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Theorem fsum 15077
 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 15043 . 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 6674 . . 3 (seq1( + , 𝐺)‘𝑀) ∈ V
3 eleq1w 2898 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛𝐴𝑗𝐴))
4 csbeq1 3869 . . . . . . . . . 10 (𝑛 = 𝑗𝑛 / 𝑘𝐵 = 𝑗 / 𝑘𝐵)
53, 4ifbieq1d 4473 . . . . . . . . 9 (𝑛 = 𝑗 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
65cbvmptv 5155 . . . . . . . 8 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝑗 / 𝑘𝐵, 0))
7 fsum.4 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
87ralrimiva 3177 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
9 nfcsb1v 3890 . . . . . . . . . . 11 𝑘𝑗 / 𝑘𝐵
109nfel1 2998 . . . . . . . . . 10 𝑘𝑗 / 𝑘𝐵 ∈ ℂ
11 csbeq1a 3880 . . . . . . . . . . 11 (𝑘 = 𝑗𝐵 = 𝑗 / 𝑘𝐵)
1211eleq1d 2900 . . . . . . . . . 10 (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑘𝐵 ∈ ℂ))
1310, 12rspc 3597 . . . . . . . . 9 (𝑗𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑗 / 𝑘𝐵 ∈ ℂ))
148, 13mpan9 510 . . . . . . . 8 ((𝜑𝑗𝐴) → 𝑗 / 𝑘𝐵 ∈ ℂ)
15 fveq2 6661 . . . . . . . . . . 11 (𝑛 = 𝑖 → (𝑓𝑛) = (𝑓𝑖))
1615csbeq1d 3870 . . . . . . . . . 10 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵)
17 csbcow 3881 . . . . . . . . . 10 (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵 = (𝑓𝑖) / 𝑘𝐵
1816, 17syl6eqr 2877 . . . . . . . . 9 (𝑛 = 𝑖(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
1918cbvmptv 5155 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑖 ∈ ℕ ↦ (𝑓𝑖) / 𝑗𝑗 / 𝑘𝐵)
206, 14, 19summo 15074 . . . . . . 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 6606 . . . . . . . . . . . 12 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
2422, 23syl 17 . . . . . . . . . . 11 (𝜑𝐹:(1...𝑀)⟶𝐴)
25 ovex 7182 . . . . . . . . . . 11 (1...𝑀) ∈ V
26 fex 6980 . . . . . . . . . . 11 ((𝐹:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐹 ∈ V)
2724, 25, 26sylancl 589 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
28 nnuz 12278 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
2921, 28eleqtrdi 2926 . . . . . . . . . . . 12 (𝜑𝑀 ∈ (ℤ‘1))
30 fsum.5 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
31 elfznn 12940 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℕ)
32 fvex 6674 . . . . . . . . . . . . . . . . 17 (𝐺𝑛) ∈ V
3330, 32eqeltrrdi 2925 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ V)
34 eqid 2824 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ ↦ 𝐶) = (𝑛 ∈ ℕ ↦ 𝐶)
3534fvmpt2 6770 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝐶 ∈ V) → ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) = 𝐶)
3631, 33, 35syl2an2 685 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) = 𝐶)
3730, 36eqtr4d 2862 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛))
3837ralrimiva 3177 . . . . . . . . . . . . 13 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛))
39 nffvmpt1 6672 . . . . . . . . . . . . . . 15 𝑛((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)
4039nfeq2 2999 . . . . . . . . . . . . . 14 𝑛(𝐺𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)
41 fveq2 6661 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
42 fveq2 6661 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘))
4341, 42eqeq12d 2840 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → ((𝐺𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) ↔ (𝐺𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)))
4440, 43rspc 3597 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑀) → (∀𝑛 ∈ (1...𝑀)(𝐺𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) → (𝐺𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)))
4538, 44mpan9 510 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1...𝑀)) → (𝐺𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘))
4629, 45seqfveq 13399 . . . . . . . . . . 11 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀))
4722, 46jca 515 . . . . . . . . . 10 (𝜑 → (𝐹:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀)))
48 f1oeq1 6595 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)–1-1-onto𝐴))
49 fveq1 6660 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
5049csbeq1d 3870 . . . . . . . . . . . . . . . 16 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = (𝐹𝑛) / 𝑘𝐵)
51 fvex 6674 . . . . . . . . . . . . . . . . 17 (𝐹𝑛) ∈ V
52 fsum.1 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
5351, 52csbie 3901 . . . . . . . . . . . . . . . 16 (𝐹𝑛) / 𝑘𝐵 = 𝐶
5450, 53syl6eq 2875 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹(𝑓𝑛) / 𝑘𝐵 = 𝐶)
5554mpteq2dv 5148 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ 𝐶))
5655seqeq3d 13381 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ 𝐶)))
5756fveq1d 6663 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀))
5857eqeq2d 2835 . . . . . . . . . . 11 (𝑓 = 𝐹 → ((seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀) ↔ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀)))
5948, 58anbi12d 633 . . . . . . . . . 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 3585 . . . . . . . . 9 (𝜑 → ∃𝑓(𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀)))
61 oveq2 7157 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
6261f1oeq2d 6602 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑀)–1-1-onto𝐴))
63 fveq2 6661 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀))
6463eqeq2d 2835 . . . . . . . . . . . 12 (𝑚 = 𝑀 → ((seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) ↔ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑀)))
6562, 64anbi12d 633 . . . . . . . . . . 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 3609 . . . . . . . . 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 587 . . . . . . . 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 5056 . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . 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 3289 . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . 15 (𝑥 = (seq1( + , 𝐺)‘𝑀) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) ↔ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
7473anbi2d 631 . . . . . . . . . . . . . 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 3289 . . . . . . . . . . . 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 3693 . . . . . . . . . 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 699 . . . . . . . . 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 649 . . . . . . . 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 460 . . . . . . 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 587 . . . . . 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 250 . . . . . 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 215 . . . . 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 484 . . . 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 6326 . . 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 690 . 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, 87syl5eq 2871 1 (𝜑 → Σ𝑘𝐴 𝐵 = (seq1( + , 𝐺)‘𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2115  ∃*wmo 2622  ∀wral 3133  ∃wrex 3134  Vcvv 3480  ⦋csb 3866   ⊆ wss 3919  ifcif 4450   class class class wbr 5052   ↦ cmpt 5132  ℩cio 6300  ⟶wf 6339  –1-1-onto→wf1o 6342  ‘cfv 6343  (class class class)co 7149  ℂcc 10533  0cc0 10535  1c1 10536   + caddc 10538  ℕcn 11634  ℤcz 11978  ℤ≥cuz 12240  ...cfz 12894  seqcseq 13373   ⇝ cli 14841  Σcsu 15042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-fz 12895  df-fzo 13038  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043 This theorem is referenced by:  sumz  15079  fsumf1o  15080  fsumcl2lem  15088  fsumadd  15096  sumsnf  15099  fsummulc2  15139  fsumconst  15145  fsumrelem  15162  gsumfsum  20165  sumsnd  41579
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