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Theorem fsum0diag2 14975
Description: Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular region 0 ≤ 𝑗, 0 ≤ 𝑘, 𝑗 + 𝑘𝑁". (Contributed by Mario Carneiro, 21-Jul-2014.)
Hypotheses
Ref Expression
fsum0diag2.1 (𝑥 = 𝑘𝐵 = 𝐴)
fsum0diag2.2 (𝑥 = (𝑘𝑗) → 𝐵 = 𝐶)
fsum0diag2.3 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗)))) → 𝐴 ∈ ℂ)
Assertion
Ref Expression
fsum0diag2 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶)
Distinct variable groups:   𝑗,𝑘,𝑥,𝑁   𝜑,𝑗,𝑘   𝐵,𝑘   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑗,𝑘)   𝐵(𝑥,𝑗)   𝐶(𝑗,𝑘)

Proof of Theorem fsum0diag2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 fznn0sub2 12868 . . . . . . 7 (𝑛 ∈ (0...(𝑁𝑗)) → ((𝑁𝑗) − 𝑛) ∈ (0...(𝑁𝑗)))
21ad2antll 725 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁𝑗)))) → ((𝑁𝑗) − 𝑛) ∈ (0...(𝑁𝑗)))
3 fsum0diag2.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗)))) → 𝐴 ∈ ℂ)
43expr 457 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑘 ∈ (0...(𝑁𝑗)) → 𝐴 ∈ ℂ))
54ralrimiv 3150 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
6 fsum0diag2.1 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝐴)
76eleq1d 2869 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
87cbvralv 3405 . . . . . . . 8 (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ ↔ ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
95, 8sylibr 235 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
109adantrr 713 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁𝑗)))) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
11 nfcsb1v 3839 . . . . . . . 8 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵
1211nfel1 2965 . . . . . . 7 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ
13 csbeq1a 3830 . . . . . . . 8 (𝑥 = ((𝑁𝑗) − 𝑛) → 𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
1413eleq1d 2869 . . . . . . 7 (𝑥 = ((𝑁𝑗) − 𝑛) → (𝐵 ∈ ℂ ↔ ((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ))
1512, 14rspc 3555 . . . . . 6 (((𝑁𝑗) − 𝑛) ∈ (0...(𝑁𝑗)) → (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ → ((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ))
162, 10, 15sylc 65 . . . . 5 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁𝑗)))) → ((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ)
1716fsum0diag 14969 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
18 nfcsb1v 3839 . . . . . . . . . 10 𝑥𝑘 / 𝑥𝐵
1918nfel1 2965 . . . . . . . . 9 𝑥𝑘 / 𝑥𝐵 ∈ ℂ
20 csbeq1a 3830 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
2120eleq1d 2869 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝑘 / 𝑥𝐵 ∈ ℂ))
2219, 21rspc 3555 . . . . . . . 8 (𝑘 ∈ (0...(𝑁𝑗)) → (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ → 𝑘 / 𝑥𝐵 ∈ ℂ))
239, 22mpan9 507 . . . . . . 7 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 / 𝑥𝐵 ∈ ℂ)
24 csbeq1 3820 . . . . . . 7 (𝑘 = ((0 + (𝑁𝑗)) − 𝑛) → 𝑘 / 𝑥𝐵 = ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
2523, 24fsumrev2 14974 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
26 elfz3nn0 12855 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
2726ad2antlr 723 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℕ0)
28 elfzelz 12762 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
2928ad2antlr 723 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℤ)
30 nn0cn 11761 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
31 zcn 11840 . . . . . . . . . . . 12 (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
32 subcl 10738 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁𝑗) ∈ ℂ)
3330, 31, 32syl2an 595 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑗 ∈ ℤ) → (𝑁𝑗) ∈ ℂ)
3427, 29, 33syl2anc 584 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ∈ ℂ)
35 addid2 10676 . . . . . . . . . 10 ((𝑁𝑗) ∈ ℂ → (0 + (𝑁𝑗)) = (𝑁𝑗))
3634, 35syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (0 + (𝑁𝑗)) = (𝑁𝑗))
3736oveq1d 7038 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) = ((𝑁𝑗) − 𝑛))
3837csbeq1d 3821 . . . . . . 7 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
3938sumeq2dv 14897 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4025, 39eqtrd 2833 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4140sumeq2dv 14897 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
42 elfz3nn0 12855 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
4342adantl 482 . . . . . . . . 9 ((𝜑𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
44 addid2 10676 . . . . . . . . 9 (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
4543, 30, 443syl 18 . . . . . . . 8 ((𝜑𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
4645oveq1d 7038 . . . . . . 7 ((𝜑𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁𝑛))
4746oveq2d 7039 . . . . . 6 ((𝜑𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁𝑛)))
4846oveq1d 7038 . . . . . . . . 9 ((𝜑𝑛 ∈ (0...𝑁)) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑛) − 𝑗))
4948adantr 481 . . . . . . . 8 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑛) − 𝑗))
5042ad2antlr 723 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑁 ∈ ℕ0)
51 elfzelz 12762 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
5251ad2antlr 723 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑛 ∈ ℤ)
53 elfzelz 12762 . . . . . . . . . 10 (𝑗 ∈ (0...(𝑁𝑛)) → 𝑗 ∈ ℤ)
5453adantl 482 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑗 ∈ ℤ)
55 zcn 11840 . . . . . . . . . 10 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
56 sub32 10774 . . . . . . . . . 10 ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5730, 55, 31, 56syl3an 1153 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5850, 52, 54, 57syl3anc 1364 . . . . . . . 8 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5949, 58eqtrd 2833 . . . . . . 7 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
6059csbeq1d 3821 . . . . . 6 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6147, 60sumeq12rdv 14901 . . . . 5 ((𝜑𝑛 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6261sumeq2dv 14897 . . . 4 (𝜑 → Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6317, 41, 623eqtr4d 2843 . . 3 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
64 fzfid 13195 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
65 elfzuz3 12759 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ𝑗))
6665adantl 482 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ𝑗))
67 elfzuz3 12759 . . . . . . . . . . 11 (𝑘 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝑘))
6867adantl 482 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ𝑘))
6968adantr 481 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ (ℤ𝑘))
70 elfzuzb 12756 . . . . . . . . 9 (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ𝑗) ∧ 𝑁 ∈ (ℤ𝑘)))
7166, 69, 70sylanbrc 583 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
72 elfzelz 12762 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
7372adantl 482 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
74 elfzel2 12760 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
7574ad2antlr 723 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
76 elfzelz 12762 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
7776ad2antlr 723 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
78 fzsubel 12797 . . . . . . . . 9 (((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗))))
7973, 75, 77, 73, 78syl22anc 835 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗))))
8071, 79mpbid 233 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗)))
81 subid 10759 . . . . . . . . 9 (𝑗 ∈ ℂ → (𝑗𝑗) = 0)
8273, 31, 813syl 18 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑗𝑗) = 0)
8382oveq1d 7038 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗𝑗)...(𝑁𝑗)) = (0...(𝑁𝑗)))
8480, 83eleqtrd 2887 . . . . . 6 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ (0...(𝑁𝑗)))
85 simpll 763 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
86 fzss2 12801 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑘) → (0...𝑘) ⊆ (0...𝑁))
8768, 86syl 17 . . . . . . . 8 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ⊆ (0...𝑁))
8887sselda 3895 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ (0...𝑁))
8985, 88, 9syl2anc 584 . . . . . 6 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
90 nfcsb1v 3839 . . . . . . . 8 𝑥(𝑘𝑗) / 𝑥𝐵
9190nfel1 2965 . . . . . . 7 𝑥(𝑘𝑗) / 𝑥𝐵 ∈ ℂ
92 csbeq1a 3830 . . . . . . . 8 (𝑥 = (𝑘𝑗) → 𝐵 = (𝑘𝑗) / 𝑥𝐵)
9392eleq1d 2869 . . . . . . 7 (𝑥 = (𝑘𝑗) → (𝐵 ∈ ℂ ↔ (𝑘𝑗) / 𝑥𝐵 ∈ ℂ))
9491, 93rspc 3555 . . . . . 6 ((𝑘𝑗) ∈ (0...(𝑁𝑗)) → (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ → (𝑘𝑗) / 𝑥𝐵 ∈ ℂ))
9584, 89, 94sylc 65 . . . . 5 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 ∈ ℂ)
9664, 95fsumcl 14927 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 ∈ ℂ)
97 oveq2 7031 . . . . 5 (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
98 oveq1 7030 . . . . . . 7 (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
9998csbeq1d 3821 . . . . . 6 (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘𝑗) / 𝑥𝐵 = (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10099adantr 481 . . . . 5 ((𝑘 = ((0 + 𝑁) − 𝑛) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 = (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10197, 100sumeq12dv 14900 . . . 4 (𝑘 = ((0 + 𝑁) − 𝑛) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10296, 101fsumrev2 14974 . . 3 (𝜑 → Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10363, 102eqtr4d 2836 . 2 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵)
104 vex 3443 . . . . . 6 𝑘 ∈ V
105104, 6csbie 3849 . . . . 5 𝑘 / 𝑥𝐵 = 𝐴
106105a1i 11 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 / 𝑥𝐵 = 𝐴)
107106sumeq2dv 14897 . . 3 (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...(𝑁𝑗))𝐴)
108107sumeq2i 14893 . 2 Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴
109 ovex 7055 . . . . . 6 (𝑘𝑗) ∈ V
110 fsum0diag2.2 . . . . . 6 (𝑥 = (𝑘𝑗) → 𝐵 = 𝐶)
111109, 110csbie 3849 . . . . 5 (𝑘𝑗) / 𝑥𝐵 = 𝐶
112111a1i 11 . . . 4 ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 = 𝐶)
113112sumeq2dv 14897 . . 3 (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
114113sumeq2i 14893 . 2 Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶
115103, 108, 1143eqtr3g 2856 1 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083  wral 3107  csb 3817  wss 3865  cfv 6232  (class class class)co 7023  cc 10388  0cc0 10390   + caddc 10393  cmin 10723  0cn0 11751  cz 11835  cuz 12097  ...cfz 12746  Σcsu 14880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-oi 8827  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-n0 11752  df-z 11836  df-uz 12098  df-rp 12244  df-fz 12747  df-fzo 12888  df-seq 13224  df-exp 13284  df-hash 13545  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-clim 14683  df-sum 14881
This theorem is referenced by:  mertens  15079  plymullem1  24491
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