MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isercoll Structured version   Visualization version   GIF version

Theorem isercoll 15017
Description: Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypotheses
Ref Expression
isercoll.z 𝑍 = (ℤ𝑀)
isercoll.m (𝜑𝑀 ∈ ℤ)
isercoll.g (𝜑𝐺:ℕ⟶𝑍)
isercoll.i ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))
isercoll.0 ((𝜑𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹𝑛) = 0)
isercoll.f ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ ℂ)
isercoll.h ((𝜑𝑘 ∈ ℕ) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))
Assertion
Ref Expression
isercoll (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑘,𝐹,𝑛   𝜑,𝑘,𝑛   𝑘,𝐺,𝑛   𝑘,𝐻,𝑛   𝑘,𝑀,𝑛   𝑛,𝑍
Allowed substitution hint:   𝑍(𝑘)

Proof of Theorem isercoll
Dummy variables 𝑗 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isercoll.z . . . . . . . . . 10 𝑍 = (ℤ𝑀)
2 uzssz 12256 . . . . . . . . . 10 (ℤ𝑀) ⊆ ℤ
31, 2eqsstri 4004 . . . . . . . . 9 𝑍 ⊆ ℤ
4 isercoll.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶𝑍)
54ffvelrnda 6846 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ 𝑍)
63, 5sseldi 3968 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℤ)
7 nnz 11996 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
87ad2antlr 723 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ∈ ℤ)
9 fzfid 13334 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝑀...𝑚) ∈ Fin)
10 ffun 6513 . . . . . . . . . . . . . . . 16 (𝐺:ℕ⟶𝑍 → Fun 𝐺)
11 funimacnv 6431 . . . . . . . . . . . . . . . 16 (Fun 𝐺 → (𝐺 “ (𝐺 “ (𝑀...𝑚))) = ((𝑀...𝑚) ∩ ran 𝐺))
124, 10, 113syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑚))) = ((𝑀...𝑚) ∩ ran 𝐺))
13 inss1 4208 . . . . . . . . . . . . . . 15 ((𝑀...𝑚) ∩ ran 𝐺) ⊆ (𝑀...𝑚)
1412, 13eqsstrdi 4024 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚))
1514ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚))
169, 15ssfid 8733 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin)
17 hashcl 13710 . . . . . . . . . . . 12 ((𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℕ0)
18 nn0z 11997 . . . . . . . . . . . 12 ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℕ0 → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℤ)
1916, 17, 183syl 18 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℤ)
20 ssid 3992 . . . . . . . . . . . . . . . . . . . 20 ℕ ⊆ ℕ
21 isercoll.m . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ ℤ)
22 isercoll.i . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))
231, 21, 4, 22isercolllem1 15014 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ℕ ⊆ ℕ) → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
2420, 23mpan2 687 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
25 ffn 6510 . . . . . . . . . . . . . . . . . . . 20 (𝐺:ℕ⟶𝑍𝐺 Fn ℕ)
26 fnresdm 6462 . . . . . . . . . . . . . . . . . . . 20 (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺)
27 isoeq1 7065 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ↾ ℕ) = 𝐺 → ((𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)) ↔ 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ))))
284, 25, 26, 274syl 19 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)) ↔ 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ))))
2924, 28mpbid 233 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
30 isof1o 7071 . . . . . . . . . . . . . . . . . 18 (𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) → 𝐺:ℕ–1-1-onto→(𝐺 “ ℕ))
31 f1ocnv 6623 . . . . . . . . . . . . . . . . . 18 (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → 𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ)
32 f1ofun 6613 . . . . . . . . . . . . . . . . . 18 (𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun 𝐺)
3329, 30, 31, 324syl 19 . . . . . . . . . . . . . . . . 17 (𝜑 → Fun 𝐺)
34 df-f1 6356 . . . . . . . . . . . . . . . . 17 (𝐺:ℕ–1-1𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun 𝐺))
354, 33, 34sylanbrc 583 . . . . . . . . . . . . . . . 16 (𝜑𝐺:ℕ–1-1𝑍)
3635ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝐺:ℕ–1-1𝑍)
37 fz1ssnn 12931 . . . . . . . . . . . . . . 15 (1...𝑛) ⊆ ℕ
38 ovex 7184 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ V
3938f1imaen 8563 . . . . . . . . . . . . . . 15 ((𝐺:ℕ–1-1𝑍 ∧ (1...𝑛) ⊆ ℕ) → (𝐺 “ (1...𝑛)) ≈ (1...𝑛))
4036, 37, 39sylancl 586 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ≈ (1...𝑛))
41 fzfid 13334 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (1...𝑛) ∈ Fin)
42 enfii 8727 . . . . . . . . . . . . . . . 16 (((1...𝑛) ∈ Fin ∧ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)) → (𝐺 “ (1...𝑛)) ∈ Fin)
4341, 40, 42syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ∈ Fin)
44 hashen 13700 . . . . . . . . . . . . . . 15 (((𝐺 “ (1...𝑛)) ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((♯‘(𝐺 “ (1...𝑛))) = (♯‘(1...𝑛)) ↔ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)))
4543, 41, 44syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → ((♯‘(𝐺 “ (1...𝑛))) = (♯‘(1...𝑛)) ↔ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)))
4640, 45mpbird 258 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (1...𝑛))) = (♯‘(1...𝑛)))
47 nnnn0 11896 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
4847ad2antlr 723 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ∈ ℕ0)
49 hashfz1 13699 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
5048, 49syl 17 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(1...𝑛)) = 𝑛)
5146, 50eqtrd 2860 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (1...𝑛))) = 𝑛)
52 elfznn 12929 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1...𝑛) → 𝑦 ∈ ℕ)
5352adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℕ)
54 zssre 11980 . . . . . . . . . . . . . . . . . . . . . 22 ℤ ⊆ ℝ
553, 54sstri 3979 . . . . . . . . . . . . . . . . . . . . 21 𝑍 ⊆ ℝ
564ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝐺:ℕ⟶𝑍)
57 ffvelrn 6844 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺:ℕ⟶𝑍𝑦 ∈ ℕ) → (𝐺𝑦) ∈ 𝑍)
5856, 52, 57syl2an 595 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ 𝑍)
5955, 58sseldi 3968 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ ℝ)
605ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ∈ 𝑍)
6155, 60sseldi 3968 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ∈ ℝ)
62 eluzelz 12245 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (ℤ‘(𝐺𝑛)) → 𝑚 ∈ ℤ)
6362ad2antlr 723 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℤ)
6463zred 12079 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℝ)
65 elfzle2 12904 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (1...𝑛) → 𝑦𝑛)
6665adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦𝑛)
6729ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
68 simpllr 772 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑛 ∈ ℕ)
69 isorel 7074 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑛 < 𝑦 ↔ (𝐺𝑛) < (𝐺𝑦)))
7067, 68, 53, 69syl12anc 834 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑛 < 𝑦 ↔ (𝐺𝑛) < (𝐺𝑦)))
7170notbid 319 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (¬ 𝑛 < 𝑦 ↔ ¬ (𝐺𝑛) < (𝐺𝑦)))
7253nnred 11645 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℝ)
7368nnred 11645 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑛 ∈ ℝ)
7472, 73lenltd 10778 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦𝑛 ↔ ¬ 𝑛 < 𝑦))
7559, 61lenltd 10778 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → ((𝐺𝑦) ≤ (𝐺𝑛) ↔ ¬ (𝐺𝑛) < (𝐺𝑦)))
7671, 74, 753bitr4d 312 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦𝑛 ↔ (𝐺𝑦) ≤ (𝐺𝑛)))
7766, 76mpbid 233 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ≤ (𝐺𝑛))
78 eluzle 12248 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (ℤ‘(𝐺𝑛)) → (𝐺𝑛) ≤ 𝑚)
7978ad2antlr 723 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ≤ 𝑚)
8059, 61, 64, 77, 79letrd 10789 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ≤ 𝑚)
8158, 1syl6eleq 2927 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ (ℤ𝑀))
82 elfz5 12893 . . . . . . . . . . . . . . . . . . . 20 (((𝐺𝑦) ∈ (ℤ𝑀) ∧ 𝑚 ∈ ℤ) → ((𝐺𝑦) ∈ (𝑀...𝑚) ↔ (𝐺𝑦) ≤ 𝑚))
8381, 63, 82syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → ((𝐺𝑦) ∈ (𝑀...𝑚) ↔ (𝐺𝑦) ≤ 𝑚))
8480, 83mpbird 258 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ (𝑀...𝑚))
8556ffnd 6511 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝐺 Fn ℕ)
8685adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝐺 Fn ℕ)
87 elpreima 6823 . . . . . . . . . . . . . . . . . . 19 (𝐺 Fn ℕ → (𝑦 ∈ (𝐺 “ (𝑀...𝑚)) ↔ (𝑦 ∈ ℕ ∧ (𝐺𝑦) ∈ (𝑀...𝑚))))
8886, 87syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦 ∈ (𝐺 “ (𝑀...𝑚)) ↔ (𝑦 ∈ ℕ ∧ (𝐺𝑦) ∈ (𝑀...𝑚))))
8953, 84, 88mpbir2and 709 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ (𝐺 “ (𝑀...𝑚)))
9089ex 413 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝑦 ∈ (1...𝑛) → 𝑦 ∈ (𝐺 “ (𝑀...𝑚))))
9190ssrdv 3976 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (1...𝑛) ⊆ (𝐺 “ (𝑀...𝑚)))
92 imass2 5962 . . . . . . . . . . . . . . 15 ((1...𝑛) ⊆ (𝐺 “ (𝑀...𝑚)) → (𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑚))))
9391, 92syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑚))))
94 ssdomg 8547 . . . . . . . . . . . . . 14 ((𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin → ((𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑚))) → (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚)))))
9516, 93, 94sylc 65 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚))))
96 hashdom 13733 . . . . . . . . . . . . . 14 (((𝐺 “ (1...𝑛)) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin) → ((♯‘(𝐺 “ (1...𝑛))) ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ↔ (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚)))))
9743, 16, 96syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → ((♯‘(𝐺 “ (1...𝑛))) ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ↔ (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚)))))
9895, 97mpbird 258 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (1...𝑛))) ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))))
9951, 98eqbrtrrd 5086 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))))
100 eluz2 12241 . . . . . . . . . . 11 ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ (ℤ𝑛) ↔ (𝑛 ∈ ℤ ∧ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℤ ∧ 𝑛 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
1018, 19, 99, 100syl3anbrc 1337 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ (ℤ𝑛))
102 fveq2 6666 . . . . . . . . . . . . 13 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
103102eleq1d 2901 . . . . . . . . . . . 12 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → ((seq1( + , 𝐻)‘𝑘) ∈ ℂ ↔ (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ))
104102fvoveq1d 7173 . . . . . . . . . . . . 13 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)))
105104breq1d 5072 . . . . . . . . . . . 12 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → ((abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))
106103, 105anbi12d 630 . . . . . . . . . . 11 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → (((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) ↔ ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
107106rspcv 3621 . . . . . . . . . 10 ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ (ℤ𝑛) → (∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
108101, 107syl 17 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
109108ralrimdva 3193 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) → ∀𝑚 ∈ (ℤ‘(𝐺𝑛))((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
110 fveq2 6666 . . . . . . . . . 10 (𝑗 = (𝐺𝑛) → (ℤ𝑗) = (ℤ‘(𝐺𝑛)))
111110raleqdv 3420 . . . . . . . . 9 (𝑗 = (𝐺𝑛) → (∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) ↔ ∀𝑚 ∈ (ℤ‘(𝐺𝑛))((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
112111rspcev 3626 . . . . . . . 8 (((𝐺𝑛) ∈ ℤ ∧ ∀𝑚 ∈ (ℤ‘(𝐺𝑛))((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)) → ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))
1136, 109, 112syl6an 680 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) → ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
114113rexlimdva 3288 . . . . . 6 (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) → ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
115 1nn 11641 . . . . . . . . 9 1 ∈ ℕ
116 ffvelrn 6844 . . . . . . . . 9 ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
1174, 115, 116sylancl 586 . . . . . . . 8 (𝜑 → (𝐺‘1) ∈ 𝑍)
118117, 1syl6eleq 2927 . . . . . . 7 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
119 eluzelz 12245 . . . . . . 7 ((𝐺‘1) ∈ (ℤ𝑀) → (𝐺‘1) ∈ ℤ)
120 eqid 2824 . . . . . . . 8 (ℤ‘(𝐺‘1)) = (ℤ‘(𝐺‘1))
121120rexuz3 14701 . . . . . . 7 ((𝐺‘1) ∈ ℤ → (∃𝑗 ∈ (ℤ‘(𝐺‘1))∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
122118, 119, 1213syl 18 . . . . . 6 (𝜑 → (∃𝑗 ∈ (ℤ‘(𝐺‘1))∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
123114, 122sylibrd 260 . . . . 5 (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) → ∃𝑗 ∈ (ℤ‘(𝐺‘1))∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
124 fzfid 13334 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝑀...𝑗) ∈ Fin)
125 funimacnv 6431 . . . . . . . . . . . 12 (Fun 𝐺 → (𝐺 “ (𝐺 “ (𝑀...𝑗))) = ((𝑀...𝑗) ∩ ran 𝐺))
1264, 10, 1253syl 18 . . . . . . . . . . 11 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑗))) = ((𝑀...𝑗) ∩ ran 𝐺))
127 inss1 4208 . . . . . . . . . . 11 ((𝑀...𝑗) ∩ ran 𝐺) ⊆ (𝑀...𝑗)
128126, 127eqsstrdi 4024 . . . . . . . . . 10 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗))
129128adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗))
130124, 129ssfid 8733 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin)
131 hashcl 13710 . . . . . . . 8 ((𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0)
132 nn0p1nn 11928 . . . . . . . 8 ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0 → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ)
133130, 131, 1323syl 18 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ)
134 eluzle 12248 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘)
135134adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘)
136130adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin)
137 nn0z 11997 . . . . . . . . . . . . . . . 16 ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0 → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℤ)
138136, 131, 1373syl 18 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℤ)
139 eluzelz 12245 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)) → 𝑘 ∈ ℤ)
140139adantl 482 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℤ)
141 zltp1le 12024 . . . . . . . . . . . . . . 15 (((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘))
142138, 140, 141syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘))
143135, 142mpbird 258 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘)
144 nn0re 11898 . . . . . . . . . . . . . . . 16 ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0 → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℝ)
145130, 131, 1443syl 18 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℝ)
146145adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℝ)
147 eluznn 12310 . . . . . . . . . . . . . . . 16 ((((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℕ)
148133, 147sylan 580 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℕ)
149148nnred 11645 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℝ)
150146, 149ltnled 10779 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ¬ 𝑘 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗))))))
151143, 150mpbid 233 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ¬ 𝑘 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))))
152 fzss2 12940 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ‘(𝐺𝑘)) → (𝑀...(𝐺𝑘)) ⊆ (𝑀...𝑗))
153 imass2 5962 . . . . . . . . . . . . . 14 ((𝑀...(𝐺𝑘)) ⊆ (𝑀...𝑗) → (𝐺 “ (𝑀...(𝐺𝑘))) ⊆ (𝐺 “ (𝑀...𝑗)))
154 imass2 5962 . . . . . . . . . . . . . 14 ((𝐺 “ (𝑀...(𝐺𝑘))) ⊆ (𝐺 “ (𝑀...𝑗)) → (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗))))
155152, 153, 1543syl 18 . . . . . . . . . . . . 13 (𝑗 ∈ (ℤ‘(𝐺𝑘)) → (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗))))
156 ssdomg 8547 . . . . . . . . . . . . . . 15 ((𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin → ((𝐺 “ (1...𝑘)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗))) → (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑗)))))
157136, 156syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝐺 “ (1...𝑘)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗))) → (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑗)))))
1584ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝐺:ℕ⟶𝑍)
159158ffvelrnda 6846 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ 𝑍)
160159, 1syl6eleq 2927 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ (ℤ𝑀))
161158, 148ffvelrnd 6847 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺𝑘) ∈ 𝑍)
1623, 161sseldi 3968 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺𝑘) ∈ ℤ)
163162adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑘) ∈ ℤ)
164 elfz5 12893 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺𝑥) ∈ (ℤ𝑀) ∧ (𝐺𝑘) ∈ ℤ) → ((𝐺𝑥) ∈ (𝑀...(𝐺𝑘)) ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
165160, 163, 164syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ((𝐺𝑥) ∈ (𝑀...(𝐺𝑘)) ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
16629ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
167 nnssre 11634 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ ⊆ ℝ
168 ressxr 10677 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ⊆ ℝ*
169167, 168sstri 3979 . . . . . . . . . . . . . . . . . . . . . 22 ℕ ⊆ ℝ*
170169a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ*)
171 imassrn 5937 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 “ ℕ) ⊆ ran 𝐺
172158adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶𝑍)
173172frnd 6517 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺𝑍)
174173, 55syl6ss 3982 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺 ⊆ ℝ)
175171, 174sstrid 3981 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ)
176175, 168syl6ss 3982 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ*)
177 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
178148adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝑘 ∈ ℕ)
179 leisorel 13811 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) ∧ (ℕ ⊆ ℝ* ∧ (𝐺 “ ℕ) ⊆ ℝ*) ∧ (𝑥 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → (𝑥𝑘 ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
180166, 170, 176, 177, 178, 179syl122anc 1373 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝑥𝑘 ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
181165, 180bitr4d 283 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ((𝐺𝑥) ∈ (𝑀...(𝐺𝑘)) ↔ 𝑥𝑘))
182181pm5.32da 579 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝑥 ∈ ℕ ∧ (𝐺𝑥) ∈ (𝑀...(𝐺𝑘))) ↔ (𝑥 ∈ ℕ ∧ 𝑥𝑘)))
183 elpreima 6823 . . . . . . . . . . . . . . . . . . 19 (𝐺 Fn ℕ → (𝑥 ∈ (𝐺 “ (𝑀...(𝐺𝑘))) ↔ (𝑥 ∈ ℕ ∧ (𝐺𝑥) ∈ (𝑀...(𝐺𝑘)))))
184158, 25, 1833syl 18 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (𝐺 “ (𝑀...(𝐺𝑘))) ↔ (𝑥 ∈ ℕ ∧ (𝐺𝑥) ∈ (𝑀...(𝐺𝑘)))))
185 fznn 12968 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℤ → (𝑥 ∈ (1...𝑘) ↔ (𝑥 ∈ ℕ ∧ 𝑥𝑘)))
186140, 185syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (1...𝑘) ↔ (𝑥 ∈ ℕ ∧ 𝑥𝑘)))
187182, 184, 1863bitr4d 312 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (𝐺 “ (𝑀...(𝐺𝑘))) ↔ 𝑥 ∈ (1...𝑘)))
188187eqrdv 2822 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝑀...(𝐺𝑘))) = (1...𝑘))
189188imaeq2d 5926 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) = (𝐺 “ (1...𝑘)))
190189sseq1d 4001 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗))) ↔ (𝐺 “ (1...𝑘)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗)))))
19135ad2antrr 722 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝐺:ℕ–1-1𝑍)
192 fz1ssnn 12931 . . . . . . . . . . . . . . . . . . 19 (1...𝑘) ⊆ ℕ
193 ovex 7184 . . . . . . . . . . . . . . . . . . . 20 (1...𝑘) ∈ V
194193f1imaen 8563 . . . . . . . . . . . . . . . . . . 19 ((𝐺:ℕ–1-1𝑍 ∧ (1...𝑘) ⊆ ℕ) → (𝐺 “ (1...𝑘)) ≈ (1...𝑘))
195191, 192, 194sylancl 586 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (1...𝑘)) ≈ (1...𝑘))
196 fzfid 13334 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (1...𝑘) ∈ Fin)
197 enfii 8727 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑘) ∈ Fin ∧ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)) → (𝐺 “ (1...𝑘)) ∈ Fin)
198196, 195, 197syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (1...𝑘)) ∈ Fin)
199 hashen 13700 . . . . . . . . . . . . . . . . . . 19 (((𝐺 “ (1...𝑘)) ∈ Fin ∧ (1...𝑘) ∈ Fin) → ((♯‘(𝐺 “ (1...𝑘))) = (♯‘(1...𝑘)) ↔ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)))
200198, 196, 199syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((♯‘(𝐺 “ (1...𝑘))) = (♯‘(1...𝑘)) ↔ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)))
201195, 200mpbird 258 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (1...𝑘))) = (♯‘(1...𝑘)))
202 nnnn0 11896 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
203 hashfz1 13699 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ0 → (♯‘(1...𝑘)) = 𝑘)
204148, 202, 2033syl 18 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(1...𝑘)) = 𝑘)
205201, 204eqtrd 2860 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (1...𝑘))) = 𝑘)
206205breq1d 5072 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((♯‘(𝐺 “ (1...𝑘))) ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ↔ 𝑘 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗))))))
207 hashdom 13733 . . . . . . . . . . . . . . . 16 (((𝐺 “ (1...𝑘)) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin) → ((♯‘(𝐺 “ (1...𝑘))) ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ↔ (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑗)))))
208198, 136, 207syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((♯‘(𝐺 “ (1...𝑘))) ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ↔ (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑗)))))
209206, 208bitr3d 282 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑘 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ↔ (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑗)))))
210157, 190, 2093imtr4d 295 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗))) → 𝑘 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗))))))
211155, 210syl5 34 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑗 ∈ (ℤ‘(𝐺𝑘)) → 𝑘 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗))))))
212151, 211mtod 199 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ¬ 𝑗 ∈ (ℤ‘(𝐺𝑘)))
213 eluzelz 12245 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ‘(𝐺‘1)) → 𝑗 ∈ ℤ)
214213ad2antlr 723 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑗 ∈ ℤ)
215 uztric 12258 . . . . . . . . . . . . 13 (((𝐺𝑘) ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ‘(𝐺𝑘)) ∨ (𝐺𝑘) ∈ (ℤ𝑗)))
216162, 214, 215syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑗 ∈ (ℤ‘(𝐺𝑘)) ∨ (𝐺𝑘) ∈ (ℤ𝑗)))
217216ord 860 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (¬ 𝑗 ∈ (ℤ‘(𝐺𝑘)) → (𝐺𝑘) ∈ (ℤ𝑗)))
218212, 217mpd 15 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺𝑘) ∈ (ℤ𝑗))
219 oveq2 7159 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝐺𝑘) → (𝑀...𝑚) = (𝑀...(𝐺𝑘)))
220219imaeq2d 5926 . . . . . . . . . . . . . . . 16 (𝑚 = (𝐺𝑘) → (𝐺 “ (𝑀...𝑚)) = (𝐺 “ (𝑀...(𝐺𝑘))))
221220imaeq2d 5926 . . . . . . . . . . . . . . 15 (𝑚 = (𝐺𝑘) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) = (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))
222221fveq2d 6670 . . . . . . . . . . . . . 14 (𝑚 = (𝐺𝑘) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) = (♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))))
223222fveq2d 6670 . . . . . . . . . . . . 13 (𝑚 = (𝐺𝑘) → (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))))
224223eleq1d 2901 . . . . . . . . . . . 12 (𝑚 = (𝐺𝑘) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ))
225223fvoveq1d 7173 . . . . . . . . . . . . 13 (𝑚 = (𝐺𝑘) → (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)))
226225breq1d 5072 . . . . . . . . . . . 12 (𝑚 = (𝐺𝑘) → ((abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)) < 𝑥))
227224, 226anbi12d 630 . . . . . . . . . . 11 (𝑚 = (𝐺𝑘) → (((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) ↔ ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)) < 𝑥)))
228227rspcv 3621 . . . . . . . . . 10 ((𝐺𝑘) ∈ (ℤ𝑗) → (∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)) < 𝑥)))
229218, 228syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)) < 𝑥)))
230189fveq2d 6670 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))) = (♯‘(𝐺 “ (1...𝑘))))
231230, 205eqtrd 2860 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))) = 𝑘)
232231fveq2d 6670 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) = (seq1( + , 𝐻)‘𝑘))
233232eleq1d 2901 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘𝑘) ∈ ℂ))
234232fvoveq1d 7173 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)))
235234breq1d 5072 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥))
236233, 235anbi12d 630 . . . . . . . . 9 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)) < 𝑥) ↔ ((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥)))
237229, 236sylibd 240 . . . . . . . 8 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥)))
238237ralrimdva 3193 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ∀𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥)))
239 fveq2 6666 . . . . . . . . 9 (𝑛 = ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) → (ℤ𝑛) = (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)))
240239raleqdv 3420 . . . . . . . 8 (𝑛 = ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) → (∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥)))
241240rspcev 3626 . . . . . . 7 ((((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ ∧ ∀𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥)) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥))
242133, 238, 241syl6an 680 . . . . . 6 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥)))
243242rexlimdva 3288 . . . . 5 (𝜑 → (∃𝑗 ∈ (ℤ‘(𝐺‘1))∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥)))
244123, 243impbid 213 . . . 4 (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ (ℤ‘(𝐺‘1))∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
245244ralbidv 3201 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ‘(𝐺‘1))∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))
246245anbi2d 628 . 2 (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥)) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ‘(𝐺‘1))∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))))
247 nnuz 12273 . . 3 ℕ = (ℤ‘1)
248 1zzd 12005 . . 3 (𝜑 → 1 ∈ ℤ)
249 seqex 13364 . . . 4 seq1( + , 𝐻) ∈ V
250249a1i 11 . . 3 (𝜑 → seq1( + , 𝐻) ∈ V)
251 eqidd 2825 . . 3 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘𝑘))
252247, 248, 250, 251clim2 14854 . 2 (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥))))
253118, 119syl 17 . . 3 (𝜑 → (𝐺‘1) ∈ ℤ)
254 seqex 13364 . . . 4 seq𝑀( + , 𝐹) ∈ V
255254a1i 11 . . 3 (𝜑 → seq𝑀( + , 𝐹) ∈ V)
256 isercoll.0 . . . 4 ((𝜑𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹𝑛) = 0)
257 isercoll.f . . . 4 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ ℂ)
258 isercoll.h . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))
2591, 21, 4, 22, 256, 257, 258isercolllem3 15016 . . 3 ((𝜑𝑚 ∈ (ℤ‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑚) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
260120, 253, 255, 259clim2 14854 . 2 (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ‘(𝐺‘1))∀𝑚 ∈ (ℤ𝑗)((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))))
261246, 252, 2603bitr4d 312 1 (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2106  wral 3142  wrex 3143  Vcvv 3499  cdif 3936  cin 3938  wss 3939   class class class wbr 5062  ccnv 5552  ran crn 5554  cres 5555  cima 5556  Fun wfun 6345   Fn wfn 6346  wf 6347  1-1wf1 6348  1-1-ontowf1o 6350  cfv 6351   Isom wiso 6352  (class class class)co 7151  cen 8498  cdom 8499  Fincfn 8501  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532  *cxr 10666   < clt 10667  cle 10668  cmin 10862  cn 11630  0cn0 11889  cz 11973  cuz 12235  +crp 12382  ...cfz 12885  seqcseq 13362  chash 13683  abscabs 14586  cli 14834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12886  df-seq 13363  df-hash 13684  df-clim 14838
This theorem is referenced by:  isercoll2  15018
  Copyright terms: Public domain W3C validator