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Theorem zisum 10736
 Description: Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.)
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
zisum (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹, ℂ)))
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 zisum
Dummy variables 𝑎 𝑏 𝑗 𝑛 𝑓 𝑔 𝑖 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpb 941 . . . . . . . 8 ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
2 eleq1w 2148 . . . . . . . . . . . . 13 (𝑛 = 𝑖 → (𝑛𝐴𝑖𝐴))
3 csbeq1 2934 . . . . . . . . . . . . 13 (𝑛 = 𝑖𝑛 / 𝑘𝐵 = 𝑖 / 𝑘𝐵)
42, 3ifbieq1d 3409 . . . . . . . . . . . 12 (𝑛 = 𝑖 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
54cbvmptv 3926 . . . . . . . . . . 11 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑖 ∈ ℤ ↦ if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
6 simpr 108 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖𝐴)
7 zisum.5 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
87ralrimiva 2446 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
98ad3antrrr 476 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
10 nfcsb1v 2961 . . . . . . . . . . . . . 14 𝑘𝑖 / 𝑘𝐵
1110nfel1 2239 . . . . . . . . . . . . 13 𝑘𝑖 / 𝑘𝐵 ∈ ℂ
12 csbeq1a 2939 . . . . . . . . . . . . . 14 (𝑘 = 𝑖𝐵 = 𝑖 / 𝑘𝐵)
1312eleq1d 2156 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ 𝑖 / 𝑘𝐵 ∈ ℂ))
1411, 13rspc 2716 . . . . . . . . . . . 12 (𝑖𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑖 / 𝑘𝐵 ∈ ℂ))
156, 9, 14sylc 61 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
16 simplr 497 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑚 ∈ ℤ)
17 zisum.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
1817ad2antrr 472 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑀 ∈ ℤ)
19 simpr 108 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑚))
20 zisum.3 . . . . . . . . . . . . 13 (𝜑𝐴𝑍)
21 zisum.1 . . . . . . . . . . . . 13 𝑍 = (ℤ𝑀)
2220, 21syl6sseq 3070 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ (ℤ𝑀))
2322ad2antrr 472 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑀))
24 zisum.dc . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝑍 DECID 𝑥𝐴)
2521raleqi 2566 . . . . . . . . . . . . . . . . . 18 (∀𝑥𝑍 DECID 𝑥𝐴 ↔ ∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴)
2624, 25sylib 120 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴)
27 eleq1w 2148 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑖 → (𝑥𝐴𝑖𝐴))
2827dcbid 786 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑖 → (DECID 𝑥𝐴DECID 𝑖𝐴))
2928cbvralv 2590 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴 ↔ ∀𝑖 ∈ (ℤ𝑀)DECID 𝑖𝐴)
3026, 29sylib 120 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑖 ∈ (ℤ𝑀)DECID 𝑖𝐴)
3130r19.21bi 2461 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
3231adantlr 461 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℤ) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
3332adantlr 461 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
3433adantlr 461 . . . . . . . . . . . 12 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
35 simp-4l 508 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → 𝜑)
36 simpr 108 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → ¬ 𝑖 ∈ (ℤ𝑀))
3722ssneld 3025 . . . . . . . . . . . . . . 15 (𝜑 → (¬ 𝑖 ∈ (ℤ𝑀) → ¬ 𝑖𝐴))
3835, 36, 37sylc 61 . . . . . . . . . . . . . 14 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → ¬ 𝑖𝐴)
3938olcd 688 . . . . . . . . . . . . 13 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → (𝑖𝐴 ∨ ¬ 𝑖𝐴))
40 df-dc 781 . . . . . . . . . . . . 13 (DECID 𝑖𝐴 ↔ (𝑖𝐴 ∨ ¬ 𝑖𝐴))
4139, 40sylibr 132 . . . . . . . . . . . 12 (((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) ∧ ¬ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
42 eluzelz 8997 . . . . . . . . . . . . . 14 (𝑖 ∈ (ℤ𝑚) → 𝑖 ∈ ℤ)
43 eluzdc 9066 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) → DECID 𝑖 ∈ (ℤ𝑀))
4418, 42, 43syl2an 283 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) → DECID 𝑖 ∈ (ℤ𝑀))
45 exmiddc 782 . . . . . . . . . . . . 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 747 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖 ∈ (ℤ𝑚)) → DECID 𝑖𝐴)
485, 15, 16, 18, 19, 23, 47, 33isumrb 10730 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
4948biimpd 142 . . . . . . . . 9 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
5049expimpd 355 . . . . . . . 8 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
511, 50syl5 32 . . . . . . 7 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
5251rexlimdva 2489 . . . . . 6 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
53 uzssz 9007 . . . . . . . . . . . . . 14 (ℤ𝑀) ⊆ ℤ
5422, 53syl6ss 3035 . . . . . . . . . . . . 13 (𝜑𝐴 ⊆ ℤ)
5554ad2antrr 472 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ⊆ ℤ)
56 1zzd 8747 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 1 ∈ ℤ)
57 simplr 497 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ)
5857nnzd 8837 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℤ)
5956, 58fzfigd 9803 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ∈ Fin)
60 simpr 108 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑓:(1...𝑚)–1-1-onto𝐴)
61 f1oeng 6454 . . . . . . . . . . . . . . 15 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
6259, 60, 61syl2anc 403 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
6362ensymd 6480 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ≈ (1...𝑚))
64 enfii 6570 . . . . . . . . . . . . 13 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
6559, 63, 64syl2anc 403 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ∈ Fin)
66 zfz1iso 10210 . . . . . . . . . . . 12 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
6755, 65, 66syl2anc 403 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
68 simpr 108 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖𝐴)
698ad3antrrr 476 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
7068, 69, 14sylc 61 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
7131adantlr 461 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
7271adantlr 461 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ (ℤ𝑀)) → DECID 𝑖𝐴)
73 breq1 3840 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴)))
74 fveq2 5289 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
7574csbeq1d 2937 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
76 csbco 2940 . . . . . . . . . . . . . . . . . 18 (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵
7775, 76syl6eqr 2138 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
7873, 77ifbieq1d 3409 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵, 0))
7978cbvmptv 3926 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵, 0))
80 eqid 2088 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↦ if(𝑗𝑚, (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗𝑚, (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵, 0))
81 simplr 497 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
8217ad2antrr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
8322ad2antrr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑀))
8460adantrr 463 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
85 simprr 499 . . . . . . . . . . . . . . 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, 85isummolem2a 10733 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))
8759adantrr 463 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (1...𝑚) ∈ Fin)
8887, 84fihasheqf1od 10162 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘(1...𝑚)) = (♯‘𝐴))
8981nnnn0d 8696 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ0)
90 hashfz1 10155 . . . . . . . . . . . . . . . . . . . . 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 2122 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘𝐴) = 𝑚)
9392breq2d 3849 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (𝑛 ≤ (♯‘𝐴) ↔ 𝑛𝑚))
9493ifbid 3408 . . . . . . . . . . . . . . . . 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 3921 . . . . . . . . . . . . . . . 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)))
96 iseqeq3 9824 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ))
9795, 96syl 14 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ))
9897fveq1d 5291 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))
9986, 98breqtrd 3861 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))
10099expr 367 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))
101100exlimdv 1747 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))
10267, 101mpd 13 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))
103 breq2 3841 . . . . . . . . . 10 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))
104102, 103syl5ibrcom 155 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
105104expimpd 355 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
106105exlimdv 1747 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
107106rexlimdva 2489 . . . . . 6 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
10852, 107jaod 672 . . . . 5 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
10917adantr 270 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → 𝑀 ∈ ℤ)
11022adantr 270 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → 𝐴 ⊆ (ℤ𝑀))
111 eleq1w 2148 . . . . . . . . . . . 12 (𝑥 = 𝑗 → (𝑥𝐴𝑗𝐴))
112111dcbid 786 . . . . . . . . . . 11 (𝑥 = 𝑗 → (DECID 𝑥𝐴DECID 𝑗𝐴))
113112cbvralv 2590 . . . . . . . . . 10 (∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴 ↔ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
11426, 113sylib 120 . . . . . . . . 9 (𝜑 → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
115114adantr 270 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
116 simpr 108 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥)
117 fveq2 5289 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
118117sseq2d 3052 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑀)))
119117raleqdv 2568 . . . . . . . . . 10 (𝑚 = 𝑀 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴))
120 iseqeq1 9822 . . . . . . . . . . 11 (𝑚 = 𝑀 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) = seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ))
121120breq1d 3847 . . . . . . . . . 10 (𝑚 = 𝑀 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
122118, 119, 1213anbi123d 1248 . . . . . . . . 9 (𝑚 = 𝑀 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥)))
123122rspcev 2722 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ (𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥)) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
124109, 110, 115, 116, 123syl13anc 1176 . . . . . . 7 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
125124orcd 687 . . . . . 6 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
126125ex 113 . . . . 5 (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))))
127108, 126impbid 127 . . . 4 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
128 eluzelz 8997 . . . . . . . 8 (𝑎 ∈ (ℤ𝑀) → 𝑎 ∈ ℤ)
129 simpr 108 . . . . . . . . . 10 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑎𝐴) → 𝑎𝐴)
1308ad2antrr 472 . . . . . . . . . 10 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑎𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
131 nfcsb1v 2961 . . . . . . . . . . . 12 𝑘𝑎 / 𝑘𝐵
132131nfel1 2239 . . . . . . . . . . 11 𝑘𝑎 / 𝑘𝐵 ∈ ℂ
133 csbeq1a 2939 . . . . . . . . . . . 12 (𝑘 = 𝑎𝐵 = 𝑎 / 𝑘𝐵)
134133eleq1d 2156 . . . . . . . . . . 11 (𝑘 = 𝑎 → (𝐵 ∈ ℂ ↔ 𝑎 / 𝑘𝐵 ∈ ℂ))
135132, 134rspc 2716 . . . . . . . . . 10 (𝑎𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑎 / 𝑘𝐵 ∈ ℂ))
136129, 130, 135sylc 61 . . . . . . . . 9 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ 𝑎𝐴) → 𝑎 / 𝑘𝐵 ∈ ℂ)
137 0cnd 7460 . . . . . . . . 9 (((𝜑𝑎 ∈ (ℤ𝑀)) ∧ ¬ 𝑎𝐴) → 0 ∈ ℂ)
138 eleq1w 2148 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
139138dcbid 786 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (DECID 𝑥𝐴DECID 𝑎𝐴))
140139cbvralv 2590 . . . . . . . . . . 11 (∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴 ↔ ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
14126, 140sylib 120 . . . . . . . . . 10 (𝜑 → ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
142141r19.21bi 2461 . . . . . . . . 9 ((𝜑𝑎 ∈ (ℤ𝑀)) → DECID 𝑎𝐴)
143136, 137, 142ifcldadc 3416 . . . . . . . 8 ((𝜑𝑎 ∈ (ℤ𝑀)) → if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0) ∈ ℂ)
144 eleq1w 2148 . . . . . . . . . 10 (𝑛 = 𝑎 → (𝑛𝐴𝑎𝐴))
145 csbeq1 2934 . . . . . . . . . 10 (𝑛 = 𝑎𝑛 / 𝑘𝐵 = 𝑎 / 𝑘𝐵)
146144, 145ifbieq1d 3409 . . . . . . . . 9 (𝑛 = 𝑎 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0))
147 eqid 2088 . . . . . . . . 9 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
148146, 147fvmptg 5364 . . . . . . . 8 ((𝑎 ∈ ℤ ∧ if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0) ∈ ℂ) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑎) = if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0))
149128, 143, 148syl2an2 561 . . . . . . 7 ((𝜑𝑎 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑎) = if(𝑎𝐴, 𝑎 / 𝑘𝐵, 0))
150149, 143eqeltrd 2164 . . . . . 6 ((𝜑𝑎 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑎) ∈ ℂ)
151 simpr 108 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ (ℤ𝑀))
15253, 151sseldi 3021 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ ℤ)
153 vex 2622 . . . . . . . . . 10 𝑗 ∈ V
154 nfv 1466 . . . . . . . . . . 11 𝑘 𝑗𝐴
155 nfcsb1v 2961 . . . . . . . . . . 11 𝑘𝑗 / 𝑘𝐵
156 nfcv 2228 . . . . . . . . . . 11 𝑘0
157154, 155, 156nfif 3415 . . . . . . . . . 10 𝑘if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0)
158 eleq1w 2148 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑘𝐴𝑗𝐴))
159 csbeq1a 2939 . . . . . . . . . . 11 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
160158, 159ifbieq1d 3409 . . . . . . . . . 10 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
161153, 157, 160csbief 2970 . . . . . . . . 9 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0)
162 simpr 108 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ 𝑗𝐴) → 𝑗𝐴)
1638ad2antrr 472 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ 𝑗𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
164155nfel1 2239 . . . . . . . . . . . 12 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
165159eleq1d 2156 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
166164, 165rspc 2716 . . . . . . . . . . 11 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
167162, 163, 166sylc 61 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ 𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
168 0cnd 7460 . . . . . . . . . 10 (((𝜑𝑗 ∈ (ℤ𝑀)) ∧ ¬ 𝑗𝐴) → 0 ∈ ℂ)
169114r19.21bi 2461 . . . . . . . . . 10 ((𝜑𝑗 ∈ (ℤ𝑀)) → DECID 𝑗𝐴)
170167, 168, 169ifcldadc 3416 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ𝑀)) → if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0) ∈ ℂ)
171161, 170syl5eqel 2174 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) ∈ ℂ)
172 nfcv 2228 . . . . . . . . . . 11 𝑛if(𝑘𝐴, 𝐵, 0)
173 nfv 1466 . . . . . . . . . . . 12 𝑘 𝑛𝐴
174 nfcsb1v 2961 . . . . . . . . . . . 12 𝑘𝑛 / 𝑘𝐵
175173, 174, 156nfif 3415 . . . . . . . . . . 11 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
176 eleq1w 2148 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑘𝐴𝑛𝐴))
177 csbeq1a 2939 . . . . . . . . . . . 12 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
178176, 177ifbieq1d 3409 . . . . . . . . . . 11 (𝑘 = 𝑛 → if(𝑘𝐴, 𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
179172, 175, 178cbvmpt 3925 . . . . . . . . . 10 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
180179eqcomi 2092 . . . . . . . . 9 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
181180fvmpts 5366 . . . . . . . 8 ((𝑗 ∈ ℤ ∧ 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) ∈ ℂ) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
182152, 171, 181syl2anc 403 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
183151, 21syl6eleqr 2181 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗𝑍)
184 zisum.4 . . . . . . . . . 10 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
185184ralrimiva 2446 . . . . . . . . 9 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
186185adantr 270 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
187 nfcsb1v 2961 . . . . . . . . . 10 𝑘𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
188187nfeq2 2240 . . . . . . . . 9 𝑘(𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
189 fveq2 5289 . . . . . . . . . 10 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
190 csbeq1a 2939 . . . . . . . . . 10 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 0) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
191189, 190eqeq12d 2102 . . . . . . . . 9 (𝑘 = 𝑗 → ((𝐹𝑘) = if(𝑘𝐴, 𝐵, 0) ↔ (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)))
192188, 191rspc 2716 . . . . . . . 8 (𝑗𝑍 → (∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0) → (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)))
193183, 186, 192sylc 61 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ𝑀)) → (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
194182, 193eqtr4d 2123 . . . . . 6 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = (𝐹𝑗))
195 addcl 7446 . . . . . . 7 ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑎 + 𝑏) ∈ ℂ)
196195adantl 271 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ)) → (𝑎 + 𝑏) ∈ ℂ)
19717, 150, 194, 196iseqfeq 9860 . . . . 5 (𝜑 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) = seq𝑀( + , 𝐹, ℂ))
198197breq1d 3847 . . . 4 (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥 ↔ seq𝑀( + , 𝐹, ℂ) ⇝ 𝑥))
199127, 198bitrd 186 . . 3 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) ↔ seq𝑀( + , 𝐹, ℂ) ⇝ 𝑥))
200199iotabidv 4988 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) = (℩𝑥seq𝑀( + , 𝐹, ℂ) ⇝ 𝑥))
201 df-isum 10706 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
202 df-fv 5010 . 2 ( ⇝ ‘seq𝑀( + , 𝐹, ℂ)) = (℩𝑥seq𝑀( + , 𝐹, ℂ) ⇝ 𝑥)
203200, 201, 2023eqtr4g 2145 1 (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹, ℂ)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 664  DECID wdc 780   ∧ w3a 924   = wceq 1289  ∃wex 1426   ∈ wcel 1438  ∀wral 2359  ∃wrex 2360  ⦋csb 2931   ⊆ wss 2997  ifcif 3389   class class class wbr 3837   ↦ cmpt 3891  ℩cio 4965  –1-1-onto→wf1o 5001  ‘cfv 5002   Isom wiso 5003  (class class class)co 5634   ≈ cen 6435  Fincfn 6437  ℂcc 7327  0cc0 7329  1c1 7330   + caddc 7332   < clt 7501   ≤ cle 7502  ℕcn 8394  ℕ0cn0 8643  ℤcz 8720  ℤ≥cuz 8988  ...cfz 9393  seqcseq4 9816  ♯chash 10147   ⇝ cli 10629  Σcsu 10705 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442 This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-frec 6138  df-1o 6163  df-oadd 6167  df-er 6272  df-en 6438  df-dom 6439  df-fin 6440  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-iseq 9818  df-seq3 9819  df-exp 9919  df-ihash 10148  df-cj 10240  df-rsqrt 10395  df-abs 10396  df-clim 10630  df-isum 10706 This theorem is referenced by:  iisum  10737  sum0  10742  isumz  10743  isumss  10745  fisumsers  10749
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