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Theorem isercoll 15014
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 12253 . . . . . . . . . 10 (ℤ𝑀) ⊆ ℤ
31, 2eqsstri 4000 . . . . . . . . 9 𝑍 ⊆ ℤ
4 isercoll.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶𝑍)
54ffvelrnda 6844 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ 𝑍)
63, 5sseldi 3964 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℤ)
7 nnz 11993 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
87ad2antlr 723 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ∈ ℤ)
9 fzfid 13331 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝑀...𝑚) ∈ Fin)
10 ffun 6511 . . . . . . . . . . . . . . . 16 (𝐺:ℕ⟶𝑍 → Fun 𝐺)
11 funimacnv 6429 . . . . . . . . . . . . . . . 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 4204 . . . . . . . . . . . . . . 15 ((𝑀...𝑚) ∩ ran 𝐺) ⊆ (𝑀...𝑚)
1412, 13eqsstrdi 4020 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚))
1514ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚))
169, 15ssfid 8730 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin)
17 hashcl 13707 . . . . . . . . . . . 12 ((𝐺 “ (𝐺 “ (𝑀...𝑚))) ∈ Fin → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℕ0)
18 nn0z 11994 . . . . . . . . . . . 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 3988 . . . . . . . . . . . . . . . . . . . 20 ℕ ⊆ ℕ
21 isercoll.m . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ ℤ)
22 isercoll.i . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) < (𝐺‘(𝑘 + 1)))
231, 21, 4, 22isercolllem1 15011 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ℕ ⊆ ℕ) → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
2420, 23mpan2 687 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ, (𝐺 “ ℕ)))
25 ffn 6508 . . . . . . . . . . . . . . . . . . . 20 (𝐺:ℕ⟶𝑍𝐺 Fn ℕ)
26 fnresdm 6460 . . . . . . . . . . . . . . . . . . . 20 (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺)
27 isoeq1 7059 . . . . . . . . . . . . . . . . . . . 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 7065 . . . . . . . . . . . . . . . . . 18 (𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) → 𝐺:ℕ–1-1-onto→(𝐺 “ ℕ))
31 f1ocnv 6621 . . . . . . . . . . . . . . . . . 18 (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → 𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ)
32 f1ofun 6611 . . . . . . . . . . . . . . . . . 18 (𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun 𝐺)
3329, 30, 31, 324syl 19 . . . . . . . . . . . . . . . . 17 (𝜑 → Fun 𝐺)
34 df-f1 6354 . . . . . . . . . . . . . . . . 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 12928 . . . . . . . . . . . . . . 15 (1...𝑛) ⊆ ℕ
38 ovex 7178 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ V
3938f1imaen 8560 . . . . . . . . . . . . . . 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 13331 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (1...𝑛) ∈ Fin)
42 enfii 8724 . . . . . . . . . . . . . . . 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 13697 . . . . . . . . . . . . . . 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 11893 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
4847ad2antlr 723 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ∈ ℕ0)
49 hashfz1 13696 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
5048, 49syl 17 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(1...𝑛)) = 𝑛)
5146, 50eqtrd 2856 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (1...𝑛))) = 𝑛)
52 elfznn 12926 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1...𝑛) → 𝑦 ∈ ℕ)
5352adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℕ)
54 zssre 11977 . . . . . . . . . . . . . . . . . . . . . 22 ℤ ⊆ ℝ
553, 54sstri 3975 . . . . . . . . . . . . . . . . . . . . 21 𝑍 ⊆ ℝ
564ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝐺:ℕ⟶𝑍)
57 ffvelrn 6842 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺:ℕ⟶𝑍𝑦 ∈ ℕ) → (𝐺𝑦) ∈ 𝑍)
5856, 52, 57syl2an 595 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ 𝑍)
5955, 58sseldi 3964 . . . . . . . . . . . . . . . . . . . 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 3964 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ∈ ℝ)
62 eluzelz 12242 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (ℤ‘(𝐺𝑛)) → 𝑚 ∈ ℤ)
6362ad2antlr 723 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℤ)
6463zred 12076 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℝ)
65 elfzle2 12901 . . . . . . . . . . . . . . . . . . . . . 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 7068 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑛 < 𝑦 ↔ (𝐺𝑛) < (𝐺𝑦)))
7067, 68, 53, 69syl12anc 832 . . . . . . . . . . . . . . . . . . . . . . 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 11642 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℝ)
7368nnred 11642 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑛 ∈ ℝ)
7472, 73lenltd 10775 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦𝑛 ↔ ¬ 𝑛 < 𝑦))
7559, 61lenltd 10775 . . . . . . . . . . . . . . . . . . . . . 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 12245 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (ℤ‘(𝐺𝑛)) → (𝐺𝑛) ≤ 𝑚)
7978ad2antlr 723 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑛) ≤ 𝑚)
8059, 61, 64, 77, 79letrd 10786 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ≤ 𝑚)
8158, 1eleqtrdi 2923 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺𝑦) ∈ (ℤ𝑀))
82 elfz5 12890 . . . . . . . . . . . . . . . . . . . 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 6509 . . . . . . . . . . . . . . . . . . . 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 6821 . . . . . . . . . . . . . . . . . . 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 3972 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (1...𝑛) ⊆ (𝐺 “ (𝑀...𝑚)))
92 imass2 5959 . . . . . . . . . . . . . . 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 8544 . . . . . . . . . . . . . 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 13730 . . . . . . . . . . . . . 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 5082 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → 𝑛 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))))
100 eluz2 12238 . . . . . . . . . . 11 ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ (ℤ𝑛) ↔ (𝑛 ∈ ℤ ∧ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ ℤ ∧ 𝑛 ≤ (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
1018, 19, 99, 100syl3anbrc 1335 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝐺𝑛))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) ∈ (ℤ𝑛))
102 fveq2 6664 . . . . . . . . . . . . 13 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
103102eleq1d 2897 . . . . . . . . . . . 12 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → ((seq1( + , 𝐻)‘𝑘) ∈ ℂ ↔ (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ))
104102fvoveq1d 7167 . . . . . . . . . . . . 13 (𝑘 = (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) → (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)))
105104breq1d 5068 . . . . . . . . . . . 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 3617 . . . . . . . . . 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 3189 . . . . . . . 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 6664 . . . . . . . . . 10 (𝑗 = (𝐺𝑛) → (ℤ𝑗) = (ℤ‘(𝐺𝑛)))
111110raleqdv 3416 . . . . . . . . 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 3622 . . . . . . . 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 3284 . . . . . 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 11638 . . . . . . . . 9 1 ∈ ℕ
116 ffvelrn 6842 . . . . . . . . 9 ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
1174, 115, 116sylancl 586 . . . . . . . 8 (𝜑 → (𝐺‘1) ∈ 𝑍)
118117, 1eleqtrdi 2923 . . . . . . 7 (𝜑 → (𝐺‘1) ∈ (ℤ𝑀))
119 eluzelz 12242 . . . . . . 7 ((𝐺‘1) ∈ (ℤ𝑀) → (𝐺‘1) ∈ ℤ)
120 eqid 2821 . . . . . . . 8 (ℤ‘(𝐺‘1)) = (ℤ‘(𝐺‘1))
121120rexuz3 14698 . . . . . . 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 13331 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝑀...𝑗) ∈ Fin)
125 funimacnv 6429 . . . . . . . . . . . 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 4204 . . . . . . . . . . 11 ((𝑀...𝑗) ∩ ran 𝐺) ⊆ (𝑀...𝑗)
128126, 127eqsstrdi 4020 . . . . . . . . . 10 (𝜑 → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗))
129128adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗))
130124, 129ssfid 8730 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) → (𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin)
131 hashcl 13707 . . . . . . . 8 ((𝐺 “ (𝐺 “ (𝑀...𝑗))) ∈ Fin → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) ∈ ℕ0)
132 nn0p1nn 11925 . . . . . . . 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 12245 . . . . . . . . . . . . . . 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 11994 . . . . . . . . . . . . . . . 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 12242 . . . . . . . . . . . . . . . 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 12021 . . . . . . . . . . . . . . 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 11895 . . . . . . . . . . . . . . . 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 12307 . . . . . . . . . . . . . . . 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 11642 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℝ)
150146, 149ltnled 10776 . . . . . . . . . . . . 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 12937 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ‘(𝐺𝑘)) → (𝑀...(𝐺𝑘)) ⊆ (𝑀...𝑗))
153 imass2 5959 . . . . . . . . . . . . . 14 ((𝑀...(𝐺𝑘)) ⊆ (𝑀...𝑗) → (𝐺 “ (𝑀...(𝐺𝑘))) ⊆ (𝐺 “ (𝑀...𝑗)))
154 imass2 5959 . . . . . . . . . . . . . 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 8544 . . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ 𝑍)
160159, 1eleqtrdi 2923 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺𝑥) ∈ (ℤ𝑀))
161158, 148ffvelrnd 6845 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺𝑘) ∈ 𝑍)
1623, 161sseldi 3964 . . . . . . . . . . . . . . . . . . . . . 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 12890 . . . . . . . . . . . . . . . . . . . . 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 11631 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ ⊆ ℝ
168 ressxr 10674 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ⊆ ℝ*
169167, 168sstri 3975 . . . . . . . . . . . . . . . . . . . . . 22 ℕ ⊆ ℝ*
170169a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆ ℝ*)
171 imassrn 5934 . . . . . . . . . . . . . . . . . . . . . . 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 6515 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺𝑍)
174173, 55sstrdi 3978 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺 ⊆ ℝ)
175171, 174sstrid 3977 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆ ℝ)
176175, 168sstrdi 3978 . . . . . . . . . . . . . . . . . . . . 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 13808 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 Isom < , < (ℕ, (𝐺 “ ℕ)) ∧ (ℕ ⊆ ℝ* ∧ (𝐺 “ ℕ) ⊆ ℝ*) ∧ (𝑥 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → (𝑥𝑘 ↔ (𝐺𝑥) ≤ (𝐺𝑘)))
180166, 170, 176, 177, 178, 179syl122anc 1371 . . . . . . . . . . . . . . . . . . . 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 6821 . . . . . . . . . . . . . . . . . . 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 12965 . . . . . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝑀...(𝐺𝑘))) = (1...𝑘))
189188imaeq2d 5923 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))) = (𝐺 “ (1...𝑘)))
190189sseq1d 3997 . . . . . . . . . . . . . 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 12928 . . . . . . . . . . . . . . . . . . 19 (1...𝑘) ⊆ ℕ
193 ovex 7178 . . . . . . . . . . . . . . . . . . . 20 (1...𝑘) ∈ V
194193f1imaen 8560 . . . . . . . . . . . . . . . . . . 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 13331 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (1...𝑘) ∈ Fin)
197 enfii 8724 . . . . . . . . . . . . . . . . . . . 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 13697 . . . . . . . . . . . . . . . . . . 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 11893 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
203 hashfz1 13696 . . . . . . . . . . . . . . . . . 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 2856 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (1...𝑘))) = 𝑘)
206205breq1d 5068 . . . . . . . . . . . . . . 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 13730 . . . . . . . . . . . . . . . 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 12242 . . . . . . . . . . . . . 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 12255 . . . . . . . . . . . . 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 858 . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝐺𝑘) → (𝑀...𝑚) = (𝑀...(𝐺𝑘)))
220219imaeq2d 5923 . . . . . . . . . . . . . . . 16 (𝑚 = (𝐺𝑘) → (𝐺 “ (𝑀...𝑚)) = (𝐺 “ (𝑀...(𝐺𝑘))))
221220imaeq2d 5923 . . . . . . . . . . . . . . 15 (𝑚 = (𝐺𝑘) → (𝐺 “ (𝐺 “ (𝑀...𝑚))) = (𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))
222221fveq2d 6668 . . . . . . . . . . . . . 14 (𝑚 = (𝐺𝑘) → (♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚)))) = (♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))))
223222fveq2d 6668 . . . . . . . . . . . . 13 (𝑚 = (𝐺𝑘) → (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))))
224223eleq1d 2897 . . . . . . . . . . . 12 (𝑚 = (𝐺𝑘) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ))
225223fvoveq1d 7167 . . . . . . . . . . . . 13 (𝑚 = (𝐺𝑘) → (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) − 𝐴)))
226225breq1d 5068 . . . . . . . . . . . 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 3617 . . . . . . . . . 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 6668 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))) = (♯‘(𝐺 “ (1...𝑘))))
231230, 205eqtrd 2856 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘))))) = 𝑘)
232231fveq2d 6668 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) = (seq1( + , 𝐻)‘𝑘))
233232eleq1d 2897 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ‘(𝐺‘1))) ∧ 𝑘 ∈ (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1))) → ((seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...(𝐺𝑘)))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘𝑘) ∈ ℂ))
234232fvoveq1d 7167 . . . . . . . . . . 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 5068 . . . . . . . . . 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 3189 . . . . . . 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 6664 . . . . . . . . 9 (𝑛 = ((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1) → (ℤ𝑛) = (ℤ‘((♯‘(𝐺 “ (𝐺 “ (𝑀...𝑗)))) + 1)))
240239raleqdv 3416 . . . . . . . 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 3622 . . . . . . 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 3284 . . . . 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 3197 . . 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 12270 . . 3 ℕ = (ℤ‘1)
248 1zzd 12002 . . 3 (𝜑 → 1 ∈ ℤ)
249 seqex 13361 . . . 4 seq1( + , 𝐻) ∈ V
250249a1i 11 . . 3 (𝜑 → seq1( + , 𝐻) ∈ V)
251 eqidd 2822 . . 3 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘𝑘))
252247, 248, 250, 251clim2 14851 . 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 13361 . . . 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 15013 . . 3 ((𝜑𝑚 ∈ (ℤ‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑚) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (𝐺 “ (𝑀...𝑚))))))
260120, 253, 255, 259clim2 14851 . 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 841   = wceq 1528  wcel 2105  wral 3138  wrex 3139  Vcvv 3495  cdif 3932  cin 3934  wss 3935   class class class wbr 5058  ccnv 5548  ran crn 5550  cres 5551  cima 5552  Fun wfun 6343   Fn wfn 6344  wf 6345  1-1wf1 6346  1-1-ontowf1o 6348  cfv 6349   Isom wiso 6350  (class class class)co 7145  cen 8495  cdom 8496  Fincfn 8498  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529  *cxr 10663   < clt 10664  cle 10665  cmin 10859  cn 11627  0cn0 11886  cz 11970  cuz 12232  +crp 12379  ...cfz 12882  seqcseq 13359  chash 13680  abscabs 14583  cli 14831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-fz 12883  df-seq 13360  df-hash 13681  df-clim 14835
This theorem is referenced by:  isercoll2  15015
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