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Theorem fsum0diag2 15949

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 13769	. . . . . . 7 ⊢ (𝑛 ∈ (0...(𝑁 − 𝑗)) → ((𝑁 − 𝑗) − 𝑛) ∈ (0...(𝑁 − 𝑗)))
2	1	ad2antll 742	. . . . . 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 462	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑘 ∈ (0...(𝑁 − 𝑗)) → 𝐴 ∈ ℂ))
5	4	ralrimiv 3154	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 ∈ ℂ)
6		fsum0diag2.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐴)
7	6	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
8	7	cbvralvw 3241	. . . . . . . 8 ⊢ (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ ↔ ∀𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 ∈ ℂ)
9	5, 8	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
10	9	adantrr 730	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗)))) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
11		nfcsb1v 3871	. . . . . . . 8 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵
12	11	nfel1 2939	. . . . . . 7 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ
13		csbeq1a 3861	. . . . . . . 8 ⊢ (𝑥 = ((𝑁 − 𝑗) − 𝑛) → 𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
14	13	eleq1d 2846	. . . . . . 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 66	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗)))) → ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ)
17	16	fsum0diag 15943	. . . 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 3871	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵
19	18	nfel1 2939	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ
20		csbeq1a 3861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
21	20	eleq1d 2846	. . . . . . . . 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 516	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ)
24		csbeq1 3850	. . . . . . 7 ⊢ (𝑘 = ((0 + (𝑁 − 𝑗)) − 𝑛) → ⦋𝑘 / 𝑥⦌𝐵 = ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
25	23, 24	fsumrev2 15948	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
26		elfz3nn0 13755	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
27	26	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑁 ∈ ℕ₀)
28		elfzelz 13656	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
29	28	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑗 ∈ ℤ)
30		nn0cn 12616	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
31		zcn 12698	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
32		subcl 11556	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁 − 𝑗) ∈ ℂ)
33	30, 31, 32	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑗 ∈ ℤ) → (𝑁 − 𝑗) ∈ ℂ)
34	27, 29, 33	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → (𝑁 − 𝑗) ∈ ℂ)
35		addlid 11493	. . . . . . . . . 10 ⊢ ((𝑁 − 𝑗) ∈ ℂ → (0 + (𝑁 − 𝑗)) = (𝑁 − 𝑗))
36	34, 35	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → (0 + (𝑁 − 𝑗)) = (𝑁 − 𝑗))
37	36	oveq1d 7435	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ((0 + (𝑁 − 𝑗)) − 𝑛) = ((𝑁 − 𝑗) − 𝑛))
38	37	csbeq1d 3851	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
39	38	sumeq2dv 15869	. . . . . 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 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
41	40	sumeq2dv 15869	. . . 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 13755	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
43	42	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
44		addlid 11493	. . . . . . . . 9 ⊢ (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
45	43, 30, 44	3syl 19	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
46	45	oveq1d 7435	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁 − 𝑛))
47	46	oveq2d 7436	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁 − 𝑛)))
48	46	oveq1d 7435	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
49	48	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
50	42	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑁 ∈ ℕ₀)
51		elfzelz 13656	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
52	51	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑛 ∈ ℤ)
53		elfzelz 13656	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...(𝑁 − 𝑛)) → 𝑗 ∈ ℤ)
54	53	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑗 ∈ ℤ)
55		zcn 12698	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
56		sub32 11592	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
57	30, 55, 31, 56	syl3an 1178	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
58	50, 52, 54, 57	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
59	49, 58	eqtrd 2796	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
60	59	csbeq1d 3851	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
61	47, 60	sumeq12rdv 15873	. . . . 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 15869	. . . 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 2806	. . 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 14116	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
65		elfzuz3 13653	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ_≥‘𝑗))
66	65	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ_≥‘𝑗))
67		elfzuz3 13653	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑘))
68	67	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝑘))
69	68	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ (ℤ_≥‘𝑘))
70		elfzuzb 13650	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑁 ∈ (ℤ_≥‘𝑘)))
71	66, 69, 70	sylanbrc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
72		elfzelz 13656	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
73	72	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
74		elfzel2 13654	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
75	74	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
76		elfzelz 13656	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
77	76	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
78		fzsubel 13694	. . . . . . . . 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 852	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗))))
80	71, 79	mpbid 235	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ ((𝑗 − 𝑗)...(𝑁 − 𝑗)))
81		subid 11577	. . . . . . . . 9 ⊢ (𝑗 ∈ ℂ → (𝑗 − 𝑗) = 0)
82	73, 31, 81	3syl 19	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑗 − 𝑗) = 0)
83	82	oveq1d 7435	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗 − 𝑗)...(𝑁 − 𝑗)) = (0...(𝑁 − 𝑗)))
84	80, 83	eleqtrd 2863	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ (0...(𝑁 − 𝑗)))
85		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
86		fzss2 13698	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑘) → (0...𝑘) ⊆ (0...𝑁))
87	68, 86	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (0...𝑘) ⊆ (0...𝑁))
88	87	sselda 3931	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ (0...𝑁))
89	85, 88, 9	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
90		nfcsb1v 3871	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵
91	90	nfel1 2939	. . . . . . 7 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ
92		csbeq1a 3861	. . . . . . . 8 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
93	92	eleq1d 2846	. . . . . . 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 66	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ)
96	64, 95	fsumcl 15899	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ)
97		oveq2 7428	. . . . 5 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
98		oveq1 7427	. . . . . . 7 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘 − 𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
99	98	csbeq1d 3851	. . . . . 6 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
100	99	adantr 486	. . . . 5 ⊢ ((𝑘 = ((0 + 𝑁) − 𝑛) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
101	97, 100	sumeq12dv 15872	. . . 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 15948	. . 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 2799	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
104		vex 3455	. . . . . 6 ⊢ 𝑘 ∈ V
105	104, 6	csbie 3882	. . . . 5 ⊢ ⦋𝑘 / 𝑥⦌𝐵 = 𝐴
106	105	a1i 11	. . . 4 ⊢ ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 = 𝐴)
107	106	sumeq2dv 15869	. . 3 ⊢ (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴)
108	107	sumeq2i 15865	. 2 ⊢ Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴
109		ovex 7453	. . . . . 6 ⊢ (𝑘 − 𝑗) ∈ V
110		fsum0diag2.2	. . . . . 6 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = 𝐶)
111	109, 110	csbie 3882	. . . . 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 15869	. . 3 ⊢ (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
114	113	sumeq2i 15865	. 2 ⊢ Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶
115	103, 108, 114	3eqtr3g 2819	1 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ⦋csb 3847 ⊆ wss 3899 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 0cc0 11200 + caddc 11203 − cmin 11541 ℕ₀cn0 12606 ℤcz 12693 ℤ_≥cuz 12965 ...cfz 13639 Σcsu 15853

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9434 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-z 12694 df-uz 12966 df-rp 13121 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-sum 15854

This theorem is used by: mertens 16055 plymullem1 26533

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