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Theorem isumshft 15781

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 𝑘,𝑍 Allowed substitution hints: 𝐴(𝑗) 𝐵(𝑘) 𝑀(𝑗,𝑘) 𝑍(𝑗)

Proof of Theorem isumshft Dummy variables 𝑚 𝑛 𝑥 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 12666	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝐾) ∈ ℤ)
4		isumshft.2	. . . . . . . . . 10 ⊢ 𝑊 = (ℤ_≥‘(𝑀 + 𝐾))
5	4	eleq2i 2825	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑊 ↔ 𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾)))
6	2	zcnd 12663	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℂ)
7		eluzelcn 12830	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾)) → 𝑚 ∈ ℂ)
8	7, 4	eleq2s 2851	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝑊 → 𝑚 ∈ ℂ)
9		isumshft.1	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ_≥‘𝑀)
10	9	fvexi 6902	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
11	10	mptex 7221	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐵) ∈ V
12	11	shftval 15017	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)))
13	6, 8, 12	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)))
14		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
15		eqid 2732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐵) = (𝑘 ∈ 𝑍 ↦ 𝐵)
16	15	fvmpt2i 7005	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ( I ‘𝐵))
17	14, 16	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ( I ‘𝐵))
18		eluzelcn 12830	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℂ)
19	18, 9	eleq2s 2851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ)
20		addcom 11396	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐾 + 𝑘) = (𝑘 + 𝐾))
21	6, 19, 20	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) = (𝑘 + 𝐾))
22		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ 𝑍)
23	22, 9	eleqtrdi 2843	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ (ℤ_≥‘𝑀))
24		eluzadd 12847	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ (ℤ_≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑘 + 𝐾) ∈ (ℤ_≥‘(𝑀 + 𝐾)))
25	23, 2, 24	syl2anr 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 𝐾) ∈ (ℤ_≥‘(𝑀 + 𝐾)))
26	21, 25	eqeltrd 2833	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) ∈ (ℤ_≥‘(𝑀 + 𝐾)))
27	26, 4	eleqtrrdi 2844	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) ∈ 𝑊)
28		isumshft.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝐾 + 𝑘) → 𝐴 = 𝐵)
29		eqid 2732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ 𝑊 ↦ 𝐴) = (𝑗 ∈ 𝑊 ↦ 𝐴)
30	28, 29	fvmpti 6994	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 + 𝑘) ∈ 𝑊 → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)) = ( I ‘𝐵))
31	27, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)) = ( I ‘𝐵))
32	17, 31	eqtr4d 2775	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)))
33	32	ralrimiva 3146	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)))
34		nffvmpt1 6899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛)
35	34	nfeq1 2918	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛))
36		fveq2 6888	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛))
37		oveq2 7413	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (𝐾 + 𝑘) = (𝐾 + 𝑛))
38	37	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)))
39	36, 38	eqeq12d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)) ↔ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛))))
40	35, 39	rspc 3600	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑘)) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛))))
41	33, 40	mpan9 507	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)))
42	41	ralrimiva 3146	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)))
43	1	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 𝑀 ∈ ℤ)
44	2	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 𝐾 ∈ ℤ)
45		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 𝑚 ∈ 𝑊)
46	45, 4	eleqtrdi 2843	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾)))
47		eluzsub 12848	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → (𝑚 − 𝐾) ∈ (ℤ_≥‘𝑀))
48	43, 44, 46, 47	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝑚 − 𝐾) ∈ (ℤ_≥‘𝑀))
49	48, 9	eleqtrrdi 2844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝑚 − 𝐾) ∈ 𝑍)
50		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 − 𝐾) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)))
51		oveq2 7413	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 − 𝐾) → (𝐾 + 𝑛) = (𝐾 + (𝑚 − 𝐾)))
52	51	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 − 𝐾) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + (𝑚 − 𝐾))))
53	50, 52	eqeq12d 2748	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 − 𝐾) → (((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)) ↔ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + (𝑚 − 𝐾)))))
54	53	rspccva 3611	. . . . . . . . . . 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))))
55	42, 49, 54	syl2an2r 683	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑚 − 𝐾)) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + (𝑚 − 𝐾))))
56		pncan3 11464	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
57	6, 8, 56	syl2an 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
58	57	fveq2d 6892	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + (𝑚 − 𝐾))) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚))
59	13, 55, 58	3eqtrrd 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = (((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)‘𝑚))
60	5, 59	sylan2br 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = (((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)‘𝑚))
61	3, 60	seqfeq 13989	. . . . . . 7 ⊢ (𝜑 → seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) = seq(𝑀 + 𝐾)( + , ((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)))
62	61	breq1d 5157	. . . . . 6 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) ⇝ 𝑥 ↔ seq(𝑀 + 𝐾)( + , ((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)) ⇝ 𝑥))
63	11	isershft 15606	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵)) ⇝ 𝑥 ↔ seq(𝑀 + 𝐾)( + , ((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)) ⇝ 𝑥))
64	1, 2, 63	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵)) ⇝ 𝑥 ↔ seq(𝑀 + 𝐾)( + , ((𝑘 ∈ 𝑍 ↦ 𝐵) shift 𝐾)) ⇝ 𝑥))
65	62, 64	bitr4d 281	. . . . 5 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) ⇝ 𝑥 ↔ seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵)) ⇝ 𝑥))
66	65	iotabidv 6524	. . . 4 ⊢ (𝜑 → (℩𝑥seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵)) ⇝ 𝑥))
67		df-fv 6548	. . . 4 ⊢ ( ⇝ ‘seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴))) = (℩𝑥seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴)) ⇝ 𝑥)
68		df-fv 6548	. . . 4 ⊢ ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵))) = (℩𝑥seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵)) ⇝ 𝑥)
69	66, 67, 68	3eqtr4g 2797	. . 3 ⊢ (𝜑 → ( ⇝ ‘seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴))) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵))))
70		eqidd 2733	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚))
71		isumshft.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑊) → 𝐴 ∈ ℂ)
72	71	fmpttd 7111	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ 𝑊 ↦ 𝐴):𝑊⟶ℂ)
73	72	ffvelcdmda 7083	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) ∈ ℂ)
74	4, 3, 70, 73	isum 15661	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ 𝑊 ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = ( ⇝ ‘seq(𝑀 + 𝐾)( + , (𝑗 ∈ 𝑊 ↦ 𝐴))))
75		eqidd 2733	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛))
76	27	ralrimiva 3146	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐾 + 𝑘) ∈ 𝑊)
77	37	eleq1d 2818	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐾 + 𝑘) ∈ 𝑊 ↔ (𝐾 + 𝑛) ∈ 𝑊))
78	77	rspccva 3611	. . . . . . 7 ⊢ ((∀𝑘 ∈ 𝑍 (𝐾 + 𝑘) ∈ 𝑊 ∧ 𝑛 ∈ 𝑍) → (𝐾 + 𝑛) ∈ 𝑊)
79	76, 78	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐾 + 𝑛) ∈ 𝑊)
80		ffvelcdm 7080	. . . . . 6 ⊢ (((𝑗 ∈ 𝑊 ↦ 𝐴):𝑊⟶ℂ ∧ (𝐾 + 𝑛) ∈ 𝑊) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)) ∈ ℂ)
81	72, 79, 80	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑗 ∈ 𝑊 ↦ 𝐴)‘(𝐾 + 𝑛)) ∈ ℂ)
82	41, 81	eqeltrd 2833	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) ∈ ℂ)
83	9, 1, 75, 82	isum 15661	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = ( ⇝ ‘seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐵))))
84	69, 74, 83	3eqtr4d 2782	. 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝑊 ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = Σ𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛))
85		sumfc 15651	. 2 ⊢ Σ𝑚 ∈ 𝑊 ((𝑗 ∈ 𝑊 ↦ 𝐴)‘𝑚) = Σ𝑗 ∈ 𝑊 𝐴
86		sumfc 15651	. 2 ⊢ Σ𝑛 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑛) = Σ𝑘 ∈ 𝑍 𝐵
87	84, 85, 86	3eqtr3g 2795	1 ⊢ (𝜑 → Σ𝑗 ∈ 𝑊 𝐴 = Σ𝑘 ∈ 𝑍 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 class class class wbr 5147 ↦ cmpt 5230 I cid 5572 ℩cio 6490 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 + caddc 11109 − cmin 11440 ℤcz 12554 ℤ_≥cuz 12818 seqcseq 13962 shift cshi 15009 ⇝ cli 15424 Σcsu 15628

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 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 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629

This theorem is referenced by: eftlub 16048 pserdv2 25933 logtayl 26159 binomcxplemnotnn0 43100

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