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Theorem zsumdc 11165
Description: Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.)
Hypotheses
Ref Expression
zisum.1 𝑍 = (ℤ𝑀)
zisum.2 (𝜑𝑀 ∈ ℤ)
zisum.3 (𝜑𝐴𝑍)
zisum.4 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
zisum.dc (𝜑 → ∀𝑥𝑍 DECID 𝑥𝐴)
zisum.5 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
zsumdc (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
Distinct variable groups:   𝐴,𝑘,𝑥   𝑥,𝐵   𝑘,𝐹,𝑥   𝑥,𝑀   𝑘,𝑍,𝑥   𝜑,𝑘,𝑥
Allowed substitution hints:   𝐵(𝑘)   𝑀(𝑘)

Proof of Theorem zsumdc
Dummy variables 𝑎 𝑏 𝑗 𝑛 𝑓 𝑔 𝑖 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpb 979 . . . . . . . 8 ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
2 eleq1w 2200 . . . . . . . . . . . . 13 (𝑛 = 𝑖 → (𝑛𝐴𝑖𝐴))
3 csbeq1 3006 . . . . . . . . . . . . 13 (𝑛 = 𝑖𝑛 / 𝑘𝐵 = 𝑖 / 𝑘𝐵)
42, 3ifbieq1d 3494 . . . . . . . . . . . 12 (𝑛 = 𝑖 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
54cbvmptv 4024 . . . . . . . . . . 11 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑖 ∈ ℤ ↦ if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
6 simpr 109 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖𝐴)
7 zisum.5 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
87ralrimiva 2505 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
98ad3antrrr 483 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
10 nfcsb1v 3035 . . . . . . . . . . . . . 14 𝑘𝑖 / 𝑘𝐵
1110nfel1 2292 . . . . . . . . . . . . 13 𝑘𝑖 / 𝑘𝐵 ∈ ℂ
12 csbeq1a 3012 . . . . . . . . . . . . . 14 (𝑘 = 𝑖𝐵 = 𝑖 / 𝑘𝐵)
1312eleq1d 2208 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ 𝑖 / 𝑘𝐵 ∈ ℂ))
1411, 13rspc 2783 . . . . . . . . . . . 12 (𝑖𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑖 / 𝑘𝐵 ∈ ℂ))
156, 9, 14sylc 62 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
16 simplr 519 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑚 ∈ ℤ)
17 zisum.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
1817ad2antrr 479 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑀 ∈ ℤ)
19 simpr 109 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑚))
20 zisum.3 . . . . . . . . . . . . 13 (𝜑𝐴𝑍)
21 zisum.1 . . . . . . . . . . . . 13 𝑍 = (ℤ𝑀)
2220, 21sseqtrdi 3145 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ (ℤ𝑀))
2322ad2antrr 479 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑀))
24 zisum.dc . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝑍 DECID 𝑥𝐴)
2521raleqi 2630 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝑍 DECID 𝑥𝐴 ↔ ∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴)
2624, 25sylib 121 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴)
27 eleq1w 2200 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑖 → (𝑥𝐴𝑖𝐴))
2827dcbid 823 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑖 → (DECID 𝑥𝐴DECID 𝑖𝐴))
2928cbvralv 2654 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴 ↔ ∀𝑖 ∈ (ℤ𝑀)DECID 𝑖𝐴)
3026, 29sylib 121 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑖 ∈ (ℤ𝑀)DECID 𝑖𝐴)
3130r19.21bi 2520 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
3231adantlr 468 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℤ) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
3332adantlr 468 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
3433adantlr 468 . . . . . . . . . . . 12 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
35 simp-4l 530 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → 𝜑)
36 simpr 109 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → ¬ 𝑖 ∈ (ℤ𝑀))
3722ssneld 3099 . . . . . . . . . . . . . . 15 (𝜑 → (¬ 𝑖 ∈ (ℤ𝑀) → ¬ 𝑖𝐴))
3835, 36, 37sylc 62 . . . . . . . . . . . . . 14 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → ¬ 𝑖𝐴)
3938olcd 723 . . . . . . . . . . . . 13 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → (𝑖𝐴 ∨ ¬ 𝑖𝐴))
40 df-dc 820 . . . . . . . . . . . . 13 (DECID 𝑖𝐴 ↔ (𝑖𝐴 ∨ ¬ 𝑖𝐴))
4139, 40sylibr 133 . . . . . . . . . . . 12 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
42 eluzelz 9347 . . . . . . . . . . . . . 14 (𝑖 ∈ (ℤ𝑚) → 𝑖 ∈ ℤ)
43 eluzdc 9416 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) → DECID 𝑖 ∈ (ℤ𝑀))
4418, 42, 43syl2an 287 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) → DECID 𝑖 ∈ (ℤ𝑀))
45 exmiddc 821 . . . . . . . . . . . . 13 (DECID 𝑖 ∈ (ℤ𝑀) → (𝑖 ∈ (ℤ𝑀) ∨ ¬ 𝑖 ∈ (ℤ𝑀)))
4644, 45syl 14 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) → (𝑖 ∈ (ℤ𝑀) ∨ ¬ 𝑖 ∈ (ℤ𝑀)))
4734, 41, 46mpjaodan 787 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) → DECID 𝑖𝐴)
485, 15, 16, 18, 19, 23, 47, 33sumrbdc 11160 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
4948biimpd 143 . . . . . . . . 9 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
5049expimpd 360 . . . . . . . 8 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
511, 50syl5 32 . . . . . . 7 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
5251rexlimdva 2549 . . . . . 6 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
53 uzssz 9357 . . . . . . . . . . . . . 14 (ℤ𝑀) ⊆ ℤ
5422, 53sstrdi 3109 . . . . . . . . . . . . 13 (𝜑𝐴 ⊆ ℤ)
5554ad2antrr 479 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ⊆ ℤ)
56 1zzd 9093 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 1 ∈ ℤ)
57 simplr 519 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ)
5857nnzd 9184 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℤ)
5956, 58fzfigd 10216 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ∈ Fin)
60 simpr 109 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑓:(1...𝑚)–1-1-onto𝐴)
61 f1oeng 6651 . . . . . . . . . . . . . . 15 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
6259, 60, 61syl2anc 408 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
6362ensymd 6677 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ≈ (1...𝑚))
64 enfii 6768 . . . . . . . . . . . . 13 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
6559, 63, 64syl2anc 408 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ∈ Fin)
66 zfz1iso 10596 . . . . . . . . . . . 12 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
6755, 65, 66syl2anc 408 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
68 simpr 109 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖𝐴)
698ad3antrrr 483 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
7068, 69, 14sylc 62 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
7131adantlr 468 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
7271adantlr 468 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
73 breq1 3932 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴)))
74 fveq2 5421 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
7574csbeq1d 3010 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
76 csbco 3013 . . . . . . . . . . . . . . . . . 18 (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵
7775, 76eqtr4di 2190 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
7873, 77ifbieq1d 3494 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵, 0))
7978cbvmptv 4024 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵, 0))
80 eqid 2139 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↦ if(𝑗𝑚, (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗𝑚, (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵, 0))
81 simplr 519 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
8217ad2antrr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
8322ad2antrr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑀))
8460adantrr 470 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
85 simprr 521 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
865, 70, 72, 79, 80, 81, 82, 83, 84, 85summodclem2a 11162 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
8759adantrr 470 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (1...𝑚) ∈ Fin)
8887, 84fihasheqf1od 10548 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘(1...𝑚)) = (♯‘𝐴))
8981nnnn0d 9042 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ0)
90 hashfz1 10541 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
9189, 90syl 14 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘(1...𝑚)) = 𝑚)
9288, 91eqtr3d 2174 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘𝐴) = 𝑚)
9392breq2d 3941 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (𝑛 ≤ (♯‘𝐴) ↔ 𝑛𝑚))
9493ifbid 3493 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
9594mpteq2dv 4019 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))
9695seqeq3d 10238 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))))
9796fveq1d 5423 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
9886, 97breqtrd 3954 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
9998expr 372 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
10099exlimdv 1791 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
10167, 100mpd 13 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
102 breq2 3933 . . . . . . . . . 10 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
103101, 102syl5ibrcom 156 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
104103expimpd 360 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
105104exlimdv 1791 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
106105rexlimdva 2549 . . . . . 6 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
10752, 106jaod 706 . . . . 5 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
10817adantr 274 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
10922adantr 274 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → 𝐴 ⊆ (ℤ𝑀))
110 eleq1w 2200 . . . . . . . . . . . 12 (𝑥 = 𝑗 → (𝑥𝐴𝑗𝐴))
111110dcbid 823 . . . . . . . . . . 11 (𝑥 = 𝑗 → (DECID 𝑥𝐴DECID 𝑗𝐴))
112111cbvralv 2654 . . . . . . . . . 10 (∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴 ↔ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
11326, 112sylib 121 . . . . . . . . 9 (𝜑 → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
114113adantr 274 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
115 simpr 109 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
116 fveq2 5421 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
117116sseq2d 3127 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑀)))
118116raleqdv 2632 . . . . . . . . . 10 (𝑚 = 𝑀 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴))
119 seqeq1 10233 . . . . . . . . . . 11 (𝑚 = 𝑀 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))))
120119breq1d 3939 . . . . . . . . . 10 (𝑚 = 𝑀 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
121117, 118, 1203anbi123d 1290 . . . . . . . . 9 (𝑚 = 𝑀 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)))
122121rspcev 2789 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ (𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
123108, 109, 114, 115, 122syl13anc 1218 . . . . . . 7 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
124123orcd 722 . . . . . 6 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
125124ex 114 . . . . 5 (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))))
126107, 125impbid 128 . . . 4 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
127 eluzelz 9347 . . . . . . . 8 (𝑎 ∈ (ℤ𝑀) → 𝑎 ∈ ℤ)
128 simpr 109 . . . . . . . . . 10 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑎𝐴) → 𝑎𝐴)
1298ad2antrr 479 . . . . . . . . . 10 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑎𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
130 nfcsb1v 3035 . . . . . . . . . . . 12 𝑘𝑎 / 𝑘𝐵
131130nfel1 2292 . . . . . . . . . . 11 𝑘𝑎 / 𝑘𝐵 ∈ ℂ
132 csbeq1a 3012 . . . . . . . . . . . 12 (𝑘 = 𝑎𝐵 = 𝑎 / 𝑘𝐵)
133132eleq1d 2208 . . . . . . . . . . 11 (𝑘 = 𝑎 → (𝐵 ∈ ℂ ↔ 𝑎 / 𝑘𝐵 ∈ ℂ))
134131, 133rspc 2783 . . . . . . . . . 10 (𝑎𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑎 / 𝑘𝐵 ∈ ℂ))
135128, 129, 134sylc 62 . . . . . . . . 9 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑎𝐴) → 𝑎 / 𝑘𝐵 ∈ ℂ)
136 0cnd 7771 . . . . . . . . 9 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ ¬ 𝑎𝐴) → 0 ∈ ℂ)
137 eleq1w 2200 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
138137dcbid 823 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (DECID 𝑥𝐴DECID 𝑎𝐴))
139138cbvralv 2654 . . . . . . . . . . 11 (∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴 ↔ ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
14026, 139sylib 121 . . . . . . . . . 10 (𝜑 → ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
141140r19.21bi 2520 . . . . . . . . 9 ((𝜑𝑎 ∈ (ℤ𝑀)) → DECID 𝑎𝐴)
142135, 136, 141ifcldadc 3501 . . . . . . . 8 ((𝜑𝑎 ∈ (ℤ𝑀)) → if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0) ∈ ℂ)
143 eleq1w 2200 . . . . . . . . . 10 (𝑛 = 𝑎 → (𝑛𝐴𝑎𝐴))
144 csbeq1 3006 . . . . . . . . . 10 (𝑛 = 𝑎𝑛 / 𝑘𝐵 = 𝑎 / 𝑘𝐵)
145143, 144ifbieq1d 3494 . . . . . . . . 9 (𝑛 = 𝑎 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0))
146 eqid 2139 . . . . . . . . 9 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
147145, 146fvmptg 5497 . . . . . . . 8 ((𝑎 ∈ ℤ ∧ if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0) ∈ ℂ) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑎) = if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0))
148127, 142, 147syl2an2 583 . . . . . . 7 ((𝜑𝑎 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑎) = if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0))
149148, 142eqeltrd 2216 . . . . . 6 ((𝜑𝑎 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑎) ∈ ℂ)
150 simpr 109 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ (ℤ𝑀))
15153, 150sseldi 3095 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ ℤ)
152 vex 2689 . . . . . . . . . 10 𝑗 ∈ V
153 nfv 1508 . . . . . . . . . . 11 𝑘 𝑗𝐴
154 nfcsb1v 3035 . . . . . . . . . . 11 𝑘𝑗 / 𝑘𝐵
155 nfcv 2281 . . . . . . . . . . 11 𝑘0
156153, 154, 155nfif 3500 . . . . . . . . . 10 𝑘if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0)
157 eleq1w 2200 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑘𝐴𝑗𝐴))
158 csbeq1a 3012 . . . . . . . . . . 11 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
159157, 158ifbieq1d 3494 . . . . . . . . . 10 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
160152, 156, 159csbief 3044 . . . . . . . . 9 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0)
161 simpr 109 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ 𝑗𝐴) → 𝑗𝐴)
1628ad2antrr 479 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ 𝑗𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
163154nfel1 2292 . . . . . . . . . . . 12 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
164158eleq1d 2208 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
165163, 164rspc 2783 . . . . . . . . . . 11 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
166161, 162, 165sylc 62 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ 𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
167 0cnd 7771 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ ¬ 𝑗𝐴) → 0 ∈ ℂ)
168113r19.21bi 2520 . . . . . . . . . 10 ((𝜑𝑗 ∈ (ℤ𝑀)) → DECID 𝑗𝐴)
169166, 167, 168ifcldadc 3501 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ𝑀)) → if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0) ∈ ℂ)
170160, 169eqeltrid 2226 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
171 nfcv 2281 . . . . . . . . . . 11 𝑛if(𝑘𝐴, 𝐵, 0)
172 nfv 1508 . . . . . . . . . . . 12 𝑘 𝑛𝐴
173 nfcsb1v 3035 . . . . . . . . . . . 12 𝑘𝑛 / 𝑘𝐵
174172, 173, 155nfif 3500 . . . . . . . . . . 11 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
175 eleq1w 2200 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑘𝐴𝑛𝐴))
176 csbeq1a 3012 . . . . . . . . . . . 12 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
177175, 176ifbieq1d 3494 . . . . . . . . . . 11 (𝑘 = 𝑛 → if(𝑘𝐴, 𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
178171, 174, 177cbvmpt 4023 . . . . . . . . . 10 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
179178eqcomi 2143 . . . . . . . . 9 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
180179fvmpts 5499 . . . . . . . 8 ((𝑗 ∈ ℤ ∧ 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) ∈ ℂ) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
181151, 170, 180syl2anc 408 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
182150, 21eleqtrrdi 2233 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗𝑍)
183 zisum.4 . . . . . . . . . 10 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
184183ralrimiva 2505 . . . . . . . . 9 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
185184adantr 274 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
186 nfcsb1v 3035 . . . . . . . . . 10 𝑘𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
187186nfeq2 2293 . . . . . . . . 9 𝑘(𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
188 fveq2 5421 . . . . . . . . . 10 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
189 csbeq1a 3012 . . . . . . . . . 10 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 0) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
190188, 189eqeq12d 2154 . . . . . . . . 9 (𝑘 = 𝑗 → ((𝐹𝑘) = if(𝑘𝐴, 𝐵, 0) ↔ (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)))
191187, 190rspc 2783 . . . . . . . 8 (𝑗𝑍 → (∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0) → (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)))
192182, 185, 191sylc 62 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ𝑀)) → (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
193181, 192eqtr4d 2175 . . . . . 6 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = (𝐹𝑗))
194 addcl 7757 . . . . . . 7 ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑎 + 𝑏) ∈ ℂ)
195194adantl 275 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ)) → (𝑎 + 𝑏) ∈ ℂ)
19617, 149, 193, 195seq3feq 10257 . . . . 5 (𝜑 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑀( + , 𝐹))
197196breq1d 3939 . . . 4 (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , 𝐹) ⇝ 𝑥))
198126, 197bitrd 187 . . 3 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))) ↔ seq𝑀( + , 𝐹) ⇝ 𝑥))
199198iotabidv 5109 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))) = (℩𝑥seq𝑀( + , 𝐹) ⇝ 𝑥))
200 df-sumdc 11135 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
201 df-fv 5131 . 2 ( ⇝ ‘seq𝑀( + , 𝐹)) = (℩𝑥seq𝑀( + , 𝐹) ⇝ 𝑥)
202199, 200, 2013eqtr4g 2197 1 (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697  DECID wdc 819  w3a 962   = wceq 1331  wex 1468  wcel 1480  wral 2416  wrex 2417  csb 3003  wss 3071  ifcif 3474   class class class wbr 3929  cmpt 3989  cio 5086  1-1-ontowf1o 5122  cfv 5123   Isom wiso 5124  (class class class)co 5774  cen 6632  Fincfn 6634  cc 7630  0cc0 7632  1c1 7633   + caddc 7635   < clt 7812  cle 7813  cn 8732  0cn0 8989  cz 9066  cuz 9338  ...cfz 9802  seqcseq 10230  chash 10533  cli 11059  Σcsu 11134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-seqfrec 10231  df-exp 10305  df-ihash 10534  df-cj 10626  df-rsqrt 10782  df-abs 10783  df-clim 11060  df-sumdc 11135
This theorem is referenced by:  isum  11166  sum0  11169  isumz  11170  isumss  11172  fsumsersdc  11176
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