esumfzf - Mathbox for Thierry Arnoux

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Theorem esumfzf 34055

Description: Formulating a partial extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)

**Hypothesis**
Ref	Expression
esumfzf.1	⊢ Ⅎ𝑘𝐹

**Assertion**
Ref	Expression
esumfzf	⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑁 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑁)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑁))

Distinct variable group: 𝑘,𝑁 Allowed substitution hint: 𝐹(𝑘)

Proof of Theorem esumfzf Dummy variables 𝑖 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 1
2		oveq2 7361	. . . . . 6 ⊢ (𝑖 = 1 → (1...𝑖) = (1...1))
3	1, 2	esumeq1d 34021	. . . . 5 ⊢ (𝑖 = 1 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...1)(𝐹‘𝑘))
4		fveq2 6826	. . . . 5 ⊢ (𝑖 = 1 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘1))
5	3, 4	eqeq12d 2745	. . . 4 ⊢ (𝑖 = 1 → (Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑖) ↔ Σ^𝑘 ∈ (1...1)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘1)))
6	5	imbi2d 340	. . 3 ⊢ (𝑖 = 1 → ((𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑖)) ↔ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...1)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘1))))
7		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑛
8		oveq2 7361	. . . . . 6 ⊢ (𝑖 = 𝑛 → (1...𝑖) = (1...𝑛))
9	7, 8	esumeq1d 34021	. . . . 5 ⊢ (𝑖 = 𝑛 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
10		fveq2 6826	. . . . 5 ⊢ (𝑖 = 𝑛 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘𝑛))
11	9, 10	eqeq12d 2745	. . . 4 ⊢ (𝑖 = 𝑛 → (Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑖) ↔ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛)))
12	11	imbi2d 340	. . 3 ⊢ (𝑖 = 𝑛 → ((𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑖)) ↔ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛))))
13		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = (𝑛 + 1)
14		oveq2 7361	. . . . . 6 ⊢ (𝑖 = (𝑛 + 1) → (1...𝑖) = (1...(𝑛 + 1)))
15	13, 14	esumeq1d 34021	. . . . 5 ⊢ (𝑖 = (𝑛 + 1) → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘))
16		fveq2 6826	. . . . 5 ⊢ (𝑖 = (𝑛 + 1) → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)))
17	15, 16	eqeq12d 2745	. . . 4 ⊢ (𝑖 = (𝑛 + 1) → (Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑖) ↔ Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1))))
18	17	imbi2d 340	. . 3 ⊢ (𝑖 = (𝑛 + 1) → ((𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑖)) ↔ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)))))
19		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑁
20		oveq2 7361	. . . . . 6 ⊢ (𝑖 = 𝑁 → (1...𝑖) = (1...𝑁))
21	19, 20	esumeq1d 34021	. . . . 5 ⊢ (𝑖 = 𝑁 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑁)(𝐹‘𝑘))
22		fveq2 6826	. . . . 5 ⊢ (𝑖 = 𝑁 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘𝑁))
23	21, 22	eqeq12d 2745	. . . 4 ⊢ (𝑖 = 𝑁 → (Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑖) ↔ Σ^𝑘 ∈ (1...𝑁)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑁)))
24	23	imbi2d 340	. . 3 ⊢ (𝑖 = 𝑁 → ((𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑖)) ↔ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...𝑁)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑁))))
25		fveq2 6826	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
26		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥{1}
27		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑘{1}
28		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑘)
29		esumfzf.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐹
30		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
31	29, 30	nffv 6836	. . . . . 6 ⊢ Ⅎ𝑘(𝐹‘𝑥)
32	25, 26, 27, 28, 31	cbvesum 34028	. . . . 5 ⊢ Σ^𝑘 ∈ {1} (𝐹‘𝑘) = Σ^𝑥 ∈ {1} (𝐹‘𝑥)
33		simpr 484	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 = 1) → 𝑥 = 1)
34	33	fveq2d 6830	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 = 1) → (𝐹‘𝑥) = (𝐹‘1))
35		1z 12524	. . . . . . 7 ⊢ 1 ∈ ℤ
36	35	a1i 11	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → 1 ∈ ℤ)
37		1nn 12158	. . . . . . 7 ⊢ 1 ∈ ℕ
38		ffvelcdm 7019	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ (0[,]+∞))
39	37, 38	mpan2 691	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝐹‘1) ∈ (0[,]+∞))
40	34, 36, 39	esumsn 34051	. . . . 5 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑥 ∈ {1} (𝐹‘𝑥) = (𝐹‘1))
41	32, 40	eqtrid 2776	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ {1} (𝐹‘𝑘) = (𝐹‘1))
42		fzsn 13488	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
43	35, 42	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
44		esumeq1 34020	. . . . 5 ⊢ ((1...1) = {1} → Σ^𝑘 ∈ (1...1)(𝐹‘𝑘) = Σ^𝑘 ∈ {1} (𝐹‘𝑘))
45	43, 44	ax-mp 5	. . . 4 ⊢ Σ^𝑘 ∈ (1...1)(𝐹‘𝑘) = Σ^𝑘 ∈ {1} (𝐹‘𝑘)
46		seq1 13940	. . . . 5 ⊢ (1 ∈ ℤ → (seq1( +_𝑒 , 𝐹)‘1) = (𝐹‘1))
47	35, 46	ax-mp 5	. . . 4 ⊢ (seq1( +_𝑒 , 𝐹)‘1) = (𝐹‘1)
48	41, 45, 47	3eqtr4g 2789	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ (1...1)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘1))
49		simpl 482	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝑛 ∈ ℕ)
50		nnuz 12797	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
51	49, 50	eleqtrdi 2838	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝑛 ∈ (ℤ_≥‘1))
52		seqp1 13942	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘1) → (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)) = ((seq1( +_𝑒 , 𝐹)‘𝑛) +_𝑒 (𝐹‘(𝑛 + 1))))
53	51, 52	syl 17	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)) = ((seq1( +_𝑒 , 𝐹)‘𝑛) +_𝑒 (𝐹‘(𝑛 + 1))))
54	53	adantr 480	. . . . . 6 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛)) → (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)) = ((seq1( +_𝑒 , 𝐹)‘𝑛) +_𝑒 (𝐹‘(𝑛 + 1))))
55		simpr 484	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛)) → Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛))
56	55	oveq1d 7368	. . . . . 6 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛)) → (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) +_𝑒 (𝐹‘(𝑛 + 1))) = ((seq1( +_𝑒 , 𝐹)‘𝑛) +_𝑒 (𝐹‘(𝑛 + 1))))
57		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
58	57	nfci 2879	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
59		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
60	29, 58, 59	nff 6652	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹:ℕ⟶(0[,]+∞)
61	57, 60	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞))
62		fzsuc 13493	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
63	51, 62	syl 17	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
64	61, 63	esumeq1d 34021	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = Σ^𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝐹‘𝑘))
65		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
66		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑘{(𝑛 + 1)}
67		ovexd 7388	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (1...𝑛) ∈ V)
68		snex 5378	. . . . . . . . . 10 ⊢ {(𝑛 + 1)} ∈ V
69	68	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → {(𝑛 + 1)} ∈ V)
70		fzp1disj 13505	. . . . . . . . . 10 ⊢ ((1...𝑛) ∩ {(𝑛 + 1)}) = ∅
71	70	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → ((1...𝑛) ∩ {(𝑛 + 1)}) = ∅)
72		simplr 768	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
73		fzssnn 13490	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → (1...𝑛) ⊆ ℕ)
74	37, 73	ax-mp 5	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
75		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
76	74, 75	sselid 3935	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
77	72, 76	ffvelcdmd 7023	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
78		simplr 768	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝐹:ℕ⟶(0[,]+∞))
79		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 ∈ {(𝑛 + 1)})
80		velsn 4595	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {(𝑛 + 1)} ↔ 𝑘 = (𝑛 + 1))
81	79, 80	sylib 218	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 = (𝑛 + 1))
82		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑛 ∈ ℕ)
83	82	peano2nnd 12164	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝑛 + 1) ∈ ℕ)
84	81, 83	eqeltrd 2828	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 ∈ ℕ)
85	78, 84	ffvelcdmd 7023	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝐹‘𝑘) ∈ (0[,]+∞))
86	61, 65, 66, 67, 69, 71, 77, 85	esumsplit 34039	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝐹‘𝑘) = (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) +_𝑒 Σ^*𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘)))
87		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{(𝑛 + 1)}
88	25, 87, 66, 28, 31	cbvesum 34028	. . . . . . . . . 10 ⊢ Σ^𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘) = Σ^𝑥 ∈ {(𝑛 + 1)} (𝐹‘𝑥)
89		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑥 = (𝑛 + 1)) → 𝑥 = (𝑛 + 1))
90	89	fveq2d 6830	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑥 = (𝑛 + 1)) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
91	49	peano2nnd 12164	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (𝑛 + 1) ∈ ℕ)
92		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝐹:ℕ⟶(0[,]+∞))
93	92, 91	ffvelcdmd 7023	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (𝐹‘(𝑛 + 1)) ∈ (0[,]+∞))
94	90, 91, 93	esumsn 34051	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑥 ∈ {(𝑛 + 1)} (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
95	88, 94	eqtrid 2776	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
96	95	oveq2d 7369	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) +_𝑒 Σ^𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘)) = (Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) +_𝑒 (𝐹‘(𝑛 + 1))))
97	64, 86, 96	3eqtrrd 2769	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) +_𝑒 (𝐹‘(𝑛 + 1))) = Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘))
98	97	adantr 480	. . . . . 6 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛)) → (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) +_𝑒 (𝐹‘(𝑛 + 1))) = Σ^*𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘))
99	54, 56, 98	3eqtr2rd 2771	. . . . 5 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛)) → Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)))
100	99	exp31 419	. . . 4 ⊢ (𝑛 ∈ ℕ → (𝐹:ℕ⟶(0[,]+∞) → (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛) → Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)))))
101	100	a2d 29	. . 3 ⊢ (𝑛 ∈ ℕ → ((𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛)) → (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)))))
102	6, 12, 18, 24, 48, 101	nnind 12165	. 2 ⊢ (𝑁 ∈ ℕ → (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ (1...𝑁)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑁)))
103	102	impcom 407	1 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑁 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑁)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Ⅎwnfc 2876 Vcvv 3438 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 {csn 4579 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 0cc0 11028 1c1 11029 + caddc 11031 +∞cpnf 11165 ℕcn 12147 ℤcz 12490 ℤ_≥cuz 12754 +_𝑒 cxad 13031 [,]cicc 13270 ...cfz 13429 seqcseq 13927 Σ^*cesum 34013

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-q 12869 df-rp 12913 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13271 df-ioc 13272 df-ico 13273 df-icc 13274 df-fz 13430 df-fzo 13577 df-fl 13715 df-mod 13793 df-seq 13928 df-exp 13988 df-fac 14200 df-bc 14229 df-hash 14257 df-shft 14993 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-limsup 15397 df-clim 15414 df-rlim 15415 df-sum 15613 df-ef 15993 df-sin 15995 df-cos 15996 df-pi 15998 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-starv 17195 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ds 17202 df-unif 17203 df-hom 17204 df-cco 17205 df-rest 17345 df-topn 17346 df-0g 17364 df-gsum 17365 df-topgen 17366 df-pt 17367 df-prds 17370 df-ordt 17424 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-ps 18491 df-tsr 18492 df-plusf 18532 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-mhm 18676 df-submnd 18677 df-grp 18834 df-minusg 18835 df-sbg 18836 df-mulg 18966 df-subg 19021 df-cntz 19215 df-cmn 19680 df-abl 19681 df-mgp 20045 df-rng 20057 df-ur 20086 df-ring 20139 df-cring 20140 df-subrng 20450 df-subrg 20474 df-abv 20713 df-lmod 20784 df-scaf 20785 df-sra 21096 df-rgmod 21097 df-psmet 21272 df-xmet 21273 df-met 21274 df-bl 21275 df-mopn 21276 df-fbas 21277 df-fg 21278 df-cnfld 21281 df-top 22798 df-topon 22815 df-topsp 22837 df-bases 22850 df-cld 22923 df-ntr 22924 df-cls 22925 df-nei 23002 df-lp 23040 df-perf 23041 df-cn 23131 df-cnp 23132 df-haus 23219 df-tx 23466 df-hmeo 23659 df-fil 23750 df-fm 23842 df-flim 23843 df-flf 23844 df-tmd 23976 df-tgp 23977 df-tsms 24031 df-trg 24064 df-xms 24225 df-ms 24226 df-tms 24227 df-nm 24487 df-ngp 24488 df-nrg 24490 df-nlm 24491 df-ii 24787 df-cncf 24788 df-limc 25784 df-dv 25785 df-log 26482 df-esum 34014

This theorem is referenced by: esumfsup 34056 esumsup 34075

W3C validator