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Theorem fsum 15673
Description: The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Mario Carneiro, 13-Jun-2019.)
Hypotheses
Ref Expression
fsum.1 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
fsum.2 (𝜑𝑀 ∈ ℕ)
fsum.3 (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)
fsum.4 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
fsum.5 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
Assertion
Ref Expression
fsum (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , 𝐺)‘𝑀))
Distinct variable groups:   𝐵,𝑛   𝐶,𝑘   𝑘,𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑘,𝑛   𝑘,𝐺,𝑛   𝑘,𝑀,𝑛
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑛)

Proof of Theorem fsum
Dummy variables 𝑓 𝑖 𝑗 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 15640 . 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))))
2 fvex 6904 . . 3 (seq1( + , 𝐺)‘𝑀) ∈ V
3 eleq1w 2815 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛𝐴𝑗𝐴))
4 csbeq1 3896 . . . . . . . . . 10 (𝑛 = 𝑗𝑛 / 𝑘𝐵 = 𝑗 / 𝑘𝐵)
53, 4ifbieq1d 4552 . . . . . . . . 9 (𝑛 = 𝑗 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
65cbvmptv 5261 . . . . . . . 8 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
7 fsum.4 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
87ralrimiva 3145 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
9 nfcsb1v 3918 . . . . . . . . . . 11 𝑘𝑗 / 𝑘𝐵
109nfel1 2918 . . . . . . . . . 10 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
11 csbeq1a 3907 . . . . . . . . . . 11 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
1211eleq1d 2817 . . . . . . . . . 10 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
1310, 12rspc 3600 . . . . . . . . 9 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
148, 13mpan9 506 . . . . . . . 8 ((𝜑𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
15 fveq2 6891 . . . . . . . . . . 11 (𝑛 = 𝑖 → (𝑓𝑛) = (𝑓𝑖))
1615csbeq1d 3897 . . . . . . . . . 10 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵)
17 csbcow 3908 . . . . . . . . . 10 (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵
1816, 17eqtr4di 2789 . . . . . . . . 9 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
1918cbvmptv 5261 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑖 ∈ ℕ ↦ (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
206, 14, 19summo 15670 . . . . . . 7 (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))))
21 fsum.2 . . . . . . . . 9 (𝜑𝑀 ∈ ℕ)
22 fsum.3 . . . . . . . . . . . 12 (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)
23 f1of 6833 . . . . . . . . . . . 12 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
2422, 23syl 17 . . . . . . . . . . 11 (𝜑𝐹:(1...𝑀)⟶𝐴)
25 ovex 7445 . . . . . . . . . . 11 (1...𝑀) ∈ V
26 fex 7230 . . . . . . . . . . 11 ((𝐹:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐹 ∈ V)
2724, 25, 26sylancl 585 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
28 nnuz 12872 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
2921, 28eleqtrdi 2842 . . . . . . . . . . . 12 (𝜑𝑀 ∈ (ℤ‘1))
30 fsum.5 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
31 elfznn 13537 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℕ)
32 fvex 6904 . . . . . . . . . . . . . . . . 17 (𝐺𝑛) ∈ V
3330, 32eqeltrrdi 2841 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ V)
34 eqid 2731 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ ↦ 𝐶) = (𝑛 ∈ ℕ ↦ 𝐶)
3534fvmpt2 7009 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝐶 ∈ V) → ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) = 𝐶)
3631, 33, 35syl2an2 683 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) = 𝐶)
3730, 36eqtr4d 2774 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛))
3837ralrimiva 3145 . . . . . . . . . . . . 13 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛))
39 nffvmpt1 6902 . . . . . . . . . . . . . . 15 𝑛((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)
4039nfeq2 2919 . . . . . . . . . . . . . 14 𝑛(𝐺𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)
41 fveq2 6891 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
42 fveq2 6891 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘))
4341, 42eqeq12d 2747 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → ((𝐺𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) ↔ (𝐺𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)))
4440, 43rspc 3600 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑀) → (∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) → (𝐺𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)))
4538, 44mpan9 506 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘))
4629, 45seqfveq 13999 . . . . . . . . . . 11 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀))
4722, 46jca 511 . . . . . . . . . 10 (𝜑 → (𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀)))
48 f1oeq1 6821 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)–1-1-onto𝐴))
49 fveq1 6890 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
5049csbeq1d 3897 . . . . . . . . . . . . . . . 16 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
51 fvex 6904 . . . . . . . . . . . . . . . . 17 (𝐹𝑛) ∈ V
52 fsum.1 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
5351, 52csbie 3929 . . . . . . . . . . . . . . . 16 (𝐹𝑛) / 𝑘𝐵 = 𝐶
5450, 53eqtrdi 2787 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = 𝐶)
5554mpteq2dv 5250 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ 𝐶))
5655seqeq3d 13981 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ 𝐶)))
5756fveq1d 6893 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀))
5857eqeq2d 2742 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀) ↔ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀)))
5948, 58anbi12d 630 . . . . . . . . . 10 (𝑓 = 𝐹 → ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀)) ↔ (𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀))))
6027, 47, 59spcedv 3588 . . . . . . . . 9 (𝜑 → ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀)))
61 oveq2 7420 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
6261f1oeq2d 6829 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑀)–1-1-onto𝐴))
63 fveq2 6891 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀))
6463eqeq2d 2742 . . . . . . . . . . . 12 (𝑚 = 𝑀 → ((seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) ↔ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀)))
6562, 64anbi12d 630 . . . . . . . . . . 11 (𝑚 = 𝑀 → ((𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ (𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀))))
6665exbidv 1923 . . . . . . . . . 10 (𝑚 = 𝑀 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀))))
6766rspcev 3612 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀))) → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
6821, 60, 67syl2anc 583 . . . . . . . 8 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
6968olcd 871 . . . . . . 7 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
70 breq2 5152 . . . . . . . . . . . . . 14 (𝑥 = (seq1( + , 𝐺)‘𝑀) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)))
7170anbi2d 628 . . . . . . . . . . . . 13 (𝑥 = (seq1( + , 𝐺)‘𝑀) → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀))))
7271rexbidv 3177 . . . . . . . . . . . 12 (𝑥 = (seq1( + , 𝐺)‘𝑀) → (∃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))) ⇝ (seq1( + , 𝐺)‘𝑀))))
73 eqeq1 2735 . . . . . . . . . . . . . . 15 (𝑥 = (seq1( + , 𝐺)‘𝑀) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) ↔ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
7473anbi2d 628 . . . . . . . . . . . . . 14 (𝑥 = (seq1( + , 𝐺)‘𝑀) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
7574exbidv 1923 . . . . . . . . . . . . 13 (𝑥 = (seq1( + , 𝐺)‘𝑀) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
7675rexbidv 3177 . . . . . . . . . . . 12 (𝑥 = (seq1( + , 𝐺)‘𝑀) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
7772, 76orbi12d 916 . . . . . . . . . . 11 (𝑥 = (seq1( + , 𝐺)‘𝑀) → ((∃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))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))))
7877moi2 3712 . . . . . . . . . 10 ((((seq1( + , 𝐺)‘𝑀) ∈ V ∧ ∃*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)))) ∧ ((∃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))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))) → 𝑥 = (seq1( + , 𝐺)‘𝑀))
792, 78mpanl1 697 . . . . . . . . 9 ((∃*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))) ∧ ((∃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))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))) → 𝑥 = (seq1( + , 𝐺)‘𝑀))
8079ancom2s 647 . . . . . . . 8 ((∃*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))) ∧ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) ∧ (∃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))))) → 𝑥 = (seq1( + , 𝐺)‘𝑀))
8180expr 456 . . . . . . 7 ((∃*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))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))) → ((∃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))) → 𝑥 = (seq1( + , 𝐺)‘𝑀)))
8220, 69, 81syl2anc 583 . . . . . 6 (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))) → 𝑥 = (seq1( + , 𝐺)‘𝑀)))
8369, 77syl5ibrcom 246 . . . . . 6 (𝜑 → (𝑥 = (seq1( + , 𝐺)‘𝑀) → (∃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)))))
8482, 83impbid 211 . . . . 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))) ↔ 𝑥 = (seq1( + , 𝐺)‘𝑀)))
8584adantr 480 . . . 4 ((𝜑 ∧ (seq1( + , 𝐺)‘𝑀) ∈ V) → ((∃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))) ↔ 𝑥 = (seq1( + , 𝐺)‘𝑀)))
8685iota5 6526 . . 3 ((𝜑 ∧ (seq1( + , 𝐺)‘𝑀) ∈ V) → (℩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)))) = (seq1( + , 𝐺)‘𝑀))
872, 86mpan2 688 . 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)))) = (seq1( + , 𝐺)‘𝑀))
881, 87eqtrid 2783 1 (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , 𝐺)‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844   = wceq 1540  wex 1780  wcel 2105  ∃*wmo 2531  wral 3060  wrex 3069  Vcvv 3473  csb 3893  wss 3948  ifcif 4528   class class class wbr 5148  cmpt 5231  cio 6493  wf 6539  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7412  cc 11114  0cc0 11116  1c1 11117   + caddc 11119  cn 12219  cz 12565  cuz 12829  ...cfz 13491  seqcseq 13973  cli 15435  Σcsu 15639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-rp 12982  df-fz 13492  df-fzo 13635  df-seq 13974  df-exp 14035  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-sum 15640
This theorem is referenced by:  sumz  15675  fsumf1o  15676  fsumcl2lem  15684  fsumadd  15693  sumsnf  15696  fsummulc2  15737  fsumconst  15743  fsumrelem  15760  gsumfsum  21216  sumsnd  44025
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