Theorem fsum2dsub 35061

Description: Lemma for breprexp 35087- Re-index a double sum, using difference of the initial indices. (Contributed by Thierry Arnoux, 7-Dec-2021.)

Hypotheses

Ref	Expression
fzsum2sub.m	⊢ (𝜑 → 𝑀 ∈ ℕ0)
fzsum2sub.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
fzsum2sub.1	⊢ (𝑖 = (𝑘 − 𝑗) → 𝐴 = 𝐵)
fzsum2sub.2	⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)
fzsum2sub.3	⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) → 𝐵 = 0)
fzsum2sub.4	⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0..^𝑗)) → 𝐵 = 0)

Assertion

Ref	Expression
fsum2dsub	⊢ (𝜑 → Σ𝑖 ∈ (0...𝑀)Σ𝑗 ∈ (1...𝑁)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵)

Distinct variable groups:   𝐴,𝑘   𝐵,𝑖   𝑖,𝑀,𝑗,𝑘   𝑖,𝑁,𝑗,𝑘   𝜑,𝑖,𝑗,𝑘

Allowed substitution hints:   𝐴(𝑖, 𝑗)   𝐵(𝑗, 𝑘)

Proof of Theorem fsum2dsub

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
2	1	elfzelzd 13571	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℤ)
3		0zd 12620	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 0 ∈ ℤ)
4		fzsum2sub.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
5	4	nn0zd 12633	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℤ)
7		simpll 779	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝜑)
8		fz1ssnn 13602	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ ℕ
9		nnssnn0 12524	. . . . . . . . . . . 12 ⊢ ℕ ⊆ ℕ0
10	8, 9	sstri 3947	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℕ0
11	10, 1	sselid 3936	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0)
12		nn0uz 12918	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
13	11, 12	eleqtrdi 2875	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (ℤ≥‘0))
14		neg0 11521	. . . . . . . . . 10 ⊢ -0 = 0
15		uzneg 12900	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘0) → -0 ∈ (ℤ≥‘-𝑗))
16	14, 15	eqeltrrid 2870	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ≥‘0) → 0 ∈ (ℤ≥‘-𝑗))
17		fzss1 13610	. . . . . . . . 9 ⊢ (0 ∈ (ℤ≥‘-𝑗) → (0...𝑀) ⊆ (-𝑗...𝑀))
18	13, 16, 17	3syl 19	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆ (-𝑗...𝑀))
19		fzssuz 13612	. . . . . . . 8 ⊢ (-𝑗...𝑀) ⊆ (ℤ≥‘-𝑗)
20	18, 19	sstrdi 3950	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...𝑀) ⊆ (ℤ≥‘-𝑗))
21	20	sselda 3938	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ≥‘-𝑗))
22	1	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝑗 ∈ (1...𝑁))
23		fzsum2sub.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)
24	7, 21, 22, 23	syl3anc 1398	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℂ)
25		fzsum2sub.1	. . . . 5 ⊢ (𝑖 = (𝑘 − 𝑗) → 𝐴 = 𝐵)
26	2, 3, 6, 24, 25	fsumshft 15856	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
27	4	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℕ0)
28	8, 1	sselid 3936	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ)
29	28	nnnn0d 12582	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℕ0)
30	27, 29	nn0addcld 12586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℕ0)
31	30	nn0red 12583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℝ)
32	31	ltp1d 12162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) < ((𝑀 + 𝑗) + 1))
33		fzdisj 13598	. . . . . . . 8 ⊢ ((𝑀 + 𝑗) < ((𝑀 + 𝑗) + 1) → ((𝑗...(𝑀 + 𝑗)) ∩ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) = ∅)
34	32, 33	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((𝑗...(𝑀 + 𝑗)) ∩ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) = ∅)
35		fzsum2sub.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
36	35	nn0zd 12633	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
37	5, 36	zaddcld 12722	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℤ)
38	37	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℤ)
39	30	nn0zd 12633	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ ℤ)
40	28	nnred 12265	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℝ)
41		nn0addge2 12568	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → 𝑗 ≤ (𝑀 + 𝑗))
42	40, 27, 41	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ≤ (𝑀 + 𝑗))
43	35	nn0red 12583	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
45	27	nn0red 12583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
46		elfzle2 13574	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...𝑁) → 𝑗 ≤ 𝑁)
47	46	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ≤ 𝑁)
48	40, 44, 45, 47	leadd2dd 11846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ≤ (𝑀 + 𝑁))
49	2, 38, 39, 42, 48	elfzd 13561	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁)))
50		fzsplit 13597	. . . . . . . 8 ⊢ ((𝑀 + 𝑗) ∈ (𝑗...(𝑀 + 𝑁)) → (𝑗...(𝑀 + 𝑁)) = ((𝑗...(𝑀 + 𝑗)) ∪ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))))
51	49, 50	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) = ((𝑗...(𝑀 + 𝑗)) ∪ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))))
52		fzfid 14029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) ∈ Fin)
53		simpll 779	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝜑)
54	1	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ (1...𝑁))
55	10, 54	sselid 3936	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑗 ∈ ℕ0)
56		fz2ssnn0 33202	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ0 → (𝑗...(𝑀 + 𝑁)) ⊆ ℕ0)
57	55, 56	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → (𝑗...(𝑀 + 𝑁)) ⊆ ℕ0)
58		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁)))
59	57, 58	sseldd 3939	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝑘 ∈ ℕ0)
60	25	eleq1d 2850	. . . . . . . . 9 ⊢ (𝑖 = (𝑘 − 𝑗) → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
61		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝜑)
62		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (ℤ≥‘-𝑗))
63		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
64	61, 62, 63, 23	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)
65	64	an32s 665	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 ∈ (ℤ≥‘-𝑗)) → 𝐴 ∈ ℂ)
66	65	ralrimiva 3159	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ∀𝑖 ∈ (ℤ≥‘-𝑗)𝐴 ∈ ℂ)
67	66	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → ∀𝑖 ∈ (ℤ≥‘-𝑗)𝐴 ∈ ℂ)
68		nnsscn 12255	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℂ
69	8, 68	sstri 3947	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ ℂ
70		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (1...𝑁))
71	69, 70	sselid 3936	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ ℂ)
72		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
73	72	nn0cnd 12584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
74	71, 73	negsubdi2d 11602	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗 − 𝑘) = (𝑘 − 𝑗))
75	70	elfzelzd 13571	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ ℤ)
76		eluzmn 12887	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (ℤ≥‘(𝑗 − 𝑘)))
77	75, 72, 76	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ (ℤ≥‘(𝑗 − 𝑘)))
78		uzneg 12900	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘(𝑗 − 𝑘)) → -(𝑗 − 𝑘) ∈ (ℤ≥‘-𝑗))
79	77, 78	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → -(𝑗 − 𝑘) ∈ (ℤ≥‘-𝑗))
80	74, 79	eqeltrrd 2866	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → (𝑘 − 𝑗) ∈ (ℤ≥‘-𝑗))
81	60, 67, 80	rspcdva 3584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
82	53, 54, 59, 81	syl21anc 851	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (𝑗...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ)
83	34, 51, 52, 82	fsumsplit 15817	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = (Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 + Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵))
84	2	zcnd 12719	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ ℂ)
85	84	addlidd 11428	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0 + 𝑗) = 𝑗)
86	85	oveq1d 7434	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗)))
87	86	eqcomd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑗)) = ((0 + 𝑗)...(𝑀 + 𝑗)))
88	87	sumeq1d 15777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
89		fzsum2sub.3	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) → 𝐵 = 0)
90	89	sumeq2dv 15779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0)
91		fzfi 14028	. . . . . . . . 9 ⊢ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin
92		sumz 15798	. . . . . . . . . 10 ⊢ (((((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ⊆ (ℤ≥‘0) ∨ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0)
93	92	olcs 890	. . . . . . . . 9 ⊢ ((((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁)) ∈ Fin → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0)
94	91, 93	ax-mp 5	. . . . . . . 8 ⊢ Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))0 = 0
95	90, 94	eqtrdi 2816	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵 = 0)
96	88, 95	oveq12d 7437	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ (𝑗...(𝑀 + 𝑗))𝐵 + Σ𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))𝐵) = (Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 + 0))
97		fzfid 14029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0 + 𝑗)...(𝑀 + 𝑗)) ∈ Fin)
98		simpll 779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝜑)
99	1	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑗 ∈ (1...𝑁))
100		elfzuz3 13567	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘𝑗))
101	100	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ≥‘𝑗))
102		eluzadd 12909	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘𝑗) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ≥‘(𝑗 + 𝑀)))
103	101, 6, 102	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) ∈ (ℤ≥‘(𝑗 + 𝑀)))
104	35	nn0cnd 12584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℂ)
105	104	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
106		zsscn 12616	. . . . . . . . . . . . . . . 16 ⊢ ℤ ⊆ ℂ
107	106, 6	sselid 3936	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ ℂ)
108	105, 107	addcomd 11429	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑁 + 𝑀) = (𝑀 + 𝑁))
109	84, 107	addcomd 11429	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗 + 𝑀) = (𝑀 + 𝑗))
110	109	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (ℤ≥‘(𝑗 + 𝑀)) = (ℤ≥‘(𝑀 + 𝑗)))
111	103, 108, 110	3eltr3d 2879	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝑀 + 𝑗)))
112	111	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝑀 + 𝑗)))
113		fzss2 13611	. . . . . . . . . . . 12 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘(𝑀 + 𝑗)) → (𝑗...(𝑀 + 𝑗)) ⊆ (𝑗...(𝑀 + 𝑁)))
114	112, 113	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → (𝑗...(𝑀 + 𝑗)) ⊆ (𝑗...(𝑀 + 𝑁)))
115		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗)))
116	86	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → ((0 + 𝑗)...(𝑀 + 𝑗)) = (𝑗...(𝑀 + 𝑗)))
117	115, 116	eleqtrd 2867	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑗)))
118	114, 117	sseldd 3939	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ (𝑗...(𝑀 + 𝑁)))
119	98, 99, 118, 59	syl21anc 851	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝑘 ∈ ℕ0)
120	98, 99, 119, 81	syl21anc 851	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))) → 𝐵 ∈ ℂ)
121	97, 120	fsumcl 15809	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 ∈ ℂ)
122	121	addridd 11427	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 + 0) = Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵)
123	83, 96, 122	3eqtrrd 2805	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)
124		fzval3 13782	. . . . . . . . . 10 ⊢ ((𝑀 + 𝑁) ∈ ℤ → (𝑗...(𝑀 + 𝑁)) = (𝑗..^((𝑀 + 𝑁) + 1)))
125	38, 124	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑗...(𝑀 + 𝑁)) = (𝑗..^((𝑀 + 𝑁) + 1)))
126	125	ineq2d 4173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∩ (𝑗..^((𝑀 + 𝑁) + 1))))
127		fzodisj 13741	. . . . . . . 8 ⊢ ((0..^𝑗) ∩ (𝑗..^((𝑀 + 𝑁) + 1))) = ∅
128	126, 127	eqtrdi 2816	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∩ (𝑗...(𝑀 + 𝑁))) = ∅)
129	38	peano2zd 12721	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℤ)
130	29	nn0ge0d 12585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 0 ≤ 𝑗)
131	129	zred 12718	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((𝑀 + 𝑁) + 1) ∈ ℝ)
132	38	zred 12718	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ∈ ℝ)
133		nn0addge2 12568	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → 𝑁 ≤ (𝑀 + 𝑁))
134	43, 4, 133	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ≤ (𝑀 + 𝑁))
135	134	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ≤ (𝑀 + 𝑁))
136	132	lep1d 12163	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (𝑀 + 𝑁) ≤ ((𝑀 + 𝑁) + 1))
137	44, 132, 131, 135, 136	letrd 11384	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑁 ≤ ((𝑀 + 𝑁) + 1))
138	40, 44, 131, 47, 137	letrd 11384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ≤ ((𝑀 + 𝑁) + 1))
139	3, 129, 2, 130, 138	elfzd 13561	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (0...((𝑀 + 𝑁) + 1)))
140		fzosplit 13740	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...((𝑀 + 𝑁) + 1)) → (0..^((𝑀 + 𝑁) + 1)) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
141	139, 140	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0..^((𝑀 + 𝑁) + 1)) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
142		fzval3 13782	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) ∈ ℤ → (0...(𝑀 + 𝑁)) = (0..^((𝑀 + 𝑁) + 1)))
143	38, 142	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) = (0..^((𝑀 + 𝑁) + 1)))
144	125	uneq2d 4122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁))) = ((0..^𝑗) ∪ (𝑗..^((𝑀 + 𝑁) + 1))))
145	141, 143, 144	3eqtr4d 2810	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) = ((0..^𝑗) ∪ (𝑗...(𝑀 + 𝑁))))
146		fzfid 14029	. . . . . . . 8 ⊢ (𝜑 → (0...(𝑀 + 𝑁)) ∈ Fin)
147	146	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0...(𝑀 + 𝑁)) ∈ Fin)
148		simpl 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝜑)
149	1	adantrl 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑗 ∈ (1...𝑁))
150		fz0ssnn0 13669	. . . . . . . . . 10 ⊢ (0...(𝑀 + 𝑁)) ⊆ ℕ0
151		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
152	150, 151	sselid 3936	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝑘 ∈ ℕ0)
153	148, 149, 152, 81	syl21anc 851	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...𝑁))) → 𝐵 ∈ ℂ)
154	153	anass1rs 668	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → 𝐵 ∈ ℂ)
155	128, 145, 147, 154	fsumsplit 15817	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵 = (Σ𝑘 ∈ (0..^𝑗)𝐵 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵))
156		fzsum2sub.4	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0..^𝑗)) → 𝐵 = 0)
157	156	sumeq2dv 15779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = Σ𝑘 ∈ (0..^𝑗)0)
158		fzofi 14030	. . . . . . . . 9 ⊢ (0..^𝑗) ∈ Fin
159		sumz 15798	. . . . . . . . . 10 ⊢ (((0..^𝑗) ⊆ (ℤ≥‘0) ∨ (0..^𝑗) ∈ Fin) → Σ𝑘 ∈ (0..^𝑗)0 = 0)
160	159	olcs 890	. . . . . . . . 9 ⊢ ((0..^𝑗) ∈ Fin → Σ𝑘 ∈ (0..^𝑗)0 = 0)
161	158, 160	ax-mp 5	. . . . . . . 8 ⊢ Σ𝑘 ∈ (0..^𝑗)0 = 0
162	157, 161	eqtrdi 2816	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (0..^𝑗)𝐵 = 0)
163	162	oveq1d 7434	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0..^𝑗)𝐵 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵))
164	52, 82	fsumcl 15809	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 ∈ ℂ)
165	164	addlidd 11428	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → (0 + Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵) = Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵)
166	155, 163, 165	3eqtrrd 2805	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ (𝑗...(𝑀 + 𝑁))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
167	123, 166	eqtrd 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑘 ∈ ((0 + 𝑗)...(𝑀 + 𝑗))𝐵 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
168	26, 167	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑁)) → Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
169	168	sumeq2dv 15779	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑁)Σ𝑖 ∈ (0...𝑀)𝐴 = Σ𝑗 ∈ (1...𝑁)Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
170		fzfid 14029	. . 3 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
171		fzfid 14029	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
172	24	anasss 472	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ (1...𝑁) ∧ 𝑖 ∈ (0...𝑀))) → 𝐴 ∈ ℂ)
173	172	ancom2s 663	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (1...𝑁))) → 𝐴 ∈ ℂ)
174	170, 171, 173	fsumcom 15851	. 2 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝑀)Σ𝑗 ∈ (1...𝑁)𝐴 = Σ𝑗 ∈ (1...𝑁)Σ𝑖 ∈ (0...𝑀)𝐴)
175	146, 171, 153	fsumcom 15851	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵 = Σ𝑗 ∈ (1...𝑁)Σ𝑘 ∈ (0...(𝑀 + 𝑁))𝐵)
176	169, 174, 175	3eqtr4d 2810	1 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝑀)Σ𝑗 ∈ (1...𝑁)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  Fincfn 8949  ℂcc 11115  ℝcr 11116  0cc0 11117  1c1 11118   + caddc 11120   < clt 11260   ≤ cle 11261   − cmin 11458  -cneg 11459  ℕcn 12250  ℕ₀cn0 12521  ℤcz 12608  ℤ_≥cuz 12880  ...cfz 13553  ..^cfzo 13701  Σcsu 15763

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-oi 9479  df-card 9941  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-div 11889  df-nn 12251  df-2 12320  df-3 12321  df-n0 12522  df-z 12609  df-uz 12881  df-rp 13035  df-fz 13554  df-fzo 13702  df-seq 14058  df-exp 14118  df-hash 14387  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-clim 15565  df-sum 15764

This theorem is used by:  breprexplemc  35086