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Theorem fsum0diag2 15834

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 13663	. . . . . . 7 ⊢ (𝑛 ∈ (0...(𝑁 − 𝑗)) → ((𝑁 − 𝑗) − 𝑛) ∈ (0...(𝑁 − 𝑗)))
2	1	ad2antll 741	. . . . . 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 461	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑘 ∈ (0...(𝑁 − 𝑗)) → 𝐴 ∈ ℂ))
5	4	ralrimiv 3162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...(𝑁 − 𝑗))𝐴 ∈ ℂ)
6		fsum0diag2.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐴)
7	6	eleq1d 2854	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ 𝐴 ∈ ℂ))
8	7	cbvralvw 3249	. . . . . . . 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 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗)))) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
11		nfcsb1v 3885	. . . . . . . 8 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵
12	11	nfel1 2947	. . . . . . 7 ⊢ Ⅎ𝑥⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ
13		csbeq1a 3875	. . . . . . . 8 ⊢ (𝑥 = ((𝑁 − 𝑗) − 𝑛) → 𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
14	13	eleq1d 2854	. . . . . . 7 ⊢ (𝑥 = ((𝑁 − 𝑗) − 𝑛) → (𝐵 ∈ ℂ ↔ ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵 ∈ ℂ))
15	12, 14	rspc 3578	. . . . . 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 15828	. . . 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 3885	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵
19	18	nfel1 2947	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ
20		csbeq1a 3875	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
21	20	eleq1d 2854	. . . . . . . . 9 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℂ ↔ ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ))
22	19, 21	rspc 3578	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 𝑗)) → (∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ → ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ))
23	9, 22	mpan9 515	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 ∈ ℂ)
24		csbeq1 3864	. . . . . . 7 ⊢ (𝑘 = ((0 + (𝑁 − 𝑗)) − 𝑛) → ⦋𝑘 / 𝑥⦌𝐵 = ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
25	23, 24	fsumrev2 15833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵)
26		elfz3nn0 13649	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
27	26	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑁 ∈ ℕ₀)
28		elfzelz 13552	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
29	28	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → 𝑗 ∈ ℤ)
30		nn0cn 12514	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
31		zcn 12596	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
32		subcl 11456	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑁 − 𝑗) ∈ ℂ)
33	30, 31, 32	syl2an 607	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑗 ∈ ℤ) → (𝑁 − 𝑗) ∈ ℂ)
34	27, 29, 33	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → (𝑁 − 𝑗) ∈ ℂ)
35		addlid 11393	. . . . . . . . . 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 7426	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ((0 + (𝑁 − 𝑗)) − 𝑛) = ((𝑁 − 𝑗) − 𝑛))
38	37	csbeq1d 3865	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 𝑗))) → ⦋((0 + (𝑁 − 𝑗)) − 𝑛) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
39	38	sumeq2dv 15753	. . . . . 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 2804	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑛 ∈ (0...(𝑁 − 𝑗))⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
41	40	sumeq2dv 15753	. . . 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 13649	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
43	42	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
44		addlid 11393	. . . . . . . . 9 ⊢ (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁)
45	43, 30, 44	3syl 19	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0 + 𝑁) = 𝑁)
46	45	oveq1d 7426	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → ((0 + 𝑁) − 𝑛) = (𝑁 − 𝑛))
47	46	oveq2d 7427	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (0...((0 + 𝑁) − 𝑛)) = (0...(𝑁 − 𝑛)))
48	46	oveq1d 7426	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑁)) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
49	48	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑛) − 𝑗))
50	42	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑁 ∈ ℕ₀)
51		elfzelz 13552	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ)
52	51	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑛 ∈ ℤ)
53		elfzelz 13552	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...(𝑁 − 𝑛)) → 𝑗 ∈ ℤ)
54	53	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → 𝑗 ∈ ℤ)
55		zcn 12596	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
56		sub32 11492	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑗 ∈ ℂ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
57	30, 55, 31, 56	syl3an 1176	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑛 ∈ ℤ ∧ 𝑗 ∈ ℤ) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
58	50, 52, 54, 57	syl3anc 1396	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ((𝑁 − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
59	49, 58	eqtrd 2804	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → (((0 + 𝑁) − 𝑛) − 𝑗) = ((𝑁 − 𝑗) − 𝑛))
60	59	csbeq1d 3865	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...(𝑁 − 𝑛))) → ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵 = ⦋((𝑁 − 𝑗) − 𝑛) / 𝑥⦌𝐵)
61	47, 60	sumeq12rdv 15758	. . . . 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 15753	. . . 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 2814	. . 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 14009	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (0...𝑘) ∈ Fin)
65		elfzuz3 13549	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑘 ∈ (ℤ_≥‘𝑗))
66	65	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (ℤ_≥‘𝑗))
67		elfzuz3 13549	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑘))
68	67	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝑘))
69	68	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ (ℤ_≥‘𝑘))
70		elfzuzb 13546	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑗...𝑁) ↔ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑁 ∈ (ℤ_≥‘𝑘)))
71	66, 69, 70	sylanbrc 594	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ (𝑗...𝑁))
72		elfzelz 13552	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℤ)
73	72	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ ℤ)
74		elfzel2 13550	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
75	74	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑁 ∈ ℤ)
76		elfzelz 13552	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
77	76	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑘 ∈ ℤ)
78		fzsubel 13588	. . . . . . . . 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 851	. . . . . . . 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 11477	. . . . . . . . 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 7426	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ((𝑗 − 𝑗)...(𝑁 − 𝑗)) = (0...(𝑁 − 𝑗)))
84	80, 83	eleqtrd 2871	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ (0...(𝑁 − 𝑗)))
85		simpll 778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝜑)
86		fzss2 13592	. . . . . . . . 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 3945	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → 𝑗 ∈ (0...𝑁))
89	85, 88, 9	syl2anc 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑥 ∈ (0...(𝑁 − 𝑗))𝐵 ∈ ℂ)
90		nfcsb1v 3885	. . . . . . . 8 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵
91	90	nfel1 2947	. . . . . . 7 ⊢ Ⅎ𝑥⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ
92		csbeq1a 3875	. . . . . . . 8 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
93	92	eleq1d 2854	. . . . . . 7 ⊢ (𝑥 = (𝑘 − 𝑗) → (𝐵 ∈ ℂ ↔ ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ))
94	91, 93	rspc 3578	. . . . . 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 15784	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 ∈ ℂ)
97		oveq2 7419	. . . . 5 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (0...𝑘) = (0...((0 + 𝑁) − 𝑛)))
98		oveq1 7418	. . . . . . 7 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → (𝑘 − 𝑗) = (((0 + 𝑁) − 𝑛) − 𝑗))
99	98	csbeq1d 3865	. . . . . 6 ⊢ (𝑘 = ((0 + 𝑁) − 𝑛) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
100	99	adantr 485	. . . . 5 ⊢ ((𝑘 = ((0 + 𝑁) − 𝑛) ∧ 𝑗 ∈ (0...𝑘)) → ⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = ⦋(((0 + 𝑁) − 𝑛) − 𝑗) / 𝑥⦌𝐵)
101	97, 100	sumeq12dv 15757	. . . 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 15833	. . 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 2807	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵)
104		vex 3467	. . . . . 6 ⊢ 𝑘 ∈ V
105	104, 6	csbie 3896	. . . . 5 ⊢ ⦋𝑘 / 𝑥⦌𝐵 = 𝐴
106	105	a1i 11	. . . 4 ⊢ ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁 − 𝑗))) → ⦋𝑘 / 𝑥⦌𝐵 = 𝐴)
107	106	sumeq2dv 15753	. . 3 ⊢ (𝑗 ∈ (0...𝑁) → Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴)
108	107	sumeq2i 15749	. 2 ⊢ Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))⦋𝑘 / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑁)Σ𝑘 ∈ (0...(𝑁 − 𝑗))𝐴
109		ovex 7444	. . . . . 6 ⊢ (𝑘 − 𝑗) ∈ V
110		fsum0diag2.2	. . . . . 6 ⊢ (𝑥 = (𝑘 − 𝑗) → 𝐵 = 𝐶)
111	109, 110	csbie 3896	. . . . 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 15753	. . 3 ⊢ (𝑘 ∈ (0...𝑁) → Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑗 ∈ (0...𝑘)𝐶)
114	113	sumeq2i 15749	. 2 ⊢ Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)⦋(𝑘 − 𝑗) / 𝑥⦌𝐵 = Σ𝑘 ∈ (0...𝑁)Σ𝑗 ∈ (0...𝑘)𝐶
115	103, 108, 114	3eqtr3g 2827	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 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ⦋csb 3861 ⊆ wss 3913 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 0cc0 11100 + caddc 11103 − cmin 11441 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 ...cfz 13535 Σcsu 15737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-rp 13017 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738

This theorem is referenced by: mertens 15940 plymullem1 26340

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