Theorem fisum0diag2 11473

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 10146	. . . . . . 7 ⊢ (𝑛 ∈ (0...(𝑁 − 𝑗)) → ((𝑁 − 𝑗) − 𝑛) ∈ (0...(𝑁 − 𝑗)))
2	1	ad2antll 491	. . . . . 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 375	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑘 ∈ (0...(𝑁 − 𝑗)) → 𝐴 ∈ ℂ))
5	4	ralrimiv 2562	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 ∈ ℂ)
6		fsum0diag2.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐴)
7	6	eleq1d 2258	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
8	7	cbvralv 2718	. . . . . . . 8 ⊢ (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ ↔ ∀𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 ∈ ℂ)
9	5, 8	sylibr 134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
10	9	adantrr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗)))) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
11		nfcsb1v 3105	. . . . . . . 8 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵
12	11	nfel1 2343	. . . . . . 7 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ
13		csbeq1a 3081	. . . . . . . 8 ⊢ (𝑥 = ((𝑁 − 𝑗) − 𝑛) → 𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
14	13	eleq1d 2258	. . . . . . 7 ⊢ (𝑥 = ((𝑁 − 𝑗) − 𝑛) → (𝐵 ∈ ℂ ↔ ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ))
15	12, 14	rspc 2850	. . . . . 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 11467	. . . 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 9283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 0 ∈ ℤ)
20	17	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ∈ ℤ)
21		elfzelz 10043	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
22	21	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ ℤ)
23	20, 22	zsubcld 9398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 − 𝑗) ∈ ℤ)
24		nfcsb1v 3105	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵
25	24	nfel1 2343	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ
26		csbeq1a 3081	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
27	26	eleq1d 2258	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ))
28	25, 27	rspc 2850	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑗)) → (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ → ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ))
29	9, 28	mpan9 281	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ)
30		csbeq1 3075	. . . . . . 7 ⊢ (𝑘 = ((0 + (𝑁 − 𝑗)) − 𝑛) → ⦋𝑘 / 𝑥⦌𝐵 = ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
31	19, 23, 29, 30	fisumrev2 11472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
32		elfz3nn0 10133	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
33	32	ad2antlr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑁 ∈ ℕ0)
34	21	ad2antlr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑗 ∈ ℤ)
35		nn0cn 9204	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
36		zcn 9276	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
37		subcl 8174	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁 − 𝑗) ∈ ℂ)
38	35, 36, 37	syl2an 289	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → (𝑁 − 𝑗) ∈ ℂ)
39	33, 34, 38	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → (𝑁 − 𝑗) ∈ ℂ)
40	39	addid2d 8125	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → (0 + (𝑁 − 𝑗)) = (𝑁 − 𝑗))
41	40	oveq1d 5906	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ((0 + (𝑁 − 𝑗)) − 𝑛) = ((𝑁 − 𝑗) − 𝑛))
42	41	csbeq1d 3079	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
43	42	sumeq2dv 11394	. . . . . 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 2222	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
45	44	sumeq2dv 11394	. . . 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 10133	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
47	46	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ0)
48		addlid 8114	. . . . . . . . 9 ⊢ (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
49	47, 35, 48	3syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
50	49	oveq1d 5906	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁 − 𝑛))
51	50	oveq2d 5907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁 − 𝑛)))
52	50	oveq1d 5906	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
53	52	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
54	46	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑁 ∈ ℕ0)
55		elfzelz 10043	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
56	55	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑛 ∈ ℤ)
57		elfzelz 10043	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...(𝑁 − 𝑛)) → 𝑗 ∈ ℤ)
58	57	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑗 ∈ ℤ)
59		zcn 9276	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
60		sub32 8209	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
61	35, 59, 36, 60	syl3an 1291	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
62	54, 56, 58, 61	syl3anc 1249	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
63	53, 62	eqtrd 2222	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
64	63	csbeq1d 3079	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
65	51, 64	sumeq12rdv 11399	. . . . 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 11394	. . . 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 2232	. . 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 9283	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
69		0zd 9283	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 0 ∈ ℤ)
70		elfzelz 10043	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
71	70	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ)
72	69, 71	fzfigd 10449	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
73		elfzuz3 10040	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ≥‘𝑗))
74	73	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ≥‘𝑗))
75		elfzuz3 10040	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → 𝑁 ∈ (ℤ≥‘𝑘))
76	75	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑘))
77	76	adantr 276	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ (ℤ≥‘𝑘))
78		elfzuzb 10037	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ≥‘𝑗) ∧ 𝑁 ∈ (ℤ≥‘𝑘)))
79	74, 77, 78	sylanbrc 417	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
80		elfzelz 10043	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
81	80	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
82	17	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
83	70	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
84		fzsubel 10078	. . . . . . . . 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 1250	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗))))
86	79, 85	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗)))
87		subid 8194	. . . . . . . . 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 5906	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗 − 𝑗)...(𝑁 − 𝑗)) = (0...(𝑁 − 𝑗)))
90	86, 89	eleqtrd 2268	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ (0...(𝑁 − 𝑗)))
91		simpll 527	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
92		fzss2 10082	. . . . . . . . 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 3170	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ (0...𝑁))
95	91, 94, 9	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
96		nfcsb1v 3105	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵
97	96	nfel1 2343	. . . . . . 7 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ
98		csbeq1a 3081	. . . . . . . 8 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
99	98	eleq1d 2258	. . . . . . 7 ⊢ (𝑥 = (𝑘 − 𝑗) → (𝐵 ∈ ℂ ↔ ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ))
100	97, 99	rspc 2850	. . . . . 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 11426	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ)
103		oveq2 5899	. . . . 5 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
104		oveq1 5898	. . . . . . 7 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘 − 𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
105	104	csbeq1d 3079	. . . . . 6 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
106	105	adantr 276	. . . . 5 ⊢ ((𝑘 = ((0 + 𝑁) − 𝑛) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
107	103, 106	sumeq12dv 11398	. . . 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 11472	. . 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 2225	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
110		vex 2755	. . . . . 6 ⊢ 𝑘 ∈ V
111	110, 6	csbie 3117	. . . . 5 ⊢ ⦋𝑘 / 𝑥⦌𝐵 = 𝐴
112	111	a1i 9	. . . 4 ⊢ ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 = 𝐴)
113	112	sumeq2dv 11394	. . 3 ⊢ (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴)
114	113	sumeq2i 11390	. 2 ⊢ Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴
115	70	adantr 276	. . . . . 6 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
116	80	adantl 277	. . . . . 6 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
117	115, 116	zsubcld 9398	. . . . 5 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ ℤ)
118		fsum0diag2.2	. . . . . 6 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = 𝐶)
119	118	adantl 277	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) ∧ 𝑥 = (𝑘 − 𝑗)) → 𝐵 = 𝐶)
120	117, 119	csbied 3118	. . . 4 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = 𝐶)
121	120	sumeq2dv 11394	. . 3 ⊢ (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
122	121	sumeq2i 11390	. 2 ⊢ Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶
123	109, 114, 122	3eqtr3g 2245	1 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2160  ∀wral 2468  ⦋csb 3072   ⊆ wss 3144  ‘cfv 5231  (class class class)co 5891  ℂcc 7827  0cc0 7829   + caddc 7832   − cmin 8146  ℕ₀cn0 9194  ℤcz 9271  ℤ_≥cuz 9546  ...cfz 10026  Σcsu 11379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-disj 3996  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-frec 6410  df-1o 6435  df-oadd 6439  df-er 6553  df-en 6759  df-dom 6760  df-fin 6761  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-fz 10027  df-fzo 10161  df-seqfrec 10464  df-exp 10538  df-ihash 10774  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-clim 11305  df-sumdc 11380

This theorem is referenced by:  mertensabs  11563