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Theorem isercoll 14617
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 11919 . . . . . . . . . 10 (ℤ𝑀) ⊆ ℤ
31, 2eqsstri 3776 . . . . . . . . 9 𝑍 ⊆ ℤ
4 isercoll.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶𝑍)
54ffvelrnda 6523 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ 𝑍)
63, 5sseldi 3742 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℤ)
7 nnz 11611 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
87ad2antlr 765 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ∈ ℤ)
9 fzfid 12986 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝑀...𝑚) ∈ Fin)
10 ffun 6209 . . . . . . . . . . . . . . . 16 (𝐺:ℕ⟶𝑍 → Fun 𝐺)
11 funimacnv 6131 . . . . . . . . . . . . . . . 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 3976 . . . . . . . . . . . . . . 15 ((𝑀...𝑚) ∩ ran 𝐺) ⊆ (𝑀...𝑚)
1412, 13syl6eqss 3796 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚))
1514ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚))
16 ssfi 8347 . . . . . . . . . . . . 13 (((𝑀...𝑚) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚)) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin)
179, 15, 16syl2anc 696 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin)
18 hashcl 13359 . . . . . . . . . . . 12 ((𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℕ0)
19 nn0z 11612 . . . . . . . . . . . 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 3765 . . . . . . . . . . . . . . . . . . . 20 ℕ ⊆ ℕ
22 isercoll.m . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ ℤ)
23 isercoll.i . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))
241, 22, 4, 23isercolllem1 14614 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ℕ ⊆ ℕ) → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
2521, 24mpan2 709 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
26 ffn 6206 . . . . . . . . . . . . . . . . . . . 20 (𝐺:ℕ⟶𝑍𝐺 Fn ℕ)
27 fnresdm 6161 . . . . . . . . . . . . . . . . . . . 20 (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺)
28 isoeq1 6731 . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
31 isof1o 6737 . . . . . . . . . . . . . . . . . 18 (𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) → 𝐺:ℕ–1-1-onto→(𝐺 “ ℕ))
32 f1ocnv 6311 . . . . . . . . . . . . . . . . . 18 (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → 𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ)
33 f1ofun 6301 . . . . . . . . . . . . . . . . . 18 (𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun 𝐺)
3430, 31, 32, 334syl 19 . . . . . . . . . . . . . . . . 17 (𝜑 → Fun 𝐺)
35 df-f1 6054 . . . . . . . . . . . . . . . . 17 (𝐺:ℕ–1-1𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun 𝐺))
364, 34, 35sylanbrc 701 . . . . . . . . . . . . . . . 16 (𝜑𝐺:ℕ–1-1𝑍)
3736ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝐺:ℕ–1-1𝑍)
38 elfznn 12583 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1...𝑛) → 𝑦 ∈ ℕ)
3938ssriv 3748 . . . . . . . . . . . . . . 15 (1...𝑛) ⊆ ℕ
40 ovex 6842 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ V
4140f1imaen 8185 . . . . . . . . . . . . . . 15 ((𝐺:ℕ–1-1𝑍 ∧ (1...𝑛) ⊆ ℕ) → (𝐺 “ (1...𝑛)) ≈ (1...𝑛))
4237, 39, 41sylancl 697 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ≈ (1...𝑛))
43 fzfid 12986 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (1...𝑛) ∈ Fin)
44 enfii 8344 . . . . . . . . . . . . . . . 16 (((1...𝑛) ∈ Fin ∧ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)) → (𝐺 “ (1...𝑛)) ∈ Fin)
4543, 42, 44syl2anc 696 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ∈ Fin)
46 hashen 13349 . . . . . . . . . . . . . . 15 (((𝐺 “ (1...𝑛)) ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((♯‘(𝐺 “ (1...𝑛))) = (♯‘(1...𝑛)) ↔ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)))
4745, 43, 46syl2anc 696 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → ((♯‘(𝐺 “ (1...𝑛))) = (♯‘(1...𝑛)) ↔ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)))
4842, 47mpbird 247 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (1...𝑛))) = (♯‘(1...𝑛)))
49 nnnn0 11511 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
5049ad2antlr 765 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ∈ ℕ0)
51 hashfz1 13348 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
5250, 51syl 17 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(1...𝑛)) = 𝑛)
5348, 52eqtrd 2794 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (1...𝑛))) = 𝑛)
5438adantl 473 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℕ)
55 zssre 11596 . . . . . . . . . . . . . . . . . . . . . 22 ℤ ⊆ ℝ
563, 55sstri 3753 . . . . . . . . . . . . . . . . . . . . 21 𝑍 ⊆ ℝ
574ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝐺:ℕ⟶𝑍)
58 ffvelrn 6521 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺:ℕ⟶𝑍𝑦 ∈ ℕ) → (𝐺𝑦) ∈ 𝑍)
5957, 38, 58syl2an 495 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ 𝑍)
6056, 59sseldi 3742 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ ℝ)
615ad2antrr 764 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ∈ 𝑍)
6256, 61sseldi 3742 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ∈ ℝ)
63 eluzelz 11909 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (ℤ‘(𝐺𝑛)) → 𝑚 ∈ ℤ)
6463ad2antlr 765 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℤ)
6564zred 11694 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℝ)
66 elfzle2 12558 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (1...𝑛) → 𝑦𝑛)
6766adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦𝑛)
6830ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
69 simpllr 817 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑛 ∈ ℕ)
70 isorel 6740 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑛 < 𝑦 ↔ (𝐺𝑛) < (𝐺𝑦)))
7168, 69, 54, 70syl12anc 1475 . . . . . . . . . . . . . . . . . . . . . . 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 11247 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℝ)
7469nnred 11247 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑛 ∈ ℝ)
7573, 74lenltd 10395 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦𝑛 ↔ ¬ 𝑛 < 𝑦))
7660, 62lenltd 10395 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → ((𝐺𝑦) ≤ (𝐺𝑛) ↔ ¬ (𝐺𝑛) < (𝐺𝑦)))
7772, 75, 763bitr4d 300 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦𝑛 ↔ (𝐺𝑦) ≤ (𝐺𝑛)))
7867, 77mpbid 222 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ≤ (𝐺𝑛))
79 eluzle 11912 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (ℤ‘(𝐺𝑛)) → (𝐺𝑛) ≤ 𝑚)
8079ad2antlr 765 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ≤ 𝑚)
8160, 62, 65, 78, 80letrd 10406 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ≤ 𝑚)
8259, 1syl6eleq 2849 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ (ℤ𝑀))
83 elfz5 12547 . . . . . . . . . . . . . . . . . . . 20 (((𝐺𝑦) ∈ (ℤ𝑀) ∧ 𝑚 ∈ ℤ) → ((𝐺𝑦) ∈ (𝑀...𝑚) ↔ (𝐺𝑦) ≤ 𝑚))
8482, 64, 83syl2anc 696 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → ((𝐺𝑦) ∈ (𝑀...𝑚) ↔ (𝐺𝑦) ≤ 𝑚))
8581, 84mpbird 247 . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝐺 Fn ℕ)
88 elpreima 6501 . . . . . . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . . . . 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 3750 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (1...𝑛) ⊆ (𝐺 “ (𝑀...𝑚)))
93 imass2 5659 . . . . . . . . . . . . . . 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 8169 . . . . . . . . . . . . . 14 ((𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin → ((𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (𝐺 “ (𝑀...𝑚))) → (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚)))))
9617, 94, 95sylc 65 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚))))
97 hashdom 13380 . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → ((♯‘(𝐺 “ (1...𝑛))) ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ↔ (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (𝐺 “ (𝑀...𝑚)))))
9996, 98mpbird 247 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (1...𝑛))) ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))))
10053, 99eqbrtrrd 4828 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))))
101 eluz2 11905 . . . . . . . . . . 11 ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ (ℤ𝑛) ↔ (𝑛 ∈ ℤ ∧ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℤ ∧ 𝑛 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
1028, 20, 100, 101syl3anbrc 1429 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ (ℤ𝑛))
103 fveq2 6353 . . . . . . . . . . . . 13 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
104103eleq1d 2824 . . . . . . . . . . . 12 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → ((seq1( + , 𝐻)‘𝑘) ∈ ℂ ↔ (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ))
105103oveq1d 6829 . . . . . . . . . . . . . 14 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → ((seq1( + , 𝐻)‘𝑘) − 𝐴) = ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴))
106105fveq2d 6357 . . . . . . . . . . . . 13 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)))
107106breq1d 4814 . . . . . . . . . . . 12 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → ((abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))
108104, 107anbi12d 749 . . . . . . . . . . 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 3445 . . . . . . . . . 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 3107 . . . . . . . 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 6353 . . . . . . . . . 10 (𝑗 = (𝐺𝑛) → (ℤ𝑗) = (ℤ‘(𝐺𝑛)))
113112raleqdv 3283 . . . . . . . . 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 3449 . . . . . . . 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 569 . . . . . . 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 3169 . . . . . 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 11243 . . . . . . . . 9 1 ∈ ℕ
118 ffvelrn 6521 . . . . . . . . 9 ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
1194, 117, 118sylancl 697 . . . . . . . 8 (𝜑 → (𝐺‘1) ∈ 𝑍)
120119, 1syl6eleq 2849 . . . . . . 7 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
121 eluzelz 11909 . . . . . . 7 ((𝐺‘1) ∈ (ℤ𝑀) → (𝐺‘1) ∈ ℤ)
122 eqid 2760 . . . . . . . 8 (ℤ‘(𝐺‘1)) = (ℤ‘(𝐺‘1))
123122rexuz3 14307 . . . . . . 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 249 . . . . 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 12986 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝑀...𝑗) ∈ Fin)
127 funimacnv 6131 . . . . . . . . . . . 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 3976 . . . . . . . . . . 11 ((𝑀...𝑗) ∩ ran 𝐺) ⊆ (𝑀...𝑗)
130128, 129syl6eqss 3796 . . . . . . . . . 10 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗))
131130adantr 472 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗))
132 ssfi 8347 . . . . . . . . 9 (((𝑀...𝑗) ∈ Fin ∧ (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗)) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin)
133126, 131, 132syl2anc 696 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin)
134 hashcl 13359 . . . . . . . 8 ((𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0)
135 nn0p1nn 11544 . . . . . . . 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 11912 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘)
138137adantl 473 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘)
139133adantr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin)
140 nn0z 11612 . . . . . . . . . . . . . . . 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 11909 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)) → 𝑘 ∈ ℤ)
143142adantl 473 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℤ)
144 zltp1le 11639 . . . . . . . . . . . . . . 15 (((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘))
145141, 143, 144syl2anc 696 . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘)
147 nn0re 11513 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℝ)
150 eluznn 11971 . . . . . . . . . . . . . . . 16 ((((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℕ)
151136, 150sylan 489 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℕ)
152151nnred 11247 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℝ)
153149, 152ltnled 10396 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ¬ 𝑘 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗))))))
154146, 153mpbid 222 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ¬ 𝑘 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))))
155 fzss2 12594 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ‘(𝐺𝑘)) → (𝑀...(𝐺𝑘)) ⊆ (𝑀...𝑗))
156 imass2 5659 . . . . . . . . . . . . . 14 ((𝑀...(𝐺𝑘)) ⊆ (𝑀...𝑗) → (𝐺 “ (𝑀...(𝐺𝑘))) ⊆ (𝐺 “ (𝑀...𝑗)))
157 imass2 5659 . . . . . . . . . . . . . 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 8169 . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝐺:ℕ⟶𝑍)
162161ffvelrnda 6523 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ 𝑍)
163162, 1syl6eleq 2849 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ (ℤ𝑀))
164161, 151ffvelrnd 6524 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺𝑘) ∈ 𝑍)
1653, 164sseldi 3742 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺𝑘) ∈ ℤ)
166165adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑘) ∈ ℤ)
167 elfz5 12547 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺𝑥) ∈ (ℤ𝑀) ∧ (𝐺𝑘) ∈ ℤ) → ((𝐺𝑥) ∈ (𝑀...(𝐺𝑘)) ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
168163, 166, 167syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ((𝐺𝑥) ∈ (𝑀...(𝐺𝑘)) ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
16930ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)))
170 nnssre 11236 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ ⊆ ℝ
171 ressxr 10295 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ⊆ ℝ*
172170, 171sstri 3753 . . . . . . . . . . . . . . . . . . . . . 22 ℕ ⊆ ℝ*
173172a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ*)
174 imassrn 5635 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 “ ℕ) ⊆ ran 𝐺
175161adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶𝑍)
176 frn 6214 . . . . . . . . . . . . . . . . . . . . . . . . 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 3756 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺 ⊆ ℝ)
179174, 178syl5ss 3755 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ)
180179, 171syl6ss 3756 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ*)
181 simpr 479 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
182151adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝑘 ∈ ℕ)
183 leisorel 13456 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) ∧ (ℕ ⊆ ℝ* ∧ (𝐺 “ ℕ) ⊆ ℝ*) ∧ (𝑥 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → (𝑥𝑘 ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
184169, 173, 180, 181, 182, 183syl122anc 1486 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝑥𝑘 ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
185168, 184bitr4d 271 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ((𝐺𝑥) ∈ (𝑀...(𝐺𝑘)) ↔ 𝑥𝑘))
186185pm5.32da 676 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝑥 ∈ ℕ ∧ (𝐺𝑥) ∈ (𝑀...(𝐺𝑘))) ↔ (𝑥 ∈ ℕ ∧ 𝑥𝑘)))
187 elpreima 6501 . . . . . . . . . . . . . . . . . . 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 12621 . . . . . . . . . . . . . . . . . . 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 300 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (𝐺 “ (𝑀...(𝐺𝑘))) ↔ 𝑥 ∈ (1...𝑘)))
192191eqrdv 2758 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝑀...(𝐺𝑘))) = (1...𝑘))
193192imaeq2d 5624 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) = (𝐺 “ (1...𝑘)))
194193sseq1d 3773 . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝐺:ℕ–1-1𝑍)
196 elfznn 12583 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℕ)
197196ssriv 3748 . . . . . . . . . . . . . . . . . . 19 (1...𝑘) ⊆ ℕ
198 ovex 6842 . . . . . . . . . . . . . . . . . . . 20 (1...𝑘) ∈ V
199198f1imaen 8185 . . . . . . . . . . . . . . . . . . 19 ((𝐺:ℕ–1-1𝑍 ∧ (1...𝑘) ⊆ ℕ) → (𝐺 “ (1...𝑘)) ≈ (1...𝑘))
200195, 197, 199sylancl 697 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (1...𝑘)) ≈ (1...𝑘))
201 fzfid 12986 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (1...𝑘) ∈ Fin)
202 enfii 8344 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑘) ∈ Fin ∧ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)) → (𝐺 “ (1...𝑘)) ∈ Fin)
203201, 200, 202syl2anc 696 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (1...𝑘)) ∈ Fin)
204 hashen 13349 . . . . . . . . . . . . . . . . . . 19 (((𝐺 “ (1...𝑘)) ∈ Fin ∧ (1...𝑘) ∈ Fin) → ((♯‘(𝐺 “ (1...𝑘))) = (♯‘(1...𝑘)) ↔ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)))
205203, 201, 204syl2anc 696 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((♯‘(𝐺 “ (1...𝑘))) = (♯‘(1...𝑘)) ↔ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)))
206200, 205mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (1...𝑘))) = (♯‘(1...𝑘)))
207 nnnn0 11511 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
208 hashfz1 13348 . . . . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (1...𝑘))) = 𝑘)
211210breq1d 4814 . . . . . . . . . . . . . . 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 13380 . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 283 . . . . . . . . . . . . 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 34 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑗 ∈ (ℤ‘(𝐺𝑘)) → 𝑘 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗))))))
217154, 216mtod 189 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ¬ 𝑗 ∈ (ℤ‘(𝐺𝑘)))
218 eluzelz 11909 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ‘(𝐺‘1)) → 𝑗 ∈ ℤ)
219218ad2antlr 765 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑗 ∈ ℤ)
220 uztric 11921 . . . . . . . . . . . . 13 (((𝐺𝑘) ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ‘(𝐺𝑘)) ∨ (𝐺𝑘) ∈ (ℤ𝑗)))
221165, 219, 220syl2anc 696 . . . . . . . . . . . 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 6822 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝐺𝑘) → (𝑀...𝑚) = (𝑀...(𝐺𝑘)))
225224imaeq2d 5624 . . . . . . . . . . . . . . . 16 (𝑚 = (𝐺𝑘) → (𝐺 “ (𝑀...𝑚)) = (𝐺 “ (𝑀...(𝐺𝑘))))
226225imaeq2d 5624 . . . . . . . . . . . . . . 15 (𝑚 = (𝐺𝑘) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) = (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))
227226fveq2d 6357 . . . . . . . . . . . . . 14 (𝑚 = (𝐺𝑘) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) = (♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))))
228227fveq2d 6357 . . . . . . . . . . . . 13 (𝑚 = (𝐺𝑘) → (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))))
229228eleq1d 2824 . . . . . . . . . . . 12 (𝑚 = (𝐺𝑘) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ))
230228oveq1d 6829 . . . . . . . . . . . . . 14 (𝑚 = (𝐺𝑘) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴) = ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴))
231230fveq2d 6357 . . . . . . . . . . . . 13 (𝑚 = (𝐺𝑘) → (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)))
232231breq1d 4814 . . . . . . . . . . . 12 (𝑚 = (𝐺𝑘) → ((abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)) < 𝑥))
233229, 232anbi12d 749 . . . . . . . . . . 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 3445 . . . . . . . . . 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 6357 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))) = (♯‘(𝐺 “ (1...𝑘))))
237236, 210eqtrd 2794 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))) = 𝑘)
238237fveq2d 6357 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) = (seq1( + , 𝐻)‘𝑘))
239238eleq1d 2824 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘𝑘) ∈ ℂ))
240238oveq1d 6829 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴) = ((seq1( + , 𝐻)‘𝑘) − 𝐴))
241240fveq2d 6357 . . . . . . . . . . 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 4814 . . . . . . . . . 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 749 . . . . . . . . 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 229 . . . . . . . 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 3107 . . . . . . 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 6353 . . . . . . . . 9 (𝑛 = ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) → (ℤ𝑛) = (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)))
247246raleqdv 3283 . . . . . . . 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 3449 . . . . . . 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 569 . . . . . 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 3169 . . . . 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 202 . . . 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 3124 . . 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 742 . 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 11936 . . 3 ℕ = (ℤ‘1)
255 1zzd 11620 . . 3 (𝜑 → 1 ∈ ℤ)
256 seqex 13017 . . . 4 seq1( + , 𝐻) ∈ V
257256a1i 11 . . 3 (𝜑 → seq1( + , 𝐻) ∈ V)
258 eqidd 2761 . . 3 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘𝑘))
259254, 255, 257, 258clim2 14454 . 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 13017 . . . 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 14616 . . 3 ((𝜑𝑚 ∈ (ℤ‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑚) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
267122, 260, 262, 266clim2 14454 . 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 300 1 (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  cin 3714  wss 3715   class class class wbr 4804  ccnv 5265  ran crn 5267  cres 5268  cima 5269  Fun wfun 6043   Fn wfn 6044  wf 6045  1-1wf1 6046  1-1-ontowf1o 6048  cfv 6049   Isom wiso 6050  (class class class)co 6814  cen 8120  cdom 8121  Fincfn 8123  cc 10146  cr 10147  0cc0 10148  1c1 10149   + caddc 10151  *cxr 10285   < clt 10286  cle 10287  cmin 10478  cn 11232  0cn0 11504  cz 11589  cuz 11899  +crp 12045  ...cfz 12539  seqcseq 13015  chash 13331  abscabs 14193  cli 14434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-xnn0 11576  df-z 11590  df-uz 11900  df-fz 12540  df-seq 13016  df-hash 13332  df-clim 14438
This theorem is referenced by:  isercoll2  14618
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