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Theorem zsum 15754
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 eleq1w 2824 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (𝑛𝐴𝑖𝐴))
2 csbeq1 3902 . . . . . . . . . . . 12 (𝑛 = 𝑖𝑛 / 𝑘𝐵 = 𝑖 / 𝑘𝐵)
31, 2ifbieq1d 4550 . . . . . . . . . . 11 (𝑛 = 𝑖 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
43cbvmptv 5255 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑖 ∈ ℤ ↦ if(𝑖𝐴, 𝑖 / 𝑘𝐵, 0))
5 simpll 767 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝜑)
6 zsum.5 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
76ralrimiva 3146 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
8 nfcsb1v 3923 . . . . . . . . . . . . . 14 𝑘𝑖 / 𝑘𝐵
98nfel1 2922 . . . . . . . . . . . . 13 𝑘𝑖 / 𝑘𝐵 ∈ ℂ
10 csbeq1a 3913 . . . . . . . . . . . . . 14 (𝑘 = 𝑖𝐵 = 𝑖 / 𝑘𝐵)
1110eleq1d 2826 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ 𝑖 / 𝑘𝐵 ∈ ℂ))
129, 11rspc 3610 . . . . . . . . . . . 12 (𝑖𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑖 / 𝑘𝐵 ∈ ℂ))
137, 12syl5 34 . . . . . . . . . . 11 (𝑖𝐴 → (𝜑𝑖 / 𝑘𝐵 ∈ ℂ))
145, 13mpan9 506 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
15 simplr 769 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑚 ∈ ℤ)
16 zsum.2 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
1716ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝑀 ∈ ℤ)
18 simpr 484 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑚))
19 zsum.3 . . . . . . . . . . . 12 (𝜑𝐴𝑍)
20 zsum.1 . . . . . . . . . . . 12 𝑍 = (ℤ𝑀)
2119, 20sseqtrdi 4024 . . . . . . . . . . 11 (𝜑𝐴 ⊆ (ℤ𝑀))
2221ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → 𝐴 ⊆ (ℤ𝑀))
234, 14, 15, 17, 18, 22sumrb 15749 . . . . . . . . 9 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
2423biimpd 229 . . . . . . . 8 (((𝜑𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
2524expimpd 453 . . . . . . 7 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
2625rexlimdva 3155 . . . . . 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 726 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴𝑍)
28 uzssz 12899 . . . . . . . . . . . . . . . 16 (ℤ𝑀) ⊆ ℤ
2920, 28eqsstri 4030 . . . . . . . . . . . . . . 15 𝑍 ⊆ ℤ
30 zssre 12620 . . . . . . . . . . . . . . 15 ℤ ⊆ ℝ
3129, 30sstri 3993 . . . . . . . . . . . . . 14 𝑍 ⊆ ℝ
32 ltso 11341 . . . . . . . . . . . . . 14 < Or ℝ
33 soss 5612 . . . . . . . . . . . . . 14 (𝑍 ⊆ ℝ → ( < Or ℝ → < Or 𝑍))
3431, 32, 33mp2 9 . . . . . . . . . . . . 13 < Or 𝑍
35 soss 5612 . . . . . . . . . . . . 13 (𝐴𝑍 → ( < Or 𝑍 → < Or 𝐴))
3627, 34, 35mpisyl 21 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → < Or 𝐴)
37 fzfi 14013 . . . . . . . . . . . . 13 (1...𝑚) ∈ Fin
38 ovex 7464 . . . . . . . . . . . . . . . 16 (1...𝑚) ∈ V
3938f1oen 9013 . . . . . . . . . . . . . . 15 (𝑓:(1...𝑚)–1-1-onto𝐴 → (1...𝑚) ≈ 𝐴)
4039adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
4140ensymd 9045 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ≈ (1...𝑚))
42 enfii 9226 . . . . . . . . . . . . 13 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
4337, 41, 42sylancr 587 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝐴 ∈ Fin)
44 fz1iso 14501 . . . . . . . . . . . 12 (( < Or 𝐴𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
4536, 43, 44syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
46 simpll 767 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
4746, 13mpan9 506 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖𝐴) → 𝑖 / 𝑘𝐵 ∈ ℂ)
48 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
4948csbeq1d 3903 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
50 csbcow 3914 . . . . . . . . . . . . . . . 16 (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵
5149, 50eqtr4di 2795 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
5251cbvmptv 5255 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑖𝑖 / 𝑘𝐵)
53 eqid 2737 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑖𝑖 / 𝑘𝐵)
54 simplr 769 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
5516ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
5621ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑀))
57 simprl 771 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
58 simprr 773 . . . . . . . . . . . . . 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 15751 . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 1933 . . . . . . . . . . 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 5147 . . . . . . . . . 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 247 . . . . . . . . 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 453 . . . . . . . 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 1933 . . . . . . 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 3155 . . . . . 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 860 . . . . 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 480 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
7021adantr 480 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → 𝐴 ⊆ (ℤ𝑀))
71 simpr 484 . . . . . . . 8 ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
72 fveq2 6906 . . . . . . . . . . 11 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
7372sseq2d 4016 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑀)))
74 seqeq1 14045 . . . . . . . . . . 11 (𝑚 = 𝑀 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))))
7574breq1d 5153 . . . . . . . . . 10 (𝑚 = 𝑀 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥))
7673, 75anbi12d 632 . . . . . . . . 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 3622 . . . . . . . 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 837 . . . . . . 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 874 . . . . . 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 412 . . . . 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 212 . . . 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 484 . . . . . . . . 9 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ (ℤ𝑀))
8328, 82sselid 3981 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 ∈ ℤ)
8482, 20eleqtrrdi 2852 . . . . . . . . . 10 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗𝑍)
85 zsum.4 . . . . . . . . . . . 12 ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
8685ralrimiva 3146 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
8786adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (ℤ𝑀)) → ∀𝑘𝑍 (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))
88 nfcsb1v 3923 . . . . . . . . . . . 12 𝑘𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
8988nfeq2 2923 . . . . . . . . . . 11 𝑘(𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)
90 fveq2 6906 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
91 csbeq1a 3913 . . . . . . . . . . . 12 (𝑘 = 𝑗 → if(𝑘𝐴, 𝐵, 0) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
9290, 91eqeq12d 2753 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝐹𝑘) = if(𝑘𝐴, 𝐵, 0) ↔ (𝐹𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0)))
9389, 92rspc 3610 . . . . . . . . . 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 6919 . . . . . . . . 9 (𝐹𝑗) ∈ V
9694, 95eqeltrrdi 2850 . . . . . . . 8 ((𝜑𝑗 ∈ (ℤ𝑀)) → 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0) ∈ V)
97 nfcv 2905 . . . . . . . . . . 11 𝑛if(𝑘𝐴, 𝐵, 0)
98 nfv 1914 . . . . . . . . . . . 12 𝑘 𝑛𝐴
99 nfcsb1v 3923 . . . . . . . . . . . 12 𝑘𝑛 / 𝑘𝐵
100 nfcv 2905 . . . . . . . . . . . 12 𝑘0
10198, 99, 100nfif 4556 . . . . . . . . . . 11 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
102 eleq1w 2824 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑘𝐴𝑛𝐴))
103 csbeq1a 3913 . . . . . . . . . . . 12 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
104102, 103ifbieq1d 4550 . . . . . . . . . . 11 (𝑘 = 𝑛 → if(𝑘𝐴, 𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
10597, 101, 104cbvmpt 5253 . . . . . . . . . 10 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
106105eqcomi 2746 . . . . . . . . 9 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
107106fvmpts 7019 . . . . . . . 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 584 . . . . . . 7 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = 𝑗 / 𝑘if(𝑘𝐴, 𝐵, 0))
109108, 94eqtr4d 2780 . . . . . 6 ((𝜑𝑗 ∈ (ℤ𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑗) = (𝐹𝑗))
11016, 109seqfeq 14068 . . . . 5 (𝜑 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑀( + , 𝐹))
111110breq1d 5153 . . . 4 (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , 𝐹) ⇝ 𝑥))
11281, 111bitrd 279 . . 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 6545 . 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 15723 . 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 6569 . 2 ( ⇝ ‘seq𝑀( + , 𝐹)) = (℩𝑥seq𝑀( + , 𝐹) ⇝ 𝑥)
116113, 114, 1153eqtr4g 2802 1 (𝜑 → Σ𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  csb 3899  wss 3951  ifcif 4525   class class class wbr 5143  cmpt 5225   Or wor 5591  cio 6512  1-1-ontowf1o 6560  cfv 6561   Isom wiso 6562  (class class class)co 7431  cen 8982  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cn 12266  cz 12613  cuz 12878  ...cfz 13547  seqcseq 14042  chash 14369  cli 15520  Σcsu 15722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723
This theorem is referenced by:  isum  15755  sum0  15757  sumz  15758  sumss  15760  fsumsers  15764
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