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Theorem zsum 14569
Description: Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.)
Hypotheses
Ref Expression
zsum.1 𝑍 = (ℤ𝑀)
zsum.2 (𝜑𝑀 ∈ ℤ)
zsum.3 (𝜑𝐴𝑍)
zsum.4 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
zsum.5 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
zsum (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝜑,𝑘   𝑘,𝑍   𝑘,𝑀
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem zsum
Dummy variables 𝑓 𝑔 𝑖 𝑗 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2791 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (𝑛𝐴𝑖𝐴))
2 csbeq1 3642 . . . . . . . . . . . 12 (𝑛 = 𝑖𝑛 / 𝑘𝐵 = 𝑖 / 𝑘𝐵)
31, 2ifbieq1d 4217 . . . . . . . . . . 11 (𝑛 = 𝑖 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
43cbvmptv 4858 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑖 ∈ ℤ ↦ if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
5 simpll 807 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝜑)
6 zsum.5 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
76ralrimiva 3068 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
8 nfcsb1v 3655 . . . . . . . . . . . . . 14 𝑘𝑖 / 𝑘𝐵
98nfel1 2881 . . . . . . . . . . . . 13 𝑘𝑖 / 𝑘𝐵 ∈ ℂ
10 csbeq1a 3648 . . . . . . . . . . . . . 14 (𝑘 = 𝑖𝐵 = 𝑖 / 𝑘𝐵)
1110eleq1d 2788 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ 𝑖 / 𝑘𝐵 ∈ ℂ))
129, 11rspc 3407 . . . . . . . . . . . 12 (𝑖𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑖 / 𝑘𝐵 ∈ ℂ))
137, 12syl5 34 . . . . . . . . . . 11 (𝑖𝐴 → (𝜑𝑖 / 𝑘𝐵 ∈ ℂ))
145, 13mpan9 487 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
15 simplr 809 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑚 ∈ ℤ)
16 zsum.2 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
1716ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑀 ∈ ℤ)
18 simpr 479 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑚))
19 zsum.3 . . . . . . . . . . . 12 (𝜑𝐴𝑍)
20 zsum.1 . . . . . . . . . . . 12 𝑍 = (ℤ𝑀)
2119, 20syl6sseq 3757 . . . . . . . . . . 11 (𝜑𝐴 ⊆ (ℤ𝑀))
2221ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑀))
234, 14, 15, 17, 18, 22sumrb 14564 . . . . . . . . 9 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
2423biimpd 219 . . . . . . . 8 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
2524expimpd 630 . . . . . . 7 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
2625rexlimdva 3133 . . . . . 6 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
2719ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴𝑍)
28 uzssz 11820 . . . . . . . . . . . . . . . 16 (ℤ𝑀) ⊆ ℤ
2920, 28eqsstri 3741 . . . . . . . . . . . . . . 15 𝑍 ⊆ ℤ
30 zssre 11497 . . . . . . . . . . . . . . 15 ℤ ⊆ ℝ
3129, 30sstri 3718 . . . . . . . . . . . . . 14 𝑍 ⊆ ℝ
32 ltso 10231 . . . . . . . . . . . . . 14 < Or ℝ
33 soss 5157 . . . . . . . . . . . . . 14 (𝑍 ⊆ ℝ → ( < Or ℝ → < Or 𝑍))
3431, 32, 33mp2 9 . . . . . . . . . . . . 13 < Or 𝑍
35 soss 5157 . . . . . . . . . . . . 13 (𝐴𝑍 → ( < Or 𝑍 → < Or 𝐴))
3627, 34, 35mpisyl 21 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → < Or 𝐴)
37 fzfi 12886 . . . . . . . . . . . . 13 (1...𝑚) ∈ Fin
38 ovex 6793 . . . . . . . . . . . . . . . 16 (1...𝑚) ∈ V
3938f1oen 8093 . . . . . . . . . . . . . . 15 (𝑓:(1...𝑚)–1-1-onto𝐴 → (1...𝑚) ≈ 𝐴)
4039adantl 473 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
4140ensymd 8123 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ≈ (1...𝑚))
42 enfii 8293 . . . . . . . . . . . . 13 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
4337, 41, 42sylancr 698 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ∈ Fin)
44 fz1iso 13359 . . . . . . . . . . . 12 (( < Or 𝐴𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
4536, 43, 44syl2anc 696 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
46 simpll 807 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
4746, 13mpan9 487 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
48 fveq2 6304 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
4948csbeq1d 3646 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
50 csbco 3649 . . . . . . . . . . . . . . . 16 (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵
5149, 50syl6eqr 2776 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
5251cbvmptv 4858 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
53 eqid 2724 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵)
54 simplr 809 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
5516ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
5621ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑀))
57 simprl 811 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
58 simprr 813 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
594, 47, 52, 53, 54, 55, 56, 57, 58summolem2a 14566 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
6059expr 644 . . . . . . . . . . . 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 ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
6160exlimdv 1974 . . . . . . . . . . 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 ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
6245, 61mpd 15 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
63 breq2 4764 . . . . . . . . . 10 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
6462, 63syl5ibrcom 237 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
6564expimpd 630 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
6665exlimdv 1974 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
6766rexlimdva 3133 . . . . . 6 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
6826, 67jaod 394 . . . . 5 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
6916adantr 472 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
7021adantr 472 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → 𝐴 ⊆ (ℤ𝑀))
71 simpr 479 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
72 fveq2 6304 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
7372sseq2d 3739 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑀)))
74 seqeq1 12919 . . . . . . . . . . 11 (𝑚 = 𝑀 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))))
7574breq1d 4770 . . . . . . . . . 10 (𝑚 = 𝑀 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
7673, 75anbi12d 749 . . . . . . . . 9 (𝑚 = 𝑀 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑀) ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)))
7776rspcev 3413 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ (𝐴 ⊆ (ℤ𝑀) ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
7869, 70, 71, 77syl12anc 1437 . . . . . . 7 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
7978orcd 406 . . . . . 6 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
8079ex 449 . . . . 5 (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))))
8168, 80impbid 202 . . . 4 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
82 simpr 479 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ (ℤ𝑀))
8328, 82sseldi 3707 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ ℤ)
8482, 20syl6eleqr 2814 . . . . . . . . . 10 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗𝑍)
85 zsum.4 . . . . . . . . . . . 12 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
8685ralrimiva 3068 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
8786adantr 472 . . . . . . . . . 10 ((𝜑𝑗 ∈ (ℤ𝑀)) → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
88 nfcsb1v 3655 . . . . . . . . . . . 12 𝑘𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
8988nfeq2 2882 . . . . . . . . . . 11 𝑘(𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
90 fveq2 6304 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
91 csbeq1a 3648 . . . . . . . . . . . 12 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 0) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
9290, 91eqeq12d 2739 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝐹𝑘) = if(𝑘𝐴, 𝐵, 0) ↔ (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)))
9389, 92rspc 3407 . . . . . . . . . 10 (𝑗𝑍 → (∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0) → (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)))
9484, 87, 93sylc 65 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ𝑀)) → (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
95 fvex 6314 . . . . . . . . 9 (𝐹𝑗) ∈ V
9694, 95syl6eqelr 2812 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) ∈ V)
97 nfcv 2866 . . . . . . . . . . 11 𝑛if(𝑘𝐴, 𝐵, 0)
98 nfv 1956 . . . . . . . . . . . 12 𝑘 𝑛𝐴
99 nfcsb1v 3655 . . . . . . . . . . . 12 𝑘𝑛 / 𝑘𝐵
100 nfcv 2866 . . . . . . . . . . . 12 𝑘0
10198, 99, 100nfif 4223 . . . . . . . . . . 11 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
102 eleq1 2791 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑘𝐴𝑛𝐴))
103 csbeq1a 3648 . . . . . . . . . . . 12 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
104102, 103ifbieq1d 4217 . . . . . . . . . . 11 (𝑘 = 𝑛 → if(𝑘𝐴, 𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
10597, 101, 104cbvmpt 4857 . . . . . . . . . 10 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
106105eqcomi 2733 . . . . . . . . 9 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
107106fvmpts 6399 . . . . . . . 8 ((𝑗 ∈ ℤ ∧ 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) ∈ V) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
10883, 96, 107syl2anc 696 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
109108, 94eqtr4d 2761 . . . . . 6 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = (𝐹𝑗))
11016, 109seqfeq 12941 . . . . 5 (𝜑 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑀( + , 𝐹))
111110breq1d 4770 . . . 4 (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , 𝐹) ⇝ 𝑥))
11281, 111bitrd 268 . . 3 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) ↔ seq𝑀( + , 𝐹) ⇝ 𝑥))
113112iotabidv 5985 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))) = (℩𝑥seq𝑀( + , 𝐹) ⇝ 𝑥))
114 df-sum 14537 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
115 df-fv 6009 . 2 ( ⇝ ‘seq𝑀( + , 𝐹)) = (℩𝑥seq𝑀( + , 𝐹) ⇝ 𝑥)
116113, 114, 1153eqtr4g 2783 1 (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1596  wex 1817  wcel 2103  wral 3014  wrex 3015  Vcvv 3304  csb 3639  wss 3680  ifcif 4194   class class class wbr 4760  cmpt 4837   Or wor 5138  cio 5962  1-1-ontowf1o 6000  cfv 6001   Isom wiso 6002  (class class class)co 6765  cen 8069  Fincfn 8072  cc 10047  cr 10048  0cc0 10049  1c1 10050   + caddc 10052   < clt 10187  cn 11133  cz 11490  cuz 11800  ...cfz 12440  seqcseq 12916  chash 13232  cli 14335  Σcsu 14536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-n0 11406  df-z 11491  df-uz 11801  df-rp 11947  df-fz 12441  df-fzo 12581  df-seq 12917  df-exp 12976  df-hash 13233  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-clim 14339  df-sum 14537
This theorem is referenced by:  isum  14570  sum0  14572  sumz  14573  sumss  14575  fsumsers  14579
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