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Theorem fsum0diag2 15668
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 13548 . . . . . . 7 (𝑛 ∈ (0...(𝑁𝑗)) → ((𝑁𝑗) − 𝑛) ∈ (0...(𝑁𝑗)))
21ad2antll 727 . . . . . 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 3142 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
6 fsum0diag2.1 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝐴)
76eleq1d 2822 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
87cbvralvw 3225 . . . . . . . 8 (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ ↔ ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
95, 8sylibr 233 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
109adantrr 715 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁𝑗)))) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
11 nfcsb1v 3880 . . . . . . . 8 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵
1211nfel1 2923 . . . . . . 7 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ
13 csbeq1a 3869 . . . . . . . 8 (𝑥 = ((𝑁𝑗) − 𝑛) → 𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
1413eleq1d 2822 . . . . . . 7 (𝑥 = ((𝑁𝑗) − 𝑛) → (𝐵 ∈ ℂ ↔ ((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ))
1512, 14rspc 3569 . . . . . 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 15662 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
18 nfcsb1v 3880 . . . . . . . . . 10 𝑥𝑘 / 𝑥𝐵
1918nfel1 2923 . . . . . . . . 9 𝑥𝑘 / 𝑥𝐵 ∈ ℂ
20 csbeq1a 3869 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
2120eleq1d 2822 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝑘 / 𝑥𝐵 ∈ ℂ))
2219, 21rspc 3569 . . . . . . . 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 3858 . . . . . . 7 (𝑘 = ((0 + (𝑁𝑗)) − 𝑛) → 𝑘 / 𝑥𝐵 = ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
2523, 24fsumrev2 15667 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
26 elfz3nn0 13535 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
2726ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℕ0)
28 elfzelz 13441 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
2928ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℤ)
30 nn0cn 12423 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
31 zcn 12504 . . . . . . . . . . . 12 (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
32 subcl 11400 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁𝑗) ∈ ℂ)
3330, 31, 32syl2an 596 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑗 ∈ ℤ) → (𝑁𝑗) ∈ ℂ)
3427, 29, 33syl2anc 584 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ∈ ℂ)
35 addid2 11338 . . . . . . . . . 10 ((𝑁𝑗) ∈ ℂ → (0 + (𝑁𝑗)) = (𝑁𝑗))
3634, 35syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (0 + (𝑁𝑗)) = (𝑁𝑗))
3736oveq1d 7372 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) = ((𝑁𝑗) − 𝑛))
3837csbeq1d 3859 . . . . . . 7 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
3938sumeq2dv 15588 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4025, 39eqtrd 2776 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4140sumeq2dv 15588 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
42 elfz3nn0 13535 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
4342adantl 482 . . . . . . . . 9 ((𝜑𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
44 addid2 11338 . . . . . . . . 9 (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
4543, 30, 443syl 18 . . . . . . . 8 ((𝜑𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
4645oveq1d 7372 . . . . . . 7 ((𝜑𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁𝑛))
4746oveq2d 7373 . . . . . 6 ((𝜑𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁𝑛)))
4846oveq1d 7372 . . . . . . . . 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 725 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑁 ∈ ℕ0)
51 elfzelz 13441 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
5251ad2antlr 725 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑛 ∈ ℤ)
53 elfzelz 13441 . . . . . . . . . 10 (𝑗 ∈ (0...(𝑁𝑛)) → 𝑗 ∈ ℤ)
5453adantl 482 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑗 ∈ ℤ)
55 zcn 12504 . . . . . . . . . 10 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
56 sub32 11435 . . . . . . . . . 10 ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5730, 55, 31, 56syl3an 1160 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5850, 52, 54, 57syl3anc 1371 . . . . . . . 8 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5949, 58eqtrd 2776 . . . . . . 7 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
6059csbeq1d 3859 . . . . . 6 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6147, 60sumeq12rdv 15592 . . . . 5 ((𝜑𝑛 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6261sumeq2dv 15588 . . . 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 2786 . . 3 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
64 fzfid 13878 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
65 elfzuz3 13438 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ𝑗))
6665adantl 482 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ𝑗))
67 elfzuz3 13438 . . . . . . . . . . 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 13435 . . . . . . . . 9 (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ𝑗) ∧ 𝑁 ∈ (ℤ𝑘)))
7166, 69, 70sylanbrc 583 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
72 elfzelz 13441 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
7372adantl 482 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
74 elfzel2 13439 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
7574ad2antlr 725 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
76 elfzelz 13441 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
7776ad2antlr 725 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
78 fzsubel 13477 . . . . . . . . 9 (((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗))))
7973, 75, 77, 73, 78syl22anc 837 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗))))
8071, 79mpbid 231 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗)))
81 subid 11420 . . . . . . . . 9 (𝑗 ∈ ℂ → (𝑗𝑗) = 0)
8273, 31, 813syl 18 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑗𝑗) = 0)
8382oveq1d 7372 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗𝑗)...(𝑁𝑗)) = (0...(𝑁𝑗)))
8480, 83eleqtrd 2840 . . . . . 6 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ (0...(𝑁𝑗)))
85 simpll 765 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
86 fzss2 13481 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑘) → (0...𝑘) ⊆ (0...𝑁))
8768, 86syl 17 . . . . . . . 8 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ⊆ (0...𝑁))
8887sselda 3944 . . . . . . 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 3880 . . . . . . . 8 𝑥(𝑘𝑗) / 𝑥𝐵
9190nfel1 2923 . . . . . . 7 𝑥(𝑘𝑗) / 𝑥𝐵 ∈ ℂ
92 csbeq1a 3869 . . . . . . . 8 (𝑥 = (𝑘𝑗) → 𝐵 = (𝑘𝑗) / 𝑥𝐵)
9392eleq1d 2822 . . . . . . 7 (𝑥 = (𝑘𝑗) → (𝐵 ∈ ℂ ↔ (𝑘𝑗) / 𝑥𝐵 ∈ ℂ))
9491, 93rspc 3569 . . . . . 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 15618 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 ∈ ℂ)
97 oveq2 7365 . . . . 5 (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
98 oveq1 7364 . . . . . . 7 (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
9998csbeq1d 3859 . . . . . 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 15591 . . . 4 (𝑘 = ((0 + 𝑁) − 𝑛) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10296, 101fsumrev2 15667 . . 3 (𝜑 → Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10363, 102eqtr4d 2779 . 2 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵)
104 vex 3449 . . . . . 6 𝑘 ∈ V
105104, 6csbie 3891 . . . . 5 𝑘 / 𝑥𝐵 = 𝐴
106105a1i 11 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 / 𝑥𝐵 = 𝐴)
107106sumeq2dv 15588 . . 3 (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...(𝑁𝑗))𝐴)
108107sumeq2i 15584 . 2 Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴
109 ovex 7390 . . . . . 6 (𝑘𝑗) ∈ V
110 fsum0diag2.2 . . . . . 6 (𝑥 = (𝑘𝑗) → 𝐵 = 𝐶)
111109, 110csbie 3891 . . . . 5 (𝑘𝑗) / 𝑥𝐵 = 𝐶
112111a1i 11 . . . 4 ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 = 𝐶)
113112sumeq2dv 15588 . . 3 (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
114113sumeq2i 15584 . 2 Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶
115103, 108, 1143eqtr3g 2799 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 205  wa 396   = wceq 1541  wcel 2106  wral 3064  csb 3855  wss 3910  cfv 6496  (class class class)co 7357  cc 11049  0cc0 11051   + caddc 11054  cmin 11385  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  Σcsu 15570
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571
This theorem is referenced by:  mertens  15771  plymullem1  25575
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