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Theorem fsum0diag2 14714
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 12640 . . . . . . 7 (𝑛 ∈ (0...(𝑁𝑗)) → ((𝑁𝑗) − 𝑛) ∈ (0...(𝑁𝑗)))
21ad2antll 767 . . . . . 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 644 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑘 ∈ (0...(𝑁𝑗)) → 𝐴 ∈ ℂ))
54ralrimiv 3103 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
6 fsum0diag2.1 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝐴)
76eleq1d 2824 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
87cbvralv 3310 . . . . . . . 8 (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ ↔ ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
95, 8sylibr 224 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
109adantrr 755 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁𝑗)))) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
11 nfcsb1v 3690 . . . . . . . 8 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵
1211nfel1 2917 . . . . . . 7 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ
13 csbeq1a 3683 . . . . . . . 8 (𝑥 = ((𝑁𝑗) − 𝑛) → 𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
1413eleq1d 2824 . . . . . . 7 (𝑥 = ((𝑁𝑗) − 𝑛) → (𝐵 ∈ ℂ ↔ ((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ))
1512, 14rspc 3443 . . . . . 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 14708 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
18 nfcsb1v 3690 . . . . . . . . . 10 𝑥𝑘 / 𝑥𝐵
1918nfel1 2917 . . . . . . . . 9 𝑥𝑘 / 𝑥𝐵 ∈ ℂ
20 csbeq1a 3683 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
2120eleq1d 2824 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝑘 / 𝑥𝐵 ∈ ℂ))
2219, 21rspc 3443 . . . . . . . 8 (𝑘 ∈ (0...(𝑁𝑗)) → (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ → 𝑘 / 𝑥𝐵 ∈ ℂ))
239, 22mpan9 487 . . . . . . 7 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 / 𝑥𝐵 ∈ ℂ)
24 csbeq1 3677 . . . . . . 7 (𝑘 = ((0 + (𝑁𝑗)) − 𝑛) → 𝑘 / 𝑥𝐵 = ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
2523, 24fsumrev2 14713 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
26 elfz3nn0 12627 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
2726ad2antlr 765 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℕ0)
28 elfzelz 12535 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
2928ad2antlr 765 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℤ)
30 nn0cn 11494 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
31 zcn 11574 . . . . . . . . . . . 12 (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
32 subcl 10472 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁𝑗) ∈ ℂ)
3330, 31, 32syl2an 495 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑗 ∈ ℤ) → (𝑁𝑗) ∈ ℂ)
3427, 29, 33syl2anc 696 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ∈ ℂ)
35 addid2 10411 . . . . . . . . . 10 ((𝑁𝑗) ∈ ℂ → (0 + (𝑁𝑗)) = (𝑁𝑗))
3634, 35syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (0 + (𝑁𝑗)) = (𝑁𝑗))
3736oveq1d 6828 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) = ((𝑁𝑗) − 𝑛))
3837csbeq1d 3681 . . . . . . 7 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
3938sumeq2dv 14632 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4025, 39eqtrd 2794 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4140sumeq2dv 14632 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
42 elfz3nn0 12627 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
4342adantl 473 . . . . . . . . 9 ((𝜑𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
44 addid2 10411 . . . . . . . . 9 (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
4543, 30, 443syl 18 . . . . . . . 8 ((𝜑𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
4645oveq1d 6828 . . . . . . 7 ((𝜑𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁𝑛))
4746oveq2d 6829 . . . . . 6 ((𝜑𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁𝑛)))
4846oveq1d 6828 . . . . . . . . 9 ((𝜑𝑛 ∈ (0...𝑁)) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑛) − 𝑗))
4948adantr 472 . . . . . . . 8 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑛) − 𝑗))
5042ad2antlr 765 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑁 ∈ ℕ0)
51 elfzelz 12535 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
5251ad2antlr 765 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑛 ∈ ℤ)
53 elfzelz 12535 . . . . . . . . . 10 (𝑗 ∈ (0...(𝑁𝑛)) → 𝑗 ∈ ℤ)
5453adantl 473 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑗 ∈ ℤ)
55 zcn 11574 . . . . . . . . . 10 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
56 sub32 10507 . . . . . . . . . 10 ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5730, 55, 31, 56syl3an 1164 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5850, 52, 54, 57syl3anc 1477 . . . . . . . 8 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5949, 58eqtrd 2794 . . . . . . 7 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
6059csbeq1d 3681 . . . . . 6 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6147, 60sumeq12rdv 14637 . . . . 5 ((𝜑𝑛 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6261sumeq2dv 14632 . . . 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 2804 . . 3 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
64 fzfid 12966 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
65 elfzuz3 12532 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ𝑗))
6665adantl 473 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ𝑗))
67 elfzuz3 12532 . . . . . . . . . . 11 (𝑘 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝑘))
6867adantl 473 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ𝑘))
6968adantr 472 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ (ℤ𝑘))
70 elfzuzb 12529 . . . . . . . . 9 (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ𝑗) ∧ 𝑁 ∈ (ℤ𝑘)))
7166, 69, 70sylanbrc 701 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
72 elfzelz 12535 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
7372adantl 473 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
74 elfzel2 12533 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
7574ad2antlr 765 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
76 elfzelz 12535 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
7776ad2antlr 765 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
78 fzsubel 12570 . . . . . . . . 9 (((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗))))
7973, 75, 77, 73, 78syl22anc 1478 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗))))
8071, 79mpbid 222 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗)))
81 subid 10492 . . . . . . . . 9 (𝑗 ∈ ℂ → (𝑗𝑗) = 0)
8273, 31, 813syl 18 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑗𝑗) = 0)
8382oveq1d 6828 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗𝑗)...(𝑁𝑗)) = (0...(𝑁𝑗)))
8480, 83eleqtrd 2841 . . . . . 6 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ (0...(𝑁𝑗)))
85 simpll 807 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
86 fzss2 12574 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑘) → (0...𝑘) ⊆ (0...𝑁))
8768, 86syl 17 . . . . . . . 8 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ⊆ (0...𝑁))
8887sselda 3744 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ (0...𝑁))
8985, 88, 9syl2anc 696 . . . . . 6 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
90 nfcsb1v 3690 . . . . . . . 8 𝑥(𝑘𝑗) / 𝑥𝐵
9190nfel1 2917 . . . . . . 7 𝑥(𝑘𝑗) / 𝑥𝐵 ∈ ℂ
92 csbeq1a 3683 . . . . . . . 8 (𝑥 = (𝑘𝑗) → 𝐵 = (𝑘𝑗) / 𝑥𝐵)
9392eleq1d 2824 . . . . . . 7 (𝑥 = (𝑘𝑗) → (𝐵 ∈ ℂ ↔ (𝑘𝑗) / 𝑥𝐵 ∈ ℂ))
9491, 93rspc 3443 . . . . . 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 14663 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 ∈ ℂ)
97 oveq2 6821 . . . . 5 (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
98 oveq1 6820 . . . . . . 7 (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
9998csbeq1d 3681 . . . . . 6 (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘𝑗) / 𝑥𝐵 = (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10099adantr 472 . . . . 5 ((𝑘 = ((0 + 𝑁) − 𝑛) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 = (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10197, 100sumeq12dv 14636 . . . 4 (𝑘 = ((0 + 𝑁) − 𝑛) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10296, 101fsumrev2 14713 . . 3 (𝜑 → Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10363, 102eqtr4d 2797 . 2 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵)
104 vex 3343 . . . . . 6 𝑘 ∈ V
105104, 6csbie 3700 . . . . 5 𝑘 / 𝑥𝐵 = 𝐴
106105a1i 11 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 / 𝑥𝐵 = 𝐴)
107106sumeq2dv 14632 . . 3 (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...(𝑁𝑗))𝐴)
108107sumeq2i 14628 . 2 Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴
109 ovex 6841 . . . . . 6 (𝑘𝑗) ∈ V
110 fsum0diag2.2 . . . . . 6 (𝑥 = (𝑘𝑗) → 𝐵 = 𝐶)
111109, 110csbie 3700 . . . . 5 (𝑘𝑗) / 𝑥𝐵 = 𝐶
112111a1i 11 . . . 4 ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 = 𝐶)
113112sumeq2dv 14632 . . 3 (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
114113sumeq2i 14628 . 2 Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶
115103, 108, 1143eqtr3g 2817 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  csb 3674  wss 3715  cfv 6049  (class class class)co 6813  cc 10126  0cc0 10128   + caddc 10131  cmin 10458  0cn0 11484  cz 11569  cuz 11879  ...cfz 12519  Σcsu 14615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-sum 14616
This theorem is referenced by:  mertens  14817  plymullem1  24169
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