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Theorem isercoll 14246
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 11583 . . . . . . . . . 10 (ℤ𝑀) ⊆ ℤ
31, 2eqsstri 3598 . . . . . . . . 9 𝑍 ⊆ ℤ
4 isercoll.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶𝑍)
54ffvelrnda 6267 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ 𝑍)
63, 5sseldi 3566 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℤ)
7 nnz 11276 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
87ad2antlr 759 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ∈ ℤ)
9 fzfid 12634 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝑀...𝑚) ∈ Fin)
10 ffun 5961 . . . . . . . . . . . . . . . 16 (𝐺:ℕ⟶𝑍 → Fun 𝐺)
11 funimacnv 5884 . . . . . . . . . . . . . . . 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 3795 . . . . . . . . . . . . . . 15 ((𝑀...𝑚) ∩ ran 𝐺) ⊆ (𝑀...𝑚)
1412, 13syl6eqss 3618 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚))
1514ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚))
16 ssfi 8065 . . . . . . . . . . . . 13 (((𝑀...𝑚) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚)) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin)
179, 15, 16syl2anc 691 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin)
18 hashcl 13009 . . . . . . . . . . . 12 ((𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℕ0)
19 nn0z 11277 . . . . . . . . . . . 12 ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℕ0 → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℤ)
2017, 18, 193syl 18 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℤ)
21 ssid 3587 . . . . . . . . . . . . . . . . . . . 20 ℕ ⊆ ℕ
22 isercoll.m . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ ℤ)
23 isercoll.i . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))
241, 22, 4, 23isercolllem1 14243 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ℕ ⊆ ℕ) → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
2521, 24mpan2 703 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
26 ffn 5958 . . . . . . . . . . . . . . . . . . . 20 (𝐺:ℕ⟶𝑍𝐺 Fn ℕ)
27 fnresdm 5914 . . . . . . . . . . . . . . . . . . . 20 (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺)
28 isoeq1 6467 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ↾ ℕ) = 𝐺 → ((𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)) ↔ 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ))))
294, 26, 27, 284syl 19 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)) ↔ 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ))))
3025, 29mpbid 221 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
31 isof1o 6473 . . . . . . . . . . . . . . . . . 18 (𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) → 𝐺:ℕ–1-1-onto→(𝐺 “ ℕ))
32 f1ocnv 6062 . . . . . . . . . . . . . . . . . 18 (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → 𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ)
33 f1ofun 6052 . . . . . . . . . . . . . . . . . 18 (𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun 𝐺)
3430, 31, 32, 334syl 19 . . . . . . . . . . . . . . . . 17 (𝜑 → Fun 𝐺)
35 df-f1 5809 . . . . . . . . . . . . . . . . 17 (𝐺:ℕ–1-1𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun 𝐺))
364, 34, 35sylanbrc 695 . . . . . . . . . . . . . . . 16 (𝜑𝐺:ℕ–1-1𝑍)
3736ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝐺:ℕ–1-1𝑍)
38 elfznn 12241 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1...𝑛) → 𝑦 ∈ ℕ)
3938ssriv 3572 . . . . . . . . . . . . . . 15 (1...𝑛) ⊆ ℕ
40 ovex 6577 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ V
4140f1imaen 7904 . . . . . . . . . . . . . . 15 ((𝐺:ℕ–1-1𝑍 ∧ (1...𝑛) ⊆ ℕ) → (𝐺 “ (1...𝑛)) ≈ (1...𝑛))
4237, 39, 41sylancl 693 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ≈ (1...𝑛))
43 fzfid 12634 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (1...𝑛) ∈ Fin)
44 enfii 8062 . . . . . . . . . . . . . . . 16 (((1...𝑛) ∈ Fin ∧ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)) → (𝐺 “ (1...𝑛)) ∈ Fin)
4543, 42, 44syl2anc 691 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ∈ Fin)
46 hashen 12997 . . . . . . . . . . . . . . 15 (((𝐺 “ (1...𝑛)) ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((#‘(𝐺 “ (1...𝑛))) = (#‘(1...𝑛)) ↔ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)))
4745, 43, 46syl2anc 691 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → ((#‘(𝐺 “ (1...𝑛))) = (#‘(1...𝑛)) ↔ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)))
4842, 47mpbird 246 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (#‘(𝐺 “ (1...𝑛))) = (#‘(1...𝑛)))
49 nnnn0 11176 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
5049ad2antlr 759 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ∈ ℕ0)
51 hashfz1 12996 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → (#‘(1...𝑛)) = 𝑛)
5250, 51syl 17 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (#‘(1...𝑛)) = 𝑛)
5348, 52eqtrd 2644 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (#‘(𝐺 “ (1...𝑛))) = 𝑛)
5438adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℕ)
55 zssre 11261 . . . . . . . . . . . . . . . . . . . . . 22 ℤ ⊆ ℝ
563, 55sstri 3577 . . . . . . . . . . . . . . . . . . . . 21 𝑍 ⊆ ℝ
574ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝐺:ℕ⟶𝑍)
58 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺:ℕ⟶𝑍𝑦 ∈ ℕ) → (𝐺𝑦) ∈ 𝑍)
5957, 38, 58syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ 𝑍)
6056, 59sseldi 3566 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ ℝ)
615ad2antrr 758 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ∈ 𝑍)
6256, 61sseldi 3566 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ∈ ℝ)
63 eluzelz 11573 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (ℤ‘(𝐺𝑛)) → 𝑚 ∈ ℤ)
6463ad2antlr 759 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℤ)
6564zred 11358 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℝ)
66 elfzle2 12216 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (1...𝑛) → 𝑦𝑛)
6766adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦𝑛)
6830ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
69 simpllr 795 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑛 ∈ ℕ)
70 isorel 6476 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑛 < 𝑦 ↔ (𝐺𝑛) < (𝐺𝑦)))
7168, 69, 54, 70syl12anc 1316 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑛 < 𝑦 ↔ (𝐺𝑛) < (𝐺𝑦)))
7271notbid 307 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (¬ 𝑛 < 𝑦 ↔ ¬ (𝐺𝑛) < (𝐺𝑦)))
7354nnred 10912 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℝ)
7469nnred 10912 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑛 ∈ ℝ)
7573, 74lenltd 10062 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦𝑛 ↔ ¬ 𝑛 < 𝑦))
7660, 62lenltd 10062 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → ((𝐺𝑦) ≤ (𝐺𝑛) ↔ ¬ (𝐺𝑛) < (𝐺𝑦)))
7772, 75, 763bitr4d 299 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦𝑛 ↔ (𝐺𝑦) ≤ (𝐺𝑛)))
7867, 77mpbid 221 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ≤ (𝐺𝑛))
79 eluzle 11576 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (ℤ‘(𝐺𝑛)) → (𝐺𝑛) ≤ 𝑚)
8079ad2antlr 759 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ≤ 𝑚)
8160, 62, 65, 78, 80letrd 10073 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ≤ 𝑚)
8259, 1syl6eleq 2698 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ (ℤ𝑀))
83 elfz5 12205 . . . . . . . . . . . . . . . . . . . 20 (((𝐺𝑦) ∈ (ℤ𝑀) ∧ 𝑚 ∈ ℤ) → ((𝐺𝑦) ∈ (𝑀...𝑚) ↔ (𝐺𝑦) ≤ 𝑚))
8482, 64, 83syl2anc 691 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → ((𝐺𝑦) ∈ (𝑀...𝑚) ↔ (𝐺𝑦) ≤ 𝑚))
8581, 84mpbird 246 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ (𝑀...𝑚))
8657, 26syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝐺 Fn ℕ)
8786adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝐺 Fn ℕ)
88 elpreima 6245 . . . . . . . . . . . . . . . . . . 19 (𝐺 Fn ℕ → (𝑦 ∈ (𝐺 “ (𝑀...𝑚)) ↔ (𝑦 ∈ ℕ ∧ (𝐺𝑦) ∈ (𝑀...𝑚))))
8987, 88syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦 ∈ (𝐺 “ (𝑀...𝑚)) ↔ (𝑦 ∈ ℕ ∧ (𝐺𝑦) ∈ (𝑀...𝑚))))
9054, 85, 89mpbir2and 959 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ (𝐺 “ (𝑀...𝑚)))
9190ex 449 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝑦 ∈ (1...𝑛) → 𝑦 ∈ (𝐺 “ (𝑀...𝑚))))
9291ssrdv 3574 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (1...𝑛) ⊆ (𝐺 “ (𝑀...𝑚)))
93 imass2 5420 . . . . . . . . . . . . . . 15 ((1...𝑛) ⊆ (𝐺 “ (𝑀...𝑚)) → (𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑚))))
9492, 93syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑚))))
95 ssdomg 7887 . . . . . . . . . . . . . 14 ((𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin → ((𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑚))) → (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚)))))
9617, 94, 95sylc 63 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚))))
97 hashdom 13029 . . . . . . . . . . . . . 14 (((𝐺 “ (1...𝑛)) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin) → ((#‘(𝐺 “ (1...𝑛))) ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ↔ (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚)))))
9845, 17, 97syl2anc 691 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → ((#‘(𝐺 “ (1...𝑛))) ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ↔ (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚)))))
9996, 98mpbird 246 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (#‘(𝐺 “ (1...𝑛))) ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))))
10053, 99eqbrtrrd 4607 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))))
101 eluz2 11569 . . . . . . . . . . 11 ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ (ℤ𝑛) ↔ (𝑛 ∈ ℤ ∧ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℤ ∧ 𝑛 ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
1028, 20, 100, 101syl3anbrc 1239 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ (ℤ𝑛))
103 fveq2 6103 . . . . . . . . . . . . 13 (𝑘 = (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
104103eleq1d 2672 . . . . . . . . . . . 12 (𝑘 = (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → ((seq1( + , 𝐻)‘𝑘) ∈ ℂ ↔ (seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ))
105103oveq1d 6564 . . . . . . . . . . . . . 14 (𝑘 = (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → ((seq1( + , 𝐻)‘𝑘) − 𝐴) = ((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴))
106105fveq2d 6107 . . . . . . . . . . . . 13 (𝑘 = (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)))
107106breq1d 4593 . . . . . . . . . . . 12 (𝑘 = (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → ((abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))
108104, 107anbi12d 743 . . . . . . . . . . 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)))
109108rspcv 3278 . . . . . . . . . 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)))
110102, 109syl 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)))
111110ralrimdva 2952 . . . . . . . 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)))
112 fveq2 6103 . . . . . . . . . 10 (𝑗 = (𝐺𝑛) → (ℤ𝑗) = (ℤ‘(𝐺𝑛)))
113112raleqdv 3121 . . . . . . . . 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)))
114113rspcev 3282 . . . . . . . 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))
1156, 111, 114syl6an 566 . . . . . . 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)))
116115rexlimdva 3013 . . . . . 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)))
117 1nn 10908 . . . . . . . . 9 1 ∈ ℕ
118 ffvelrn 6265 . . . . . . . . 9 ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
1194, 117, 118sylancl 693 . . . . . . . 8 (𝜑 → (𝐺‘1) ∈ 𝑍)
120119, 1syl6eleq 2698 . . . . . . 7 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
121 eluzelz 11573 . . . . . . 7 ((𝐺‘1) ∈ (ℤ𝑀) → (𝐺‘1) ∈ ℤ)
122 eqid 2610 . . . . . . . 8 (ℤ‘(𝐺‘1)) = (ℤ‘(𝐺‘1))
123122rexuz3 13936 . . . . . . 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)))
124120, 121, 1233syl 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)))
125116, 124sylibrd 248 . . . . 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)))
126 fzfid 12634 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝑀...𝑗) ∈ Fin)
127 funimacnv 5884 . . . . . . . . . . . 12 (Fun 𝐺 → (𝐺 “ (𝐺 “ (𝑀...𝑗))) = ((𝑀...𝑗) ∩ ran 𝐺))
1284, 10, 1273syl 18 . . . . . . . . . . 11 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑗))) = ((𝑀...𝑗) ∩ ran 𝐺))
129 inss1 3795 . . . . . . . . . . 11 ((𝑀...𝑗) ∩ ran 𝐺) ⊆ (𝑀...𝑗)
130128, 129syl6eqss 3618 . . . . . . . . . 10 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗))
131130adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗))
132 ssfi 8065 . . . . . . . . 9 (((𝑀...𝑗) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗)) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin)
133126, 131, 132syl2anc 691 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin)
134 hashcl 13009 . . . . . . . 8 ((𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0)
135 nn0p1nn 11209 . . . . . . . 8 ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0 → ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ)
136133, 134, 1353syl 18 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ)
137 eluzle 11576 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)) → ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘)
138137adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘)
139133adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin)
140 nn0z 11277 . . . . . . . . . . . . . . . 16 ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0 → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℤ)
141139, 134, 1403syl 18 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℤ)
142 eluzelz 11573 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)) → 𝑘 ∈ ℤ)
143142adantl 481 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℤ)
144 zltp1le 11304 . . . . . . . . . . . . . . 15 (((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘))
145141, 143, 144syl2anc 691 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘))
146138, 145mpbird 246 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘)
147 nn0re 11178 . . . . . . . . . . . . . . . 16 ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0 → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℝ)
148133, 134, 1473syl 18 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℝ)
149148adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℝ)
150 eluznn 11634 . . . . . . . . . . . . . . . 16 ((((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℕ)
151136, 150sylan 487 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℕ)
152151nnred 10912 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℝ)
153149, 152ltnled 10063 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ¬ 𝑘 ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗))))))
154146, 153mpbid 221 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ¬ 𝑘 ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))))
155 fzss2 12252 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ‘(𝐺𝑘)) → (𝑀...(𝐺𝑘)) ⊆ (𝑀...𝑗))
156 imass2 5420 . . . . . . . . . . . . . 14 ((𝑀...(𝐺𝑘)) ⊆ (𝑀...𝑗) → (𝐺 “ (𝑀...(𝐺𝑘))) ⊆ (𝐺 “ (𝑀...𝑗)))
157 imass2 5420 . . . . . . . . . . . . . 14 ((𝐺 “ (𝑀...(𝐺𝑘))) ⊆ (𝐺 “ (𝑀...𝑗)) → (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗))))
158155, 156, 1573syl 18 . . . . . . . . . . . . 13 (𝑗 ∈ (ℤ‘(𝐺𝑘)) → (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗))))
159 ssdomg 7887 . . . . . . . . . . . . . . 15 ((𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin → ((𝐺 “ (1...𝑘)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑗))) → (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑗)))))
160139, 159syl 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)))))
1614ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝐺:ℕ⟶𝑍)
162161ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ 𝑍)
163162, 1syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ (ℤ𝑀))
164161, 151ffvelrnd 6268 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺𝑘) ∈ 𝑍)
1653, 164sseldi 3566 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺𝑘) ∈ ℤ)
166165adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑘) ∈ ℤ)
167 elfz5 12205 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺𝑥) ∈ (ℤ𝑀) ∧ (𝐺𝑘) ∈ ℤ) → ((𝐺𝑥) ∈ (𝑀...(𝐺𝑘)) ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
168163, 166, 167syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ((𝐺𝑥) ∈ (𝑀...(𝐺𝑘)) ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
16930ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
170 nnssre 10901 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ ⊆ ℝ
171 ressxr 9962 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ⊆ ℝ*
172170, 171sstri 3577 . . . . . . . . . . . . . . . . . . . . . 22 ℕ ⊆ ℝ*
173172a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ*)
174 imassrn 5396 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 “ ℕ) ⊆ ran 𝐺
175161adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶𝑍)
176 frn 5966 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐺:ℕ⟶𝑍 → ran 𝐺𝑍)
177175, 176syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺𝑍)
178177, 56syl6ss 3580 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺 ⊆ ℝ)
179174, 178syl5ss 3579 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ)
180179, 171syl6ss 3580 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ*)
181 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
182151adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝑘 ∈ ℕ)
183 leisorel 13101 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) ∧ (ℕ ⊆ ℝ* ∧ (𝐺 “ ℕ) ⊆ ℝ*) ∧ (𝑥 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → (𝑥𝑘 ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
184169, 173, 180, 181, 182, 183syl122anc 1327 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝑥𝑘 ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
185168, 184bitr4d 270 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ((𝐺𝑥) ∈ (𝑀...(𝐺𝑘)) ↔ 𝑥𝑘))
186185pm5.32da 671 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝑥 ∈ ℕ ∧ (𝐺𝑥) ∈ (𝑀...(𝐺𝑘))) ↔ (𝑥 ∈ ℕ ∧ 𝑥𝑘)))
187 elpreima 6245 . . . . . . . . . . . . . . . . . . 19 (𝐺 Fn ℕ → (𝑥 ∈ (𝐺 “ (𝑀...(𝐺𝑘))) ↔ (𝑥 ∈ ℕ ∧ (𝐺𝑥) ∈ (𝑀...(𝐺𝑘)))))
188161, 26, 1873syl 18 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (𝐺 “ (𝑀...(𝐺𝑘))) ↔ (𝑥 ∈ ℕ ∧ (𝐺𝑥) ∈ (𝑀...(𝐺𝑘)))))
189 fznn 12278 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℤ → (𝑥 ∈ (1...𝑘) ↔ (𝑥 ∈ ℕ ∧ 𝑥𝑘)))
190143, 189syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (1...𝑘) ↔ (𝑥 ∈ ℕ ∧ 𝑥𝑘)))
191186, 188, 1903bitr4d 299 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (𝐺 “ (𝑀...(𝐺𝑘))) ↔ 𝑥 ∈ (1...𝑘)))
192191eqrdv 2608 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝑀...(𝐺𝑘))) = (1...𝑘))
193192imaeq2d 5385 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) = (𝐺 “ (1...𝑘)))
194193sseq1d 3595 . . . . . . . . . . . . . 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)))))
19536ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝐺:ℕ–1-1𝑍)
196 elfznn 12241 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℕ)
197196ssriv 3572 . . . . . . . . . . . . . . . . . . 19 (1...𝑘) ⊆ ℕ
198 ovex 6577 . . . . . . . . . . . . . . . . . . . 20 (1...𝑘) ∈ V
199198f1imaen 7904 . . . . . . . . . . . . . . . . . . 19 ((𝐺:ℕ–1-1𝑍 ∧ (1...𝑘) ⊆ ℕ) → (𝐺 “ (1...𝑘)) ≈ (1...𝑘))
200195, 197, 199sylancl 693 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (1...𝑘)) ≈ (1...𝑘))
201 fzfid 12634 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (1...𝑘) ∈ Fin)
202 enfii 8062 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑘) ∈ Fin ∧ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)) → (𝐺 “ (1...𝑘)) ∈ Fin)
203201, 200, 202syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (1...𝑘)) ∈ Fin)
204 hashen 12997 . . . . . . . . . . . . . . . . . . 19 (((𝐺 “ (1...𝑘)) ∈ Fin ∧ (1...𝑘) ∈ Fin) → ((#‘(𝐺 “ (1...𝑘))) = (#‘(1...𝑘)) ↔ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)))
205203, 201, 204syl2anc 691 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (1...𝑘))) = (#‘(1...𝑘)) ↔ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)))
206200, 205mpbird 246 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (1...𝑘))) = (#‘(1...𝑘)))
207 nnnn0 11176 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
208 hashfz1 12996 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ0 → (#‘(1...𝑘)) = 𝑘)
209151, 207, 2083syl 18 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(1...𝑘)) = 𝑘)
210206, 209eqtrd 2644 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (1...𝑘))) = 𝑘)
211210breq1d 4593 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (1...𝑘))) ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ↔ 𝑘 ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗))))))
212 hashdom 13029 . . . . . . . . . . . . . . . 16 (((𝐺 “ (1...𝑘)) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin) → ((#‘(𝐺 “ (1...𝑘))) ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ↔ (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑗)))))
213203, 139, 212syl2anc 691 . . . . . . . . . . . . . . 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)))))
214211, 213bitr3d 269 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑘 ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ↔ (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑗)))))
215160, 194, 2143imtr4d 282 . . . . . . . . . . . . 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))))))
216158, 215syl5 33 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑗 ∈ (ℤ‘(𝐺𝑘)) → 𝑘 ≤ (#‘(𝐺 “ (𝐺 “ (𝑀...𝑗))))))
217154, 216mtod 188 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ¬ 𝑗 ∈ (ℤ‘(𝐺𝑘)))
218 eluzelz 11573 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ‘(𝐺‘1)) → 𝑗 ∈ ℤ)
219218ad2antlr 759 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑗 ∈ ℤ)
220 uztric 11585 . . . . . . . . . . . . 13 (((𝐺𝑘) ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ‘(𝐺𝑘)) ∨ (𝐺𝑘) ∈ (ℤ𝑗)))
221165, 219, 220syl2anc 691 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑗 ∈ (ℤ‘(𝐺𝑘)) ∨ (𝐺𝑘) ∈ (ℤ𝑗)))
222221ord 391 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (¬ 𝑗 ∈ (ℤ‘(𝐺𝑘)) → (𝐺𝑘) ∈ (ℤ𝑗)))
223217, 222mpd 15 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺𝑘) ∈ (ℤ𝑗))
224 oveq2 6557 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝐺𝑘) → (𝑀...𝑚) = (𝑀...(𝐺𝑘)))
225224imaeq2d 5385 . . . . . . . . . . . . . . . 16 (𝑚 = (𝐺𝑘) → (𝐺 “ (𝑀...𝑚)) = (𝐺 “ (𝑀...(𝐺𝑘))))
226225imaeq2d 5385 . . . . . . . . . . . . . . 15 (𝑚 = (𝐺𝑘) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) = (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))
227226fveq2d 6107 . . . . . . . . . . . . . 14 (𝑚 = (𝐺𝑘) → (#‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) = (#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))))
228227fveq2d 6107 . . . . . . . . . . . . 13 (𝑚 = (𝐺𝑘) → (seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) = (seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))))
229228eleq1d 2672 . . . . . . . . . . . 12 (𝑚 = (𝐺𝑘) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ))
230228oveq1d 6564 . . . . . . . . . . . . . 14 (𝑚 = (𝐺𝑘) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴) = ((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴))
231230fveq2d 6107 . . . . . . . . . . . . 13 (𝑚 = (𝐺𝑘) → (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)))
232231breq1d 4593 . . . . . . . . . . . 12 (𝑚 = (𝐺𝑘) → ((abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)) < 𝑥))
233229, 232anbi12d 743 . . . . . . . . . . 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)))
234233rspcv 3278 . . . . . . . . . 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)))
235223, 234syl 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)))
236193fveq2d 6107 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))) = (#‘(𝐺 “ (1...𝑘))))
237236, 210eqtrd 2644 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))) = 𝑘)
238237fveq2d 6107 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) = (seq1( + , 𝐻)‘𝑘))
239238eleq1d 2672 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘𝑘) ∈ ℂ))
240238oveq1d 6564 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴) = ((seq1( + , 𝐻)‘𝑘) − 𝐴))
241240fveq2d 6107 . . . . . . . . . . 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)))
242241breq1d 4593 . . . . . . . . . 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))
243239, 242anbi12d 743 . . . . . . . . 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)))
244235, 243sylibd 228 . . . . . . . 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)))
245244ralrimdva 2952 . . . . . . 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)))
246 fveq2 6103 . . . . . . . . 9 (𝑛 = ((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) → (ℤ𝑛) = (ℤ‘((#‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)))
247246raleqdv 3121 . . . . . . . 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)))
248247rspcev 3282 . . . . . . 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))
249136, 245, 248syl6an 566 . . . . . 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)))
250249rexlimdva 3013 . . . . 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)))
251125, 250impbid 201 . . . 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)))
252251ralbidv 2969 . . 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)))
253252anbi2d 736 . 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))))
254 nnuz 11599 . . 3 ℕ = (ℤ‘1)
255 1zzd 11285 . . 3 (𝜑 → 1 ∈ ℤ)
256 seqex 12665 . . . 4 seq1( + , 𝐻) ∈ V
257256a1i 11 . . 3 (𝜑 → seq1( + , 𝐻) ∈ V)
258 eqidd 2611 . . 3 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘𝑘))
259254, 255, 257, 258clim2 14083 . 2 (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥))))
260120, 121syl 17 . . 3 (𝜑 → (𝐺‘1) ∈ ℤ)
261 seqex 12665 . . . 4 seq𝑀( + , 𝐹) ∈ V
262261a1i 11 . . 3 (𝜑 → seq𝑀( + , 𝐹) ∈ V)
263 isercoll.0 . . . 4 ((𝜑𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹𝑛) = 0)
264 isercoll.f . . . 4 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ ℂ)
265 isercoll.h . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐻𝑘) = (𝐹‘(𝐺𝑘)))
2661, 22, 4, 23, 263, 264, 265isercolllem3 14245 . . 3 ((𝜑𝑚 ∈ (ℤ‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑚) = (seq1( + , 𝐻)‘(#‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
267122, 260, 262, 266clim2 14083 . 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))))
268253, 259, 2673bitr4d 299 1 (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  cdif 3537  cin 3539  wss 3540   class class class wbr 4583  ccnv 5037  ran crn 5039  cres 5040  cima 5041  Fun wfun 5798   Fn wfn 5799  wf 5800  1-1wf1 5801  1-1-ontowf1o 5803  cfv 5804   Isom wiso 5805  (class class class)co 6549  cen 7838  cdom 7839  Fincfn 7841  cc 9813  cr 9814  0cc0 9815  1c1 9816   + caddc 9818  *cxr 9952   < clt 9953  cle 9954  cmin 10145  cn 10897  0cn0 11169  cz 11254  cuz 11563  +crp 11708  ...cfz 12197  seqcseq 12663  #chash 12979  abscabs 13822  cli 14063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-hash 12980  df-clim 14067
This theorem is referenced by:  isercoll2  14247
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