Theorem fisum0diag2 11344

Description: Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular region 0 ≤ 𝑗, 0 ≤ 𝑘, 𝑗 + 𝑘 ≤ 𝑁." (Contributed by Mario Carneiro, 21-Jul-2014.)

Hypotheses

Ref	Expression
fsum0diag2.1	⊢ (𝑥 = 𝑘 → 𝐵 = 𝐴)
fsum0diag2.2	⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = 𝐶)
fsum0diag2.3	⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗)))) → 𝐴 ∈ ℂ)
fisum0diag2.n	⊢ (𝜑 → 𝑁 ∈ ℤ)

Assertion

Ref	Expression
fisum0diag2	⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (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.

Step	Hyp	Ref	Expression
1		fznn0sub2 10027	. . . . . . 7 ⊢ (𝑛 ∈ (0...(𝑁 − 𝑗)) → ((𝑁 − 𝑗) − 𝑛) ∈ (0...(𝑁 − 𝑗)))
2	1	ad2antll 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗)))) → ((𝑁 − 𝑗) − 𝑛) ∈ (0...(𝑁 − 𝑗)))
3		fsum0diag2.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗)))) → 𝐴 ∈ ℂ)
4	3	expr 373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑘 ∈ (0...(𝑁 − 𝑗)) → 𝐴 ∈ ℂ))
5	4	ralrimiv 2529	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 ∈ ℂ)
6		fsum0diag2.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐴)
7	6	eleq1d 2226	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
8	7	cbvralv 2680	. . . . . . . 8 ⊢ (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ ↔ ∀𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 ∈ ℂ)
9	5, 8	sylibr 133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
10	9	adantrr 471	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗)))) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
11		nfcsb1v 3064	. . . . . . . 8 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵
12	11	nfel1 2310	. . . . . . 7 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ
13		csbeq1a 3040	. . . . . . . 8 ⊢ (𝑥 = ((𝑁 − 𝑗) − 𝑛) → 𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
14	13	eleq1d 2226	. . . . . . 7 ⊢ (𝑥 = ((𝑁 − 𝑗) − 𝑛) → (𝐵 ∈ ℂ ↔ ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ))
15	12, 14	rspc 2810	. . . . . 6 ⊢ (((𝑁 − 𝑗) − 𝑛) ∈ (0...(𝑁 − 𝑗)) → (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ → ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ))
16	2, 10, 15	sylc 62	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗)))) → ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ)
17		fisum0diag2.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	16, 17	fisum0diag 11338	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...𝑁)Σ𝑗 ∈ (0...(𝑁 − 𝑛))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
19		0zd 9179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 0 ∈ ℤ)
20	17	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
21		elfzelz 9928	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
22	21	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ ℤ)
23	20, 22	zsubcld 9291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 − 𝑗) ∈ ℤ)
24		nfcsb1v 3064	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵
25	24	nfel1 2310	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ
26		csbeq1a 3040	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
27	26	eleq1d 2226	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ))
28	25, 27	rspc 2810	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑗)) → (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ → ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ))
29	9, 28	mpan9 279	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ)
30		csbeq1 3034	. . . . . . 7 ⊢ (𝑘 = ((0 + (𝑁 − 𝑗)) − 𝑛) → ⦋𝑘 / 𝑥⦌𝐵 = ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
31	19, 23, 29, 30	fisumrev2 11343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
32		elfz3nn0 10017	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
33	32	ad2antlr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑁 ∈ ℕ0)
34	21	ad2antlr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑗 ∈ ℤ)
35		nn0cn 9100	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
36		zcn 9172	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
37		subcl 8074	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁 − 𝑗) ∈ ℂ)
38	35, 36, 37	syl2an 287	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → (𝑁 − 𝑗) ∈ ℂ)
39	33, 34, 38	syl2anc 409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → (𝑁 − 𝑗) ∈ ℂ)
40	39	addid2d 8025	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → (0 + (𝑁 − 𝑗)) = (𝑁 − 𝑗))
41	40	oveq1d 5839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ((0 + (𝑁 − 𝑗)) − 𝑛) = ((𝑁 − 𝑗) − 𝑛))
42	41	csbeq1d 3038	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
43	42	sumeq2dv 11265	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
44	31, 43	eqtrd 2190	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
45	44	sumeq2dv 11265	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑁)Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
46		elfz3nn0 10017	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
47	46	adantl 275	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
48		addid2 8014	. . . . . . . . 9 ⊢ (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
49	47, 35, 48	3syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
50	49	oveq1d 5839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁 − 𝑛))
51	50	oveq2d 5840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁 − 𝑛)))
52	50	oveq1d 5839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
53	52	adantr 274	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
54	46	ad2antlr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑁 ∈ ℕ0)
55		elfzelz 9928	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
56	55	ad2antlr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑛 ∈ ℤ)
57		elfzelz 9928	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...(𝑁 − 𝑛)) → 𝑗 ∈ ℤ)
58	57	adantl 275	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑗 ∈ ℤ)
59		zcn 9172	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
60		sub32 8109	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
61	35, 59, 36, 60	syl3an 1262	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
62	54, 56, 58, 61	syl3anc 1220	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
63	53, 62	eqtrd 2190	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
64	63	csbeq1d 3038	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
65	51, 64	sumeq12rdv 11270	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...(𝑁 − 𝑛))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
66	65	sumeq2dv 11265	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (0...𝑁)Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...𝑁)Σ𝑗 ∈ (0...(𝑁 − 𝑛))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
67	18, 45, 66	3eqtr4d 2200	. . 3 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...𝑁)Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
68		0zd 9179	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
69		0zd 9179	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 0 ∈ ℤ)
70		elfzelz 9928	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
71	70	adantl 275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ)
72	69, 71	fzfigd 10330	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
73		elfzuz3 9925	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ≥‘𝑗))
74	73	adantl 275	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ≥‘𝑗))
75		elfzuz3 9925	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → 𝑁 ∈ (ℤ≥‘𝑘))
76	75	adantl 275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑘))
77	76	adantr 274	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ (ℤ≥‘𝑘))
78		elfzuzb 9922	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ≥‘𝑗) ∧ 𝑁 ∈ (ℤ≥‘𝑘)))
79	74, 77, 78	sylanbrc 414	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
80		elfzelz 9928	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
81	80	adantl 275	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
82	17	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
83	70	ad2antlr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
84		fzsubel 9962	. . . . . . . . 9 ⊢ (((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗))))
85	81, 82, 83, 81, 84	syl22anc 1221	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗))))
86	79, 85	mpbid 146	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗)))
87		subid 8094	. . . . . . . . 9 ⊢ (𝑗 ∈ ℂ → (𝑗 − 𝑗) = 0)
88	81, 36, 87	3syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑗 − 𝑗) = 0)
89	88	oveq1d 5839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗 − 𝑗)...(𝑁 − 𝑗)) = (0...(𝑁 − 𝑗)))
90	86, 89	eleqtrd 2236	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ (0...(𝑁 − 𝑗)))
91		simpll 519	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
92		fzss2 9966	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑘) → (0...𝑘) ⊆ (0...𝑁))
93	76, 92	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (0...𝑘) ⊆ (0...𝑁))
94	93	sselda 3128	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ (0...𝑁))
95	91, 94, 9	syl2anc 409	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
96		nfcsb1v 3064	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵
97	96	nfel1 2310	. . . . . . 7 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ
98		csbeq1a 3040	. . . . . . . 8 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
99	98	eleq1d 2226	. . . . . . 7 ⊢ (𝑥 = (𝑘 − 𝑗) → (𝐵 ∈ ℂ ↔ ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ))
100	97, 99	rspc 2810	. . . . . 6 ⊢ ((𝑘 − 𝑗) ∈ (0...(𝑁 − 𝑗)) → (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ))
101	90, 95, 100	sylc 62	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ)
102	72, 101	fsumcl 11297	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ)
103		oveq2 5832	. . . . 5 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
104		oveq1 5831	. . . . . . 7 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘 − 𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
105	104	csbeq1d 3038	. . . . . 6 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
106	105	adantr 274	. . . . 5 ⊢ ((𝑘 = ((0 + 𝑁) − 𝑛) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
107	103, 106	sumeq12dv 11269	. . . 4 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
108	68, 17, 102, 107	fisumrev2 11343	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...𝑁)Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
109	67, 108	eqtr4d 2193	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
110		vex 2715	. . . . . 6 ⊢ 𝑘 ∈ V
111	110, 6	csbie 3076	. . . . 5 ⊢ ⦋𝑘 / 𝑥⦌𝐵 = 𝐴
112	111	a1i 9	. . . 4 ⊢ ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 = 𝐴)
113	112	sumeq2dv 11265	. . 3 ⊢ (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴)
114	113	sumeq2i 11261	. 2 ⊢ Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴
115	70	adantr 274	. . . . . 6 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
116	80	adantl 275	. . . . . 6 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
117	115, 116	zsubcld 9291	. . . . 5 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ ℤ)
118		fsum0diag2.2	. . . . . 6 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = 𝐶)
119	118	adantl 275	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) ∧ 𝑥 = (𝑘 − 𝑗)) → 𝐵 = 𝐶)
120	117, 119	csbied 3077	. . . 4 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = 𝐶)
121	120	sumeq2dv 11265	. . 3 ⊢ (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
122	121	sumeq2i 11261	. 2 ⊢ Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶
123	109, 114, 122	3eqtr3g 2213	1 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335   ∈ wcel 2128  ∀wral 2435  ⦋csb 3031   ⊆ wss 3102  ‘cfv 5170  (class class class)co 5824  ℂcc 7730  0cc0 7732   + caddc 7735   − cmin 8046  ℕ₀cn0 9090  ℤcz 9167  ℤ_≥cuz 9439  ...cfz 9912  Σcsu 11250

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-disj 3943  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-frec 6338  df-1o 6363  df-oadd 6367  df-er 6480  df-en 6686  df-dom 6687  df-fin 6688  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fz 9913  df-fzo 10042  df-seqfrec 10345  df-exp 10419  df-ihash 10650  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-clim 11176  df-sumdc 11251

This theorem is referenced by:  mertensabs  11434