MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsum0diag2 Structured version   Visualization version   GIF version

Theorem fsum0diag2 15134
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 13013 . . . . . . 7 (𝑛 ∈ (0...(𝑁𝑗)) → ((𝑁𝑗) − 𝑛) ∈ (0...(𝑁𝑗)))
21ad2antll 728 . . . . . 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 460 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑘 ∈ (0...(𝑁𝑗)) → 𝐴 ∈ ℂ))
54ralrimiv 3151 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
6 fsum0diag2.1 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝐴)
76eleq1d 2877 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
87cbvralvw 3399 . . . . . . . 8 (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ ↔ ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
95, 8sylibr 237 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
109adantrr 716 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁𝑗)))) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
11 nfcsb1v 3855 . . . . . . . 8 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵
1211nfel1 2974 . . . . . . 7 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ
13 csbeq1a 3845 . . . . . . . 8 (𝑥 = ((𝑁𝑗) − 𝑛) → 𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
1413eleq1d 2877 . . . . . . 7 (𝑥 = ((𝑁𝑗) − 𝑛) → (𝐵 ∈ ℂ ↔ ((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ))
1512, 14rspc 3562 . . . . . 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 15128 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
18 nfcsb1v 3855 . . . . . . . . . 10 𝑥𝑘 / 𝑥𝐵
1918nfel1 2974 . . . . . . . . 9 𝑥𝑘 / 𝑥𝐵 ∈ ℂ
20 csbeq1a 3845 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
2120eleq1d 2877 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝑘 / 𝑥𝐵 ∈ ℂ))
2219, 21rspc 3562 . . . . . . . 8 (𝑘 ∈ (0...(𝑁𝑗)) → (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ → 𝑘 / 𝑥𝐵 ∈ ℂ))
239, 22mpan9 510 . . . . . . 7 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 / 𝑥𝐵 ∈ ℂ)
24 csbeq1 3834 . . . . . . 7 (𝑘 = ((0 + (𝑁𝑗)) − 𝑛) → 𝑘 / 𝑥𝐵 = ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
2523, 24fsumrev2 15133 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
26 elfz3nn0 13000 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
2726ad2antlr 726 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℕ0)
28 elfzelz 12906 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
2928ad2antlr 726 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℤ)
30 nn0cn 11899 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
31 zcn 11978 . . . . . . . . . . . 12 (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
32 subcl 10878 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁𝑗) ∈ ℂ)
3330, 31, 32syl2an 598 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑗 ∈ ℤ) → (𝑁𝑗) ∈ ℂ)
3427, 29, 33syl2anc 587 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ∈ ℂ)
35 addid2 10816 . . . . . . . . . 10 ((𝑁𝑗) ∈ ℂ → (0 + (𝑁𝑗)) = (𝑁𝑗))
3634, 35syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (0 + (𝑁𝑗)) = (𝑁𝑗))
3736oveq1d 7154 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) = ((𝑁𝑗) − 𝑛))
3837csbeq1d 3835 . . . . . . 7 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
3938sumeq2dv 15056 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4025, 39eqtrd 2836 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4140sumeq2dv 15056 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
42 elfz3nn0 13000 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
4342adantl 485 . . . . . . . . 9 ((𝜑𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
44 addid2 10816 . . . . . . . . 9 (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
4543, 30, 443syl 18 . . . . . . . 8 ((𝜑𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
4645oveq1d 7154 . . . . . . 7 ((𝜑𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁𝑛))
4746oveq2d 7155 . . . . . 6 ((𝜑𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁𝑛)))
4846oveq1d 7154 . . . . . . . . 9 ((𝜑𝑛 ∈ (0...𝑁)) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑛) − 𝑗))
4948adantr 484 . . . . . . . 8 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑛) − 𝑗))
5042ad2antlr 726 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑁 ∈ ℕ0)
51 elfzelz 12906 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
5251ad2antlr 726 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑛 ∈ ℤ)
53 elfzelz 12906 . . . . . . . . . 10 (𝑗 ∈ (0...(𝑁𝑛)) → 𝑗 ∈ ℤ)
5453adantl 485 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑗 ∈ ℤ)
55 zcn 11978 . . . . . . . . . 10 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
56 sub32 10913 . . . . . . . . . 10 ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5730, 55, 31, 56syl3an 1157 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5850, 52, 54, 57syl3anc 1368 . . . . . . . 8 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
5949, 58eqtrd 2836 . . . . . . 7 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
6059csbeq1d 3835 . . . . . 6 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6147, 60sumeq12rdv 15060 . . . . 5 ((𝜑𝑛 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6261sumeq2dv 15056 . . . 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 2846 . . 3 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
64 fzfid 13340 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
65 elfzuz3 12903 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ𝑗))
6665adantl 485 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ𝑗))
67 elfzuz3 12903 . . . . . . . . . . 11 (𝑘 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝑘))
6867adantl 485 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ𝑘))
6968adantr 484 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ (ℤ𝑘))
70 elfzuzb 12900 . . . . . . . . 9 (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ𝑗) ∧ 𝑁 ∈ (ℤ𝑘)))
7166, 69, 70sylanbrc 586 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
72 elfzelz 12906 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
7372adantl 485 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
74 elfzel2 12904 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
7574ad2antlr 726 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
76 elfzelz 12906 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
7776ad2antlr 726 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
78 fzsubel 12942 . . . . . . . . 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 235 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗)))
81 subid 10898 . . . . . . . . 9 (𝑗 ∈ ℂ → (𝑗𝑗) = 0)
8273, 31, 813syl 18 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑗𝑗) = 0)
8382oveq1d 7154 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗𝑗)...(𝑁𝑗)) = (0...(𝑁𝑗)))
8480, 83eleqtrd 2895 . . . . . 6 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ (0...(𝑁𝑗)))
85 simpll 766 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
86 fzss2 12946 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑘) → (0...𝑘) ⊆ (0...𝑁))
8768, 86syl 17 . . . . . . . 8 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ⊆ (0...𝑁))
8887sselda 3918 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ (0...𝑁))
8985, 88, 9syl2anc 587 . . . . . 6 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
90 nfcsb1v 3855 . . . . . . . 8 𝑥(𝑘𝑗) / 𝑥𝐵
9190nfel1 2974 . . . . . . 7 𝑥(𝑘𝑗) / 𝑥𝐵 ∈ ℂ
92 csbeq1a 3845 . . . . . . . 8 (𝑥 = (𝑘𝑗) → 𝐵 = (𝑘𝑗) / 𝑥𝐵)
9392eleq1d 2877 . . . . . . 7 (𝑥 = (𝑘𝑗) → (𝐵 ∈ ℂ ↔ (𝑘𝑗) / 𝑥𝐵 ∈ ℂ))
9491, 93rspc 3562 . . . . . 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 15086 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 ∈ ℂ)
97 oveq2 7147 . . . . 5 (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
98 oveq1 7146 . . . . . . 7 (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
9998csbeq1d 3835 . . . . . 6 (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘𝑗) / 𝑥𝐵 = (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10099adantr 484 . . . . 5 ((𝑘 = ((0 + 𝑁) − 𝑛) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 = (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10197, 100sumeq12dv 15059 . . . 4 (𝑘 = ((0 + 𝑁) − 𝑛) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10296, 101fsumrev2 15133 . . 3 (𝜑 → Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10363, 102eqtr4d 2839 . 2 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵)
104 vex 3447 . . . . . 6 𝑘 ∈ V
105104, 6csbie 3866 . . . . 5 𝑘 / 𝑥𝐵 = 𝐴
106105a1i 11 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 / 𝑥𝐵 = 𝐴)
107106sumeq2dv 15056 . . 3 (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...(𝑁𝑗))𝐴)
108107sumeq2i 15052 . 2 Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴
109 ovex 7172 . . . . . 6 (𝑘𝑗) ∈ V
110 fsum0diag2.2 . . . . . 6 (𝑥 = (𝑘𝑗) → 𝐵 = 𝐶)
111109, 110csbie 3866 . . . . 5 (𝑘𝑗) / 𝑥𝐵 = 𝐶
112111a1i 11 . . . 4 ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 = 𝐶)
113112sumeq2dv 15056 . . 3 (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
114113sumeq2i 15052 . 2 Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶
115103, 108, 1143eqtr3g 2859 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 209  wa 399   = wceq 1538  wcel 2112  wral 3109  csb 3831  wss 3884  cfv 6328  (class class class)co 7139  cc 10528  0cc0 10530   + caddc 10533  cmin 10863  0cn0 11889  cz 11973  cuz 12235  ...cfz 12889  Σcsu 15038
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-sum 15039
This theorem is referenced by:  mertens  15238  plymullem1  24815
  Copyright terms: Public domain W3C validator