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Theorem fsum0diag2 15710

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)

**Assertion**
Ref	Expression
fsum0diag2	⊢ (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 fsum0diag2 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fznn0sub2 13555	. . . . . . 7 ⊢ (𝑛 ∈ (0...(𝑁 − 𝑗)) → ((𝑁 − 𝑗) − 𝑛) ∈ (0...(𝑁 − 𝑗)))
2	1	ad2antll 730	. . . . . 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 456	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑘 ∈ (0...(𝑁 − 𝑗)) → 𝐴 ∈ ℂ))
5	4	ralrimiv 3128	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 ∈ ℂ)
6		fsum0diag2.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐴)
7	6	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
8	7	cbvralvw 3215	. . . . . . . 8 ⊢ (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ ↔ ∀𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 ∈ ℂ)
9	5, 8	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
10	9	adantrr 718	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗)))) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
11		nfcsb1v 3874	. . . . . . . 8 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵
12	11	nfel1 2916	. . . . . . 7 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ
13		csbeq1a 3864	. . . . . . . 8 ⊢ (𝑥 = ((𝑁 − 𝑗) − 𝑛) → 𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
14	13	eleq1d 2822	. . . . . . 7 ⊢ (𝑥 = ((𝑁 − 𝑗) − 𝑛) → (𝐵 ∈ ℂ ↔ ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ))
15	12, 14	rspc 3565	. . . . . 6 ⊢ (((𝑁 − 𝑗) − 𝑛) ∈ (0...(𝑁 − 𝑗)) → (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ → ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ))
16	2, 10, 15	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗)))) → ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ)
17	16	fsum0diag 15704	. . . 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)
18		nfcsb1v 3874	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵
19	18	nfel1 2916	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ
20		csbeq1a 3864	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
21	20	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ))
22	19, 21	rspc 3565	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑗)) → (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ → ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ))
23	9, 22	mpan9 506	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ)
24		csbeq1 3853	. . . . . . 7 ⊢ (𝑘 = ((0 + (𝑁 − 𝑗)) − 𝑛) → ⦋𝑘 / 𝑥⦌𝐵 = ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
25	23, 24	fsumrev2 15709	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
26		elfz3nn0 13541	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
27	26	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑁 ∈ ℕ₀)
28		elfzelz 13444	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
29	28	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑗 ∈ ℤ)
30		nn0cn 12415	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
31		zcn 12497	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
32		subcl 11383	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁 − 𝑗) ∈ ℂ)
33	30, 31, 32	syl2an 597	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑗 ∈ ℤ) → (𝑁 − 𝑗) ∈ ℂ)
34	27, 29, 33	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → (𝑁 − 𝑗) ∈ ℂ)
35		addlid 11320	. . . . . . . . . 10 ⊢ ((𝑁 − 𝑗) ∈ ℂ → (0 + (𝑁 − 𝑗)) = (𝑁 − 𝑗))
36	34, 35	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → (0 + (𝑁 − 𝑗)) = (𝑁 − 𝑗))
37	36	oveq1d 7375	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ((0 + (𝑁 − 𝑗)) − 𝑛) = ((𝑁 − 𝑗) − 𝑛))
38	37	csbeq1d 3854	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
39	38	sumeq2dv 15629	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
40	25, 39	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
41	40	sumeq2dv 15629	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑁)Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
42		elfz3nn0 13541	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
43	42	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
44		addlid 11320	. . . . . . . . 9 ⊢ (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
45	43, 30, 44	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
46	45	oveq1d 7375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁 − 𝑛))
47	46	oveq2d 7376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁 − 𝑛)))
48	46	oveq1d 7375	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
49	48	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
50	42	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑁 ∈ ℕ₀)
51		elfzelz 13444	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
52	51	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑛 ∈ ℤ)
53		elfzelz 13444	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...(𝑁 − 𝑛)) → 𝑗 ∈ ℤ)
54	53	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑗 ∈ ℤ)
55		zcn 12497	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
56		sub32 11419	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
57	30, 55, 31, 56	syl3an 1161	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
58	50, 52, 54, 57	syl3anc 1374	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
59	49, 58	eqtrd 2772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
60	59	csbeq1d 3854	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
61	47, 60	sumeq12rdv 15634	. . . . 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)
62	61	sumeq2dv 15629	. . . 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)
63	17, 41, 62	3eqtr4d 2782	. . 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)
64		fzfid 13900	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
65		elfzuz3 13441	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ_≥‘𝑗))
66	65	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ_≥‘𝑗))
67		elfzuz3 13441	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑘))
68	67	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝑘))
69	68	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ (ℤ_≥‘𝑘))
70		elfzuzb 13438	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑁 ∈ (ℤ_≥‘𝑘)))
71	66, 69, 70	sylanbrc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
72		elfzelz 13444	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
73	72	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
74		elfzel2 13442	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
75	74	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
76		elfzelz 13444	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
77	76	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
78		fzsubel 13480	. . . . . . . . 9 ⊢ (((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗))))
79	73, 75, 77, 73, 78	syl22anc 839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗))))
80	71, 79	mpbid 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗)))
81		subid 11404	. . . . . . . . 9 ⊢ (𝑗 ∈ ℂ → (𝑗 − 𝑗) = 0)
82	73, 31, 81	3syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑗 − 𝑗) = 0)
83	82	oveq1d 7375	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗 − 𝑗)...(𝑁 − 𝑗)) = (0...(𝑁 − 𝑗)))
84	80, 83	eleqtrd 2839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ (0...(𝑁 − 𝑗)))
85		simpll 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
86		fzss2 13484	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑘) → (0...𝑘) ⊆ (0...𝑁))
87	68, 86	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (0...𝑘) ⊆ (0...𝑁))
88	87	sselda 3934	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ (0...𝑁))
89	85, 88, 9	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
90		nfcsb1v 3874	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵
91	90	nfel1 2916	. . . . . . 7 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ
92		csbeq1a 3864	. . . . . . . 8 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
93	92	eleq1d 2822	. . . . . . 7 ⊢ (𝑥 = (𝑘 − 𝑗) → (𝐵 ∈ ℂ ↔ ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ))
94	91, 93	rspc 3565	. . . . . 6 ⊢ ((𝑘 − 𝑗) ∈ (0...(𝑁 − 𝑗)) → (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ))
95	84, 89, 94	sylc 65	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ)
96	64, 95	fsumcl 15660	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ)
97		oveq2 7368	. . . . 5 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
98		oveq1 7367	. . . . . . 7 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘 − 𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
99	98	csbeq1d 3854	. . . . . 6 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
100	99	adantr 480	. . . . 5 ⊢ ((𝑘 = ((0 + 𝑁) − 𝑛) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
101	97, 100	sumeq12dv 15633	. . . 4 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...((0 + 𝑁) − 𝑛))⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
102	96, 101	fsumrev2 15709	. . 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)
103	63, 102	eqtr4d 2775	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
104		vex 3445	. . . . . 6 ⊢ 𝑘 ∈ V
105	104, 6	csbie 3885	. . . . 5 ⊢ ⦋𝑘 / 𝑥⦌𝐵 = 𝐴
106	105	a1i 11	. . . 4 ⊢ ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 = 𝐴)
107	106	sumeq2dv 15629	. . 3 ⊢ (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴)
108	107	sumeq2i 15625	. 2 ⊢ Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴
109		ovex 7393	. . . . . 6 ⊢ (𝑘 − 𝑗) ∈ V
110		fsum0diag2.2	. . . . . 6 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = 𝐶)
111	109, 110	csbie 3885	. . . . 5 ⊢ ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = 𝐶
112	111	a1i 11	. . . 4 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = 𝐶)
113	112	sumeq2dv 15629	. . 3 ⊢ (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
114	113	sumeq2i 15625	. 2 ⊢ Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶
115	103, 108, 114	3eqtr3g 2795	1 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⦋csb 3850 ⊆ wss 3902 ‘cfv 6493 (class class class)co 7360 ℂcc 11028 0cc0 11030 + caddc 11033 − cmin 11368 ℕ₀cn0 12405 ℤcz 12492 ℤ_≥cuz 12755 ...cfz 13427 Σcsu 15613

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-n0 12406 df-z 12493 df-uz 12756 df-rp 12910 df-fz 13428 df-fzo 13575 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-sum 15614

This theorem is referenced by: mertens 15813 plymullem1 26179

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