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Theorem zsumdc 11376
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 995 . . . . . . . 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 2238 . . . . . . . . . . . . 13 (𝑛 = 𝑖 → (𝑛𝐴𝑖𝐴))
3 csbeq1 3060 . . . . . . . . . . . . 13 (𝑛 = 𝑖𝑛 / 𝑘𝐵 = 𝑖 / 𝑘𝐵)
42, 3ifbieq1d 3556 . . . . . . . . . . . 12 (𝑛 = 𝑖 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
54cbvmptv 4096 . . . . . . . . . . 11 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑖 ∈ ℤ ↦ if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
6 simpr 110 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖𝐴)
7 zisum.5 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
87ralrimiva 2550 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
98ad3antrrr 492 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
10 nfcsb1v 3090 . . . . . . . . . . . . . 14 𝑘𝑖 / 𝑘𝐵
1110nfel1 2330 . . . . . . . . . . . . 13 𝑘𝑖 / 𝑘𝐵 ∈ ℂ
12 csbeq1a 3066 . . . . . . . . . . . . . 14 (𝑘 = 𝑖𝐵 = 𝑖 / 𝑘𝐵)
1312eleq1d 2246 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ 𝑖 / 𝑘𝐵 ∈ ℂ))
1411, 13rspc 2835 . . . . . . . . . . . 12 (𝑖𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑖 / 𝑘𝐵 ∈ ℂ))
156, 9, 14sylc 62 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
16 simplr 528 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑚 ∈ ℤ)
17 zisum.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
1817ad2antrr 488 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑀 ∈ ℤ)
19 simpr 110 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑚))
20 zisum.3 . . . . . . . . . . . . 13 (𝜑𝐴𝑍)
21 zisum.1 . . . . . . . . . . . . 13 𝑍 = (ℤ𝑀)
2220, 21sseqtrdi 3203 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ (ℤ𝑀))
2322ad2antrr 488 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑀))
24 zisum.dc . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝑍 DECID 𝑥𝐴)
2521raleqi 2676 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝑍 DECID 𝑥𝐴 ↔ ∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴)
2624, 25sylib 122 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴)
27 eleq1w 2238 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑖 → (𝑥𝐴𝑖𝐴))
2827dcbid 838 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑖 → (DECID 𝑥𝐴DECID 𝑖𝐴))
2928cbvralv 2703 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴 ↔ ∀𝑖 ∈ (ℤ𝑀)DECID 𝑖𝐴)
3026, 29sylib 122 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑖 ∈ (ℤ𝑀)DECID 𝑖𝐴)
3130r19.21bi 2565 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
3231adantlr 477 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℤ) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
3332adantlr 477 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
3433adantlr 477 . . . . . . . . . . . 12 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
35 simp-4l 541 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → 𝜑)
36 simpr 110 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → ¬ 𝑖 ∈ (ℤ𝑀))
3722ssneld 3157 . . . . . . . . . . . . . . 15 (𝜑 → (¬ 𝑖 ∈ (ℤ𝑀) → ¬ 𝑖𝐴))
3835, 36, 37sylc 62 . . . . . . . . . . . . . 14 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → ¬ 𝑖𝐴)
3938olcd 734 . . . . . . . . . . . . 13 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → (𝑖𝐴 ∨ ¬ 𝑖𝐴))
40 df-dc 835 . . . . . . . . . . . . 13 (DECID 𝑖𝐴 ↔ (𝑖𝐴 ∨ ¬ 𝑖𝐴))
4139, 40sylibr 134 . . . . . . . . . . . 12 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
42 eluzelz 9526 . . . . . . . . . . . . . 14 (𝑖 ∈ (ℤ𝑚) → 𝑖 ∈ ℤ)
43 eluzdc 9599 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) → DECID 𝑖 ∈ (ℤ𝑀))
4418, 42, 43syl2an 289 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) → DECID 𝑖 ∈ (ℤ𝑀))
45 exmiddc 836 . . . . . . . . . . . . 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 798 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) → DECID 𝑖𝐴)
485, 15, 16, 18, 19, 23, 47, 33sumrbdc 11371 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
4948biimpd 144 . . . . . . . . 9 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
5049expimpd 363 . . . . . . . 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 2594 . . . . . 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 9536 . . . . . . . . . . . . . 14 (ℤ𝑀) ⊆ ℤ
5422, 53sstrdi 3167 . . . . . . . . . . . . 13 (𝜑𝐴 ⊆ ℤ)
5554ad2antrr 488 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ⊆ ℤ)
56 1zzd 9269 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 1 ∈ ℤ)
57 simplr 528 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ)
5857nnzd 9363 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℤ)
5956, 58fzfigd 10417 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ∈ Fin)
60 simpr 110 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑓:(1...𝑚)–1-1-onto𝐴)
61 f1oeng 6751 . . . . . . . . . . . . . . 15 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
6259, 60, 61syl2anc 411 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
6362ensymd 6777 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ≈ (1...𝑚))
64 enfii 6868 . . . . . . . . . . . . 13 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
6559, 63, 64syl2anc 411 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ∈ Fin)
66 zfz1iso 10805 . . . . . . . . . . . 12 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
6755, 65, 66syl2anc 411 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
68 simpr 110 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖𝐴)
698ad3antrrr 492 . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
7271adantlr 477 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
73 breq1 4003 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴)))
74 fveq2 5511 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
7574csbeq1d 3064 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
76 csbco 3067 . . . . . . . . . . . . . . . . . 18 (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵
7775, 76eqtr4di 2228 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
7873, 77ifbieq1d 3556 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵, 0))
7978cbvmptv 4096 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵, 0))
80 eqid 2177 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↦ if(𝑗𝑚, (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗𝑚, (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵, 0))
81 simplr 528 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
8217ad2antrr 488 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
8322ad2antrr 488 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑀))
8460adantrr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
85 simprr 531 . . . . . . . . . . . . . . 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 11373 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (1...𝑚) ∈ Fin)
8887, 84fihasheqf1od 10753 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘(1...𝑚)) = (♯‘𝐴))
8981nnnn0d 9218 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ0)
90 hashfz1 10747 . . . . . . . . . . . . . . . . . . . . 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 2212 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘𝐴) = 𝑚)
9392breq2d 4012 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (𝑛 ≤ (♯‘𝐴) ↔ 𝑛𝑚))
9493ifbid 3555 . . . . . . . . . . . . . . . . 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 4091 . . . . . . . . . . . . . . . 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 10439 . . . . . . . . . . . . . . 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 5513 . . . . . . . . . . . . . 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 4026 . . . . . . . . . . . . 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 375 . . . . . . . . . . . 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 1819 . . . . . . . . . . 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 4004 . . . . . . . . . 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 157 . . . . . . . . 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 363 . . . . . . . 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 1819 . . . . . . 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 2594 . . . . . 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 717 . . . . 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 276 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
10922adantr 276 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → 𝐴 ⊆ (ℤ𝑀))
110 eleq1w 2238 . . . . . . . . . . . 12 (𝑥 = 𝑗 → (𝑥𝐴𝑗𝐴))
111110dcbid 838 . . . . . . . . . . 11 (𝑥 = 𝑗 → (DECID 𝑥𝐴DECID 𝑗𝐴))
112111cbvralv 2703 . . . . . . . . . 10 (∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴 ↔ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
11326, 112sylib 122 . . . . . . . . 9 (𝜑 → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
114113adantr 276 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
115 simpr 110 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
116 fveq2 5511 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
117116sseq2d 3185 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑀)))
118116raleqdv 2678 . . . . . . . . . 10 (𝑚 = 𝑀 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴))
119 seqeq1 10434 . . . . . . . . . . 11 (𝑚 = 𝑀 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))))
120119breq1d 4010 . . . . . . . . . 10 (𝑚 = 𝑀 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
121117, 118, 1203anbi123d 1312 . . . . . . . . 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 2841 . . . . . . . 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 1240 . . . . . . 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 733 . . . . . 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 115 . . . . 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 129 . . . 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 9526 . . . . . . . 8 (𝑎 ∈ (ℤ𝑀) → 𝑎 ∈ ℤ)
128 simpr 110 . . . . . . . . . 10 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑎𝐴) → 𝑎𝐴)
1298ad2antrr 488 . . . . . . . . . 10 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑎𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
130 nfcsb1v 3090 . . . . . . . . . . . 12 𝑘𝑎 / 𝑘𝐵
131130nfel1 2330 . . . . . . . . . . 11 𝑘𝑎 / 𝑘𝐵 ∈ ℂ
132 csbeq1a 3066 . . . . . . . . . . . 12 (𝑘 = 𝑎𝐵 = 𝑎 / 𝑘𝐵)
133132eleq1d 2246 . . . . . . . . . . 11 (𝑘 = 𝑎 → (𝐵 ∈ ℂ ↔ 𝑎 / 𝑘𝐵 ∈ ℂ))
134131, 133rspc 2835 . . . . . . . . . 10 (𝑎𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑎 / 𝑘𝐵 ∈ ℂ))
135128, 129, 134sylc 62 . . . . . . . . 9 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑎𝐴) → 𝑎 / 𝑘𝐵 ∈ ℂ)
136 0cnd 7941 . . . . . . . . 9 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ ¬ 𝑎𝐴) → 0 ∈ ℂ)
137 eleq1w 2238 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
138137dcbid 838 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (DECID 𝑥𝐴DECID 𝑎𝐴))
139138cbvralv 2703 . . . . . . . . . . 11 (∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴 ↔ ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
14026, 139sylib 122 . . . . . . . . . 10 (𝜑 → ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
141140r19.21bi 2565 . . . . . . . . 9 ((𝜑𝑎 ∈ (ℤ𝑀)) → DECID 𝑎𝐴)
142135, 136, 141ifcldadc 3563 . . . . . . . 8 ((𝜑𝑎 ∈ (ℤ𝑀)) → if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0) ∈ ℂ)
143 eleq1w 2238 . . . . . . . . . 10 (𝑛 = 𝑎 → (𝑛𝐴𝑎𝐴))
144 csbeq1 3060 . . . . . . . . . 10 (𝑛 = 𝑎𝑛 / 𝑘𝐵 = 𝑎 / 𝑘𝐵)
145143, 144ifbieq1d 3556 . . . . . . . . 9 (𝑛 = 𝑎 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0))
146 eqid 2177 . . . . . . . . 9 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
147145, 146fvmptg 5588 . . . . . . . 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 594 . . . . . . 7 ((𝜑𝑎 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑎) = if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0))
149148, 142eqeltrd 2254 . . . . . 6 ((𝜑𝑎 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑎) ∈ ℂ)
150 simpr 110 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ (ℤ𝑀))
15153, 150sselid 3153 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ ℤ)
152 vex 2740 . . . . . . . . . 10 𝑗 ∈ V
153 nfv 1528 . . . . . . . . . . 11 𝑘 𝑗𝐴
154 nfcsb1v 3090 . . . . . . . . . . 11 𝑘𝑗 / 𝑘𝐵
155 nfcv 2319 . . . . . . . . . . 11 𝑘0
156153, 154, 155nfif 3562 . . . . . . . . . 10 𝑘if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0)
157 eleq1w 2238 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑘𝐴𝑗𝐴))
158 csbeq1a 3066 . . . . . . . . . . 11 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
159157, 158ifbieq1d 3556 . . . . . . . . . 10 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
160152, 156, 159csbief 3101 . . . . . . . . 9 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0)
161 simpr 110 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ 𝑗𝐴) → 𝑗𝐴)
1628ad2antrr 488 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ 𝑗𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
163154nfel1 2330 . . . . . . . . . . . 12 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
164158eleq1d 2246 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
165163, 164rspc 2835 . . . . . . . . . . 11 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
166161, 162, 165sylc 62 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ 𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
167 0cnd 7941 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ ¬ 𝑗𝐴) → 0 ∈ ℂ)
168113r19.21bi 2565 . . . . . . . . . 10 ((𝜑𝑗 ∈ (ℤ𝑀)) → DECID 𝑗𝐴)
169166, 167, 168ifcldadc 3563 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ𝑀)) → if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0) ∈ ℂ)
170160, 169eqeltrid 2264 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
171 nfcv 2319 . . . . . . . . . . 11 𝑛if(𝑘𝐴, 𝐵, 0)
172 nfv 1528 . . . . . . . . . . . 12 𝑘 𝑛𝐴
173 nfcsb1v 3090 . . . . . . . . . . . 12 𝑘𝑛 / 𝑘𝐵
174172, 173, 155nfif 3562 . . . . . . . . . . 11 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
175 eleq1w 2238 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑘𝐴𝑛𝐴))
176 csbeq1a 3066 . . . . . . . . . . . 12 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
177175, 176ifbieq1d 3556 . . . . . . . . . . 11 (𝑘 = 𝑛 → if(𝑘𝐴, 𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
178171, 174, 177cbvmpt 4095 . . . . . . . . . 10 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
179178eqcomi 2181 . . . . . . . . 9 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
180179fvmpts 5590 . . . . . . . 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 411 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
182150, 21eleqtrrdi 2271 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗𝑍)
183 zisum.4 . . . . . . . . . 10 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
184183ralrimiva 2550 . . . . . . . . 9 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
185184adantr 276 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
186 nfcsb1v 3090 . . . . . . . . . 10 𝑘𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
187186nfeq2 2331 . . . . . . . . 9 𝑘(𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
188 fveq2 5511 . . . . . . . . . 10 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
189 csbeq1a 3066 . . . . . . . . . 10 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 0) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
190188, 189eqeq12d 2192 . . . . . . . . 9 (𝑘 = 𝑗 → ((𝐹𝑘) = if(𝑘𝐴, 𝐵, 0) ↔ (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)))
191187, 190rspc 2835 . . . . . . . 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 2213 . . . . . 6 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = (𝐹𝑗))
194 addcl 7927 . . . . . . 7 ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑎 + 𝑏) ∈ ℂ)
195194adantl 277 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ)) → (𝑎 + 𝑏) ∈ ℂ)
19617, 149, 193, 195seq3feq 10458 . . . . 5 (𝜑 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑀( + , 𝐹))
197196breq1d 4010 . . . 4 (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , 𝐹) ⇝ 𝑥))
198126, 197bitrd 188 . . 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 5195 . 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 11346 . 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 5220 . 2 ( ⇝ ‘seq𝑀( + , 𝐹)) = (℩𝑥seq𝑀( + , 𝐹) ⇝ 𝑥)
202199, 200, 2013eqtr4g 2235 1 (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  csb 3057  wss 3129  ifcif 3534   class class class wbr 4000  cmpt 4061  cio 5172  1-1-ontowf1o 5211  cfv 5212   Isom wiso 5213  (class class class)co 5869  cen 6732  Fincfn 6734  cc 7800  0cc0 7802  1c1 7803   + caddc 7805   < clt 7982  cle 7983  cn 8908  0cn0 9165  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431  chash 10739  cli 11270  Σcsu 11345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346
This theorem is referenced by:  isum  11377  sum0  11380  isumz  11381  isumss  11383  fsumsersdc  11387
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