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Theorem fisum0diag2 11167
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)
fisum0diag2.n (𝜑𝑁 ∈ ℤ)
Assertion
Ref Expression
fisum0diag2 (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 fisum0diag2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 fznn0sub2 9856 . . . . . . 7 (𝑛 ∈ (0...(𝑁𝑗)) → ((𝑁𝑗) − 𝑛) ∈ (0...(𝑁𝑗)))
21ad2antll 480 . . . . . 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 370 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑘 ∈ (0...(𝑁𝑗)) → 𝐴 ∈ ℂ))
54ralrimiv 2479 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
6 fsum0diag2.1 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝐴)
76eleq1d 2184 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
87cbvralv 2629 . . . . . . . 8 (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ ↔ ∀𝑘 ∈ (0...(𝑁𝑗))𝐴 ∈ ℂ)
95, 8sylibr 133 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
109adantrr 468 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁𝑗)))) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
11 nfcsb1v 3003 . . . . . . . 8 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵
1211nfel1 2267 . . . . . . 7 𝑥((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ
13 csbeq1a 2981 . . . . . . . 8 (𝑥 = ((𝑁𝑗) − 𝑛) → 𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
1413eleq1d 2184 . . . . . . 7 (𝑥 = ((𝑁𝑗) − 𝑛) → (𝐵 ∈ ℂ ↔ ((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ))
1512, 14rspc 2755 . . . . . 6 (((𝑁𝑗) − 𝑛) ∈ (0...(𝑁𝑗)) → (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ → ((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ))
162, 10, 15sylc 62 . . . . 5 ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁𝑗)))) → ((𝑁𝑗) − 𝑛) / 𝑥𝐵 ∈ ℂ)
17 fisum0diag2.n . . . . 5 (𝜑𝑁 ∈ ℤ)
1816, 17fisum0diag 11161 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
19 0zd 9020 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → 0 ∈ ℤ)
2017adantr 272 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
21 elfzelz 9757 . . . . . . . . 9 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
2221adantl 273 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑗 ∈ ℤ)
2320, 22zsubcld 9132 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑁𝑗) ∈ ℤ)
24 nfcsb1v 3003 . . . . . . . . . 10 𝑥𝑘 / 𝑥𝐵
2524nfel1 2267 . . . . . . . . 9 𝑥𝑘 / 𝑥𝐵 ∈ ℂ
26 csbeq1a 2981 . . . . . . . . . 10 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
2726eleq1d 2184 . . . . . . . . 9 (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝑘 / 𝑥𝐵 ∈ ℂ))
2825, 27rspc 2755 . . . . . . . 8 (𝑘 ∈ (0...(𝑁𝑗)) → (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ → 𝑘 / 𝑥𝐵 ∈ ℂ))
299, 28mpan9 277 . . . . . . 7 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 / 𝑥𝐵 ∈ ℂ)
30 csbeq1 2976 . . . . . . 7 (𝑘 = ((0 + (𝑁𝑗)) − 𝑛) → 𝑘 / 𝑥𝐵 = ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
3119, 23, 29, 30fisumrev2 11166 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵)
32 elfz3nn0 9846 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
3332ad2antlr 478 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℕ0)
3421ad2antlr 478 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℤ)
35 nn0cn 8941 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
36 zcn 9013 . . . . . . . . . . . 12 (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
37 subcl 7925 . . . . . . . . . . . 12 ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁𝑗) ∈ ℂ)
3835, 36, 37syl2an 285 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑗 ∈ ℤ) → (𝑁𝑗) ∈ ℂ)
3933, 34, 38syl2anc 406 . . . . . . . . . 10 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ∈ ℂ)
4039addid2d 7876 . . . . . . . . 9 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → (0 + (𝑁𝑗)) = (𝑁𝑗))
4140oveq1d 5755 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) = ((𝑁𝑗) − 𝑛))
4241csbeq1d 2979 . . . . . . 7 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁𝑗))) → ((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4342sumeq2dv 11088 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑛 ∈ (0...(𝑁𝑗))((0 + (𝑁𝑗)) − 𝑛) / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4431, 43eqtrd 2148 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
4544sumeq2dv 11088 . . . 4 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑛 ∈ (0...(𝑁𝑗))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
46 elfz3nn0 9846 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
4746adantl 273 . . . . . . . . 9 ((𝜑𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
48 addid2 7865 . . . . . . . . 9 (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
4947, 35, 483syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
5049oveq1d 5755 . . . . . . 7 ((𝜑𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁𝑛))
5150oveq2d 5756 . . . . . 6 ((𝜑𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁𝑛)))
5250oveq1d 5755 . . . . . . . . 9 ((𝜑𝑛 ∈ (0...𝑁)) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑛) − 𝑗))
5352adantr 272 . . . . . . . 8 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑛) − 𝑗))
5446ad2antlr 478 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑁 ∈ ℕ0)
55 elfzelz 9757 . . . . . . . . . 10 (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
5655ad2antlr 478 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑛 ∈ ℤ)
57 elfzelz 9757 . . . . . . . . . 10 (𝑗 ∈ (0...(𝑁𝑛)) → 𝑗 ∈ ℤ)
5857adantl 273 . . . . . . . . 9 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → 𝑗 ∈ ℤ)
59 zcn 9013 . . . . . . . . . 10 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
60 sub32 7960 . . . . . . . . . 10 ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
6135, 59, 36, 60syl3an 1241 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
6254, 56, 58, 61syl3anc 1199 . . . . . . . 8 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → ((𝑁𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
6353, 62eqtrd 2148 . . . . . . 7 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁𝑗) − 𝑛))
6463csbeq1d 2979 . . . . . 6 (((𝜑𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = ((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6551, 64sumeq12rdv 11093 . . . . 5 ((𝜑𝑛 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6665sumeq2dv 11088 . . . 4 (𝜑 → Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑛))((𝑁𝑗) − 𝑛) / 𝑥𝐵)
6718, 45, 663eqtr4d 2158 . . 3 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
68 0zd 9020 . . . 4 (𝜑 → 0 ∈ ℤ)
69 0zd 9020 . . . . . 6 ((𝜑𝑘 ∈ (0...𝑁)) → 0 ∈ ℤ)
70 elfzelz 9757 . . . . . . 7 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
7170adantl 273 . . . . . 6 ((𝜑𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ)
7269, 71fzfigd 10155 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
73 elfzuz3 9754 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ𝑗))
7473adantl 273 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ𝑗))
75 elfzuz3 9754 . . . . . . . . . . 11 (𝑘 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝑘))
7675adantl 273 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ𝑘))
7776adantr 272 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ (ℤ𝑘))
78 elfzuzb 9751 . . . . . . . . 9 (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ𝑗) ∧ 𝑁 ∈ (ℤ𝑘)))
7974, 77, 78sylanbrc 411 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
80 elfzelz 9757 . . . . . . . . . 10 (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
8180adantl 273 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
8217ad2antrr 477 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
8370ad2antlr 478 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
84 fzsubel 9791 . . . . . . . . 9 (((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗))))
8581, 82, 83, 81, 84syl22anc 1200 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗))))
8679, 85mpbid 146 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ ((𝑗𝑗)...(𝑁𝑗)))
87 subid 7945 . . . . . . . . 9 (𝑗 ∈ ℂ → (𝑗𝑗) = 0)
8881, 36, 873syl 17 . . . . . . . 8 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑗𝑗) = 0)
8988oveq1d 5755 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗𝑗)...(𝑁𝑗)) = (0...(𝑁𝑗)))
9086, 89eleqtrd 2194 . . . . . 6 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ (0...(𝑁𝑗)))
91 simpll 501 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
92 fzss2 9795 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑘) → (0...𝑘) ⊆ (0...𝑁))
9376, 92syl 14 . . . . . . . 8 ((𝜑𝑘 ∈ (0...𝑁)) → (0...𝑘) ⊆ (0...𝑁))
9493sselda 3065 . . . . . . 7 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ (0...𝑁))
9591, 94, 9syl2anc 406 . . . . . 6 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ)
96 nfcsb1v 3003 . . . . . . . 8 𝑥(𝑘𝑗) / 𝑥𝐵
9796nfel1 2267 . . . . . . 7 𝑥(𝑘𝑗) / 𝑥𝐵 ∈ ℂ
98 csbeq1a 2981 . . . . . . . 8 (𝑥 = (𝑘𝑗) → 𝐵 = (𝑘𝑗) / 𝑥𝐵)
9998eleq1d 2184 . . . . . . 7 (𝑥 = (𝑘𝑗) → (𝐵 ∈ ℂ ↔ (𝑘𝑗) / 𝑥𝐵 ∈ ℂ))
10097, 99rspc 2755 . . . . . 6 ((𝑘𝑗) ∈ (0...(𝑁𝑗)) → (∀𝑥 ∈ (0...(𝑁𝑗))𝐵 ∈ ℂ → (𝑘𝑗) / 𝑥𝐵 ∈ ℂ))
10190, 95, 100sylc 62 . . . . 5 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 ∈ ℂ)
10272, 101fsumcl 11120 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 ∈ ℂ)
103 oveq2 5748 . . . . 5 (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
104 oveq1 5747 . . . . . . 7 (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
105104csbeq1d 2979 . . . . . 6 (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘𝑗) / 𝑥𝐵 = (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
106105adantr 272 . . . . 5 ((𝑘 = ((0 + 𝑁) − 𝑛) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 = (((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
107103, 106sumeq12dv 11092 . . . 4 (𝑘 = ((0 + 𝑁) − 𝑛) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10868, 17, 102, 107fisumrev2 11166 . . 3 (𝜑 → Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑛 ∈ (0...𝑁𝑗 ∈ (0...((0 + 𝑁) − 𝑛))(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥𝐵)
10967, 108eqtr4d 2151 . 2 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵)
110 vex 2661 . . . . . 6 𝑘 ∈ V
111110, 6csbie 3013 . . . . 5 𝑘 / 𝑥𝐵 = 𝐴
112111a1i 9 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 / 𝑥𝐵 = 𝐴)
113112sumeq2dv 11088 . . 3 (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑘 ∈ (0...(𝑁𝑗))𝐴)
114113sumeq2i 11084 . 2 Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝑘 / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴
11570adantr 272 . . . . . 6 ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
11680adantl 273 . . . . . 6 ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
117115, 116zsubcld 9132 . . . . 5 ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ ℤ)
118 fsum0diag2.2 . . . . . 6 (𝑥 = (𝑘𝑗) → 𝐵 = 𝐶)
119118adantl 273 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) ∧ 𝑥 = (𝑘𝑗)) → 𝐵 = 𝐶)
120117, 119csbied 3014 . . . 4 ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) / 𝑥𝐵 = 𝐶)
121120sumeq2dv 11088 . . 3 (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
122121sumeq2i 11084 . 2 Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)(𝑘𝑗) / 𝑥𝐵 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶
123109, 114, 1223eqtr3g 2171 1 (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  wral 2391  csb 2973  wss 3039  cfv 5091  (class class class)co 5740  cc 7582  0cc0 7584   + caddc 7587  cmin 7897  0cn0 8931  cz 9008  cuz 9278  ...cfz 9741  Σcsu 11073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-disj 3875  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-fzo 9871  df-seqfrec 10170  df-exp 10244  df-ihash 10473  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-clim 10999  df-sumdc 11074
This theorem is referenced by:  mertensabs  11257
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