . . . . 7 (𝑞 ∈ (ℤ‘(𝑜 + 1)) → 𝑞 ∈ ℂ)
103102ad2antlr 727 . . . . . 6 ((((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) ∧ 𝑝 = (𝑞𝑜)) → 𝑞 ∈ ℂ)
104 nncn 12271 . . . . . . . 8 (𝑜 ∈ ℕ → 𝑜 ∈ ℂ)
105104adantl 481 . . . . . . 7 ((𝜑𝑜 ∈ ℕ) → 𝑜 ∈ ℂ)
106105ad2antrr 726 . . . . . 6 ((((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) ∧ 𝑝 = (𝑞𝑜)) → 𝑜 ∈ ℂ)
107 simpr 484 . . . . . 6 ((((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) ∧ 𝑝 = (𝑞𝑜)) → 𝑝 = (𝑞𝑜))
108103, 106, 107rsubrotld 42291 . . . . 5 ((((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) ∧ 𝑝 = (𝑞𝑜)) → 𝑞 = (𝑜 + 𝑝))
109101, 108rspcedeq2vd 3629 . . . 4 (((𝜑𝑜 ∈ ℕ) ∧ 𝑞 ∈ (ℤ‘(𝑜 + 1))) → ∃𝑝 ∈ ℕ 𝑞 = (𝑜 + 𝑝))
110 oveq1 7437 . . . . . 6 (𝑞 = (𝑜 + 𝑝) → (𝑞 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))
111110eqeq2d 2745 . . . . 5 (𝑞 = (𝑜 + 𝑝) → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
112111adantl 481 . . . 4 (((𝜑𝑜 ∈ ℕ) ∧ 𝑞 = (𝑜 + 𝑝)) → ((𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
11389, 109, 112rexxfrd 5414 . . 3 ((𝜑𝑜 ∈ ℕ) → (∃𝑞 ∈ (ℤ‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
114113rexbidva 3174 . 2 (𝜑 → (∃𝑜 ∈ ℕ ∃𝑞 ∈ (ℤ‘(𝑜 + 1))(𝑜 · 𝐴) = (𝑞 · 𝐴) ↔ ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴)))
11576, 114mpbid 232 1 (𝜑 → ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067   class class class wbr 5147  cmpt 5230  wf 6558  1-1wf1 6559  cfv 6562  (class class class)co 7430  cdom 8981  csdm 8982  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   < clt 11292  cle 11293  cmin 11489  cn 12263  cz 12610  cuz 12875  Basecbs 17244  Mgmcmgm 18663  .gcmg 19097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-seq 14039  df-hash 14366  df-mgm 18665  df-mulg 19098
This theorem is referenced by:  fidomncyc  42521
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