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Theorem isumshft 11511

Description: Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)

**Hypotheses**
Ref	Expression
isumshft.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
isumshft.2	⊢ 𝑊 = (ℤ_≥‘(𝑀 + 𝐾))
isumshft.3	⊢ (𝑗 = (𝐾 + 𝑘) → 𝐴 = 𝐵)
isumshft.4	⊢ (𝜑 → 𝐾 ∈ ℤ)
isumshft.5	⊢ (𝜑 → 𝑀 ∈ ℤ)
isumshft.6	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑊) → 𝐴 ∈ ℂ)

**Assertion**
Ref	Expression
isumshft	⊢ (𝜑 → Σ𝑗 ∈ 𝑊 𝐴 = Σ𝑘 ∈ 𝑍 𝐵)

Distinct variable groups: 𝐴,𝑘 𝑗,𝑘,𝐾 𝜑,𝑗,𝑘 𝑗,𝑊,𝑘 𝐵,𝑗 𝑗,𝑍,𝑘 Allowed substitution hints: 𝐴(𝑗) 𝐵(𝑘) 𝑀(𝑗,𝑘)

Proof of Theorem isumshft Dummy variables 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumshft.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		isumshft.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℤ)
3	1, 2	zaddcld 9392	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
4		isumshft.2	. . . . . . . . . . . . 13 ⊢ 𝑊 = (ℤ_≥‘(𝑀 + 𝐾))
5	4	eleq2i 2254	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ (ℤ_≥‘(𝑀 + 𝐾)))
6	5	biimpri 133	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ_≥‘(𝑀 + 𝐾)) → 𝑥 ∈ 𝑊)
7	6	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → 𝑥 ∈ 𝑊)
8		isumshft.6	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑊) → 𝐴 ∈ ℂ)
9	8	ralrimiva 2560	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑗 ∈ 𝑊 𝐴 ∈ ℂ)
10	9	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → ∀𝑗 ∈ 𝑊 𝐴 ∈ ℂ)
11		nfcsb1v 3102	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐴
12	11	nfel1 2340	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⦋𝑥 / 𝑗⦌𝐴 ∈ ℂ
13		csbeq1a 3078	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑥 → 𝐴 = ⦋𝑥 / 𝑗⦌𝐴)
14	13	eleq1d 2256	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑥 → (𝐴 ∈ ℂ ↔ ⦋𝑥 / 𝑗⦌𝐴 ∈ ℂ))
15	12, 14	rspc 2847	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑊 → (∀𝑗 ∈ 𝑊 𝐴 ∈ ℂ → ⦋𝑥 / 𝑗⦌𝐴 ∈ ℂ))
16	7, 10, 15	sylc 62	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → ⦋𝑥 / 𝑗⦌𝐴 ∈ ℂ)
17		eqid 2187	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑊 ↦ 𝐴) = (𝑗 ∈ 𝑊 ↦ 𝐴)
18	17	fvmpts 5607	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑊 ∧ ⦋𝑥 / 𝑗⦌𝐴 ∈ ℂ) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑗⦌𝐴)
19	7, 16, 18	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑥) = ⦋𝑥 / 𝑗⦌𝐴)
20	19, 16	eqeltrd 2264	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑥) ∈ ℂ)
21	4	eleq2i 2254	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑊 ↔ 𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾)))
22	2	zcnd 9389	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℂ)
23		eluzelcn 9552	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾)) → 𝑚 ∈ ℂ)
24	23, 4	eleq2s 2282	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝑊 → 𝑚 ∈ ℂ)
25		zex 9275	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ V
26		isumshft.1	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = (ℤ_≥‘𝑀)
27		uzssz 9560	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
28	26, 27	eqsstri 3199	. . . . . . . . . . . . . 14 ⊢ 𝑍 ⊆ ℤ
29	25, 28	ssexi 4153	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
30	29	mptex 5755	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐵) ∈ V
31	30	shftval 10847	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)))
32	22, 24, 31	syl2an 289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)))
33		eqidd 2188	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) = (𝑘 ∈ 𝑍 ↦ 𝐵))
34		isumshft.3	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝐾 + 𝑘) → 𝐴 = 𝐵)
35	34	eleq1d 2256	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝐾 + 𝑘) → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
36	9	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∀𝑗 ∈ 𝑊 𝐴 ∈ ℂ)
37	1	adantr 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ∈ ℤ)
38	2	adantr 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐾 ∈ ℤ)
39	37, 38	zaddcld 9392	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + 𝐾) ∈ ℤ)
40		eluzelz 9550	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℤ)
41	40, 26	eleq2s 2282	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
42	41	adantl 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ)
43	38, 42	zaddcld 9392	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) ∈ ℤ)
44	37	zred 9388	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ∈ ℝ)
45	42	zred 9388	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℝ)
46	38	zred 9388	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐾 ∈ ℝ)
47	26	eleq2i 2254	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ_≥‘𝑀))
48	47	biimpi 120	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ (ℤ_≥‘𝑀))
49	48	adantl 277	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (ℤ_≥‘𝑀))
50		eluzle 9553	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑘)
51	49, 50	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ≤ 𝑘)
52	44, 45, 46, 51	leadd1dd 8529	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + 𝐾) ≤ (𝑘 + 𝐾))
53	42	zcnd 9389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℂ)
54	38	zcnd 9389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐾 ∈ ℂ)
55	53, 54	addcomd 8121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 𝐾) = (𝐾 + 𝑘))
56	52, 55	breqtrd 4041	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + 𝐾) ≤ (𝐾 + 𝑘))
57		eluz2 9547	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 + 𝑘) ∈ (ℤ_≥‘(𝑀 + 𝐾)) ↔ ((𝑀 + 𝐾) ∈ ℤ ∧ (𝐾 + 𝑘) ∈ ℤ ∧ (𝑀 + 𝐾) ≤ (𝐾 + 𝑘)))
58	39, 43, 56, 57	syl3anbrc 1182	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) ∈ (ℤ_≥‘(𝑀 + 𝐾)))
59	58, 4	eleqtrrdi 2281	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) ∈ 𝑊)
60	35, 36, 59	rspcdva 2858	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
61	33, 60	fvmpt2d 5615	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = 𝐵)
62		eqidd 2188	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑗 ∈ 𝑊 ↦ 𝐴) = (𝑗 ∈ 𝑊 ↦ 𝐴))
63	34	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 = (𝐾 + 𝑘)) → 𝐴 = 𝐵)
64	62, 63, 59, 60	fvmptd 5610	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)) = 𝐵)
65	61, 64	eqtr4d 2223	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)))
66	65	ralrimiva 2560	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)))
67		nffvmpt1 5538	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛)
68	67	nfeq1 2339	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛))
69		fveq2 5527	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛))
70		oveq2 5896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝐾 + 𝑘) = (𝐾 + 𝑛))
71	70	fveq2d 5531	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)))
72	69, 71	eqeq12d 2202	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)) ↔ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛))))
73	68, 72	rspc 2847	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛))))
74	66, 73	mpan9 281	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)))
75	74	ralrimiva 2560	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)))
76	75	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ∀𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)))
77	1	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 𝑀 ∈ ℤ)
78	2	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 𝐾 ∈ ℤ)
79		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 𝑚 ∈ 𝑊)
80	79, 4	eleqtrdi 2280	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾)))
81		eluzsub 9570	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → (𝑚 − 𝐾) ∈ (ℤ_≥‘𝑀))
82	77, 78, 80, 81	syl3anc 1248	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝑚 − 𝐾) ∈ (ℤ_≥‘𝑀))
83	82, 26	eleqtrrdi 2281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝑚 − 𝐾) ∈ 𝑍)
84		fveq2 5527	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 − 𝐾) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)))
85		oveq2 5896	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 − 𝐾) → (𝐾 + 𝑛) = (𝐾 + (𝑚 − 𝐾)))
86	85	fveq2d 5531	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 − 𝐾) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + (𝑚 − 𝐾))))
87	84, 86	eqeq12d 2202	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 − 𝐾) → (((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)) ↔ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + (𝑚 − 𝐾)))))
88	87	rspccva 2852	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)) ∧ (𝑚 − 𝐾) ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + (𝑚 − 𝐾))))
89	76, 83, 88	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + (𝑚 − 𝐾))))
90		pncan3 8178	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
91	22, 24, 90	syl2an 289	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
92	91	fveq2d 5531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + (𝑚 − 𝐾))) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚))
93	32, 89, 92	3eqtrrd 2225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = (((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)‘𝑚))
94	21, 93	sylan2br 288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = (((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)‘𝑚))
95		addcl 7949	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
96	95	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
97	3, 20, 94, 96	seq3feq 10485	. . . . . . 7 ⊢ (𝜑 → seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) = seq(𝑀 + 𝐾)( + , ((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)))
98	97	breq1d 4025	. . . . . 6 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) ⇝ 𝑥 ↔ seq(𝑀 + 𝐾)( + , ((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)) ⇝ 𝑥))
99	30	a1i 9	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ∈ V)
100	26	eleq2i 2254	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑍 ↔ 𝑥 ∈ (ℤ_≥‘𝑀))
101	100	biimpri 133	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℤ_≥‘𝑀) → 𝑥 ∈ 𝑍)
102	101	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → 𝑥 ∈ 𝑍)
103	60	ralrimiva 2560	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝐵 ∈ ℂ)
104	103	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → ∀𝑘 ∈ 𝑍 𝐵 ∈ ℂ)
105		nfcsb1v 3102	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐵
106	105	nfel1 2340	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐵 ∈ ℂ
107		csbeq1a 3078	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑘⦌𝐵)
108	107	eleq1d 2256	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (𝐵 ∈ ℂ ↔ ⦋𝑥 / 𝑘⦌𝐵 ∈ ℂ))
109	106, 108	rspc 2847	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐵 ∈ ℂ → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℂ))
110	102, 104, 109	sylc 62	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℂ)
111		eqid 2187	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐵) = (𝑘 ∈ 𝑍 ↦ 𝐵)
112	111	fvmpts 5607	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑍 ∧ ⦋𝑥 / 𝑘⦌𝐵 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐵)
113	102, 110, 112	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐵)
114	113, 110	eqeltrd 2264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑥) ∈ ℂ)
115	99, 1, 2, 114, 96	iser3shft 11367	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵)) ⇝ 𝑥 ↔ seq(𝑀 + 𝐾)( + , ((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)) ⇝ 𝑥))
116	98, 115	bitr4d 191	. . . . 5 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) ⇝ 𝑥 ↔ seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵)) ⇝ 𝑥))
117	116	iotabidv 5211	. . . 4 ⊢ (𝜑 → (℩𝑥seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵)) ⇝ 𝑥))
118		df-fv 5236	. . . 4 ⊢ ( ⇝ ‘seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴))) = (℩𝑥seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) ⇝ 𝑥)
119		df-fv 5236	. . . 4 ⊢ ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵))) = (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵)) ⇝ 𝑥)
120	117, 118, 119	3eqtr4g 2245	. . 3 ⊢ (𝜑 → ( ⇝ ‘seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴))) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵))))
121		eqidd 2188	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚))
122	8	fmpttd 5684	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ 𝑊 ↦ 𝐴):𝑊⟶ℂ)
123	122	ffvelcdmda 5664	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) ∈ ℂ)
124	4, 3, 121, 123	isum 11406	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ 𝑊 ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = ( ⇝ ‘seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴))))
125		eqidd 2188	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛))
126	122	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑗 ∈ 𝑊 ↦ 𝐴):𝑊⟶ℂ)
127		eluzelcn 9552	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℂ)
128	127, 26	eleq2s 2282	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ)
129		addcom 8107	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐾 + 𝑘) = (𝑘 + 𝐾))
130	22, 128, 129	syl2an 289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) = (𝑘 + 𝐾))
131		id 19	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ 𝑍)
132	131, 26	eleqtrdi 2280	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ (ℤ_≥‘𝑀))
133		eluzadd 9569	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ_≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑘 + 𝐾) ∈ (ℤ_≥‘(𝑀 + 𝐾)))
134	132, 2, 133	syl2anr 290	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 𝐾) ∈ (ℤ_≥‘(𝑀 + 𝐾)))
135	130, 134	eqeltrd 2264	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) ∈ (ℤ_≥‘(𝑀 + 𝐾)))
136	135, 4	eleqtrrdi 2281	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) ∈ 𝑊)
137	136	ralrimiva 2560	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐾 + 𝑘) ∈ 𝑊)
138	70	eleq1d 2256	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐾 + 𝑘) ∈ 𝑊 ↔ (𝐾 + 𝑛) ∈ 𝑊))
139	138	rspccva 2852	. . . . . . 7 ⊢ ((∀𝑘 ∈ 𝑍 (𝐾 + 𝑘) ∈ 𝑊 ∧ 𝑛 ∈ 𝑍) → (𝐾 + 𝑛) ∈ 𝑊)
140	137, 139	sylan 283	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐾 + 𝑛) ∈ 𝑊)
141	126, 140	ffvelcdmd 5665	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)) ∈ ℂ)
142	74, 141	eqeltrd 2264	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) ∈ ℂ)
143	26, 1, 125, 142	isum 11406	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵))))
144	120, 124, 143	3eqtr4d 2230	. 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝑊 ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = Σ𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛))
145		sumfct 11395	. . 3 ⊢ (∀𝑗 ∈ 𝑊 𝐴 ∈ ℂ → Σ𝑚 ∈ 𝑊 ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = Σ𝑗 ∈ 𝑊 𝐴)
146	9, 145	syl 14	. 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝑊 ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = Σ𝑗 ∈ 𝑊 𝐴)
147		sumfct 11395	. . 3 ⊢ (∀𝑘 ∈ 𝑍 𝐵 ∈ ℂ → Σ𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = Σ𝑘 ∈ 𝑍 𝐵)
148	103, 147	syl 14	. 2 ⊢ (𝜑 → Σ𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = Σ𝑘 ∈ 𝑍 𝐵)
149	144, 146, 148	3eqtr3d 2228	1 ⊢ (𝜑 → Σ𝑗 ∈ 𝑊 𝐴 = Σ𝑘 ∈ 𝑍 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 ∀wral 2465 Vcvv 2749 ⦋csb 3069 class class class wbr 4015 ↦ cmpt 4076 ℩cio 5188 ⟶wf 5224 ‘cfv 5228 (class class class)co 5888 ℂcc 7822 + caddc 7827 ≤ cle 8006 − cmin 8141 ℤcz 9266 ℤ_≥cuz 9541 seqcseq 10458 shift cshi 10836 ⇝ cli 11299 Σcsu 11374

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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-frec 6405 df-1o 6430 df-oadd 6434 df-er 6548 df-en 6754 df-dom 6755 df-fin 6756 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-fz 10022 df-fzo 10156 df-seqfrec 10459 df-exp 10533 df-ihash 10769 df-shft 10837 df-cj 10864 df-rsqrt 11020 df-abs 11021 df-clim 11300 df-sumdc 11375

This theorem is referenced by: eftlub 11711

W3C validator