esumfzf - Mathbox for Thierry Arnoux

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Theorem esumfzf 34213

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 1916	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 1
2		oveq2 7375	. . . . . 6 ⊢ (𝑖 = 1 → (1...𝑖) = (1...1))
3	1, 2	esumeq1d 34179	. . . . 5 ⊢ (𝑖 = 1 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...1)(𝐹‘𝑘))
4		fveq2 6840	. . . . 5 ⊢ (𝑖 = 1 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘1))
5	3, 4	eqeq12d 2752	. . . 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 1916	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑛
8		oveq2 7375	. . . . . 6 ⊢ (𝑖 = 𝑛 → (1...𝑖) = (1...𝑛))
9	7, 8	esumeq1d 34179	. . . . 5 ⊢ (𝑖 = 𝑛 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
10		fveq2 6840	. . . . 5 ⊢ (𝑖 = 𝑛 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘𝑛))
11	9, 10	eqeq12d 2752	. . . 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 1916	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = (𝑛 + 1)
14		oveq2 7375	. . . . . 6 ⊢ (𝑖 = (𝑛 + 1) → (1...𝑖) = (1...(𝑛 + 1)))
15	13, 14	esumeq1d 34179	. . . . 5 ⊢ (𝑖 = (𝑛 + 1) → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘))
16		fveq2 6840	. . . . 5 ⊢ (𝑖 = (𝑛 + 1) → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)))
17	15, 16	eqeq12d 2752	. . . 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 1916	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑁
20		oveq2 7375	. . . . . 6 ⊢ (𝑖 = 𝑁 → (1...𝑖) = (1...𝑁))
21	19, 20	esumeq1d 34179	. . . . 5 ⊢ (𝑖 = 𝑁 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑁)(𝐹‘𝑘))
22		fveq2 6840	. . . . 5 ⊢ (𝑖 = 𝑁 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘𝑁))
23	21, 22	eqeq12d 2752	. . . 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 6840	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
26		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥{1}
27		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑘{1}
28		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑘)
29		esumfzf.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐹
30		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
31	29, 30	nffv 6850	. . . . . 6 ⊢ Ⅎ𝑘(𝐹‘𝑥)
32	25, 26, 27, 28, 31	cbvesum 34186	. . . . 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 6844	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 = 1) → (𝐹‘𝑥) = (𝐹‘1))
35		1z 12557	. . . . . . 7 ⊢ 1 ∈ ℤ
36	35	a1i 11	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → 1 ∈ ℤ)
37		1nn 12185	. . . . . . 7 ⊢ 1 ∈ ℕ
38		ffvelcdm 7033	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ (0[,]+∞))
39	37, 38	mpan2 692	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝐹‘1) ∈ (0[,]+∞))
40	34, 36, 39	esumsn 34209	. . . . 5 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑥 ∈ {1} (𝐹‘𝑥) = (𝐹‘1))
41	32, 40	eqtrid 2783	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ {1} (𝐹‘𝑘) = (𝐹‘1))
42		fzsn 13520	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
43	35, 42	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
44		esumeq1 34178	. . . . 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 13976	. . . . 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 2796	. . 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 12827	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
51	49, 50	eleqtrdi 2846	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝑛 ∈ (ℤ_≥‘1))
52		seqp1 13978	. . . . . . . 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 7382	. . . . . 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 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
58	57	nfci 2886	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
59		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
60	29, 58, 59	nff 6664	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹:ℕ⟶(0[,]+∞)
61	57, 60	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞))
62		fzsuc 13525	. . . . . . . . . 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 34179	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = Σ^𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝐹‘𝑘))
65		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
66		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑘{(𝑛 + 1)}
67		ovexd 7402	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (1...𝑛) ∈ V)
68		snex 5381	. . . . . . . . . 10 ⊢ {(𝑛 + 1)} ∈ V
69	68	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → {(𝑛 + 1)} ∈ V)
70		fzp1disj 13537	. . . . . . . . . 10 ⊢ ((1...𝑛) ∩ {(𝑛 + 1)}) = ∅
71	70	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → ((1...𝑛) ∩ {(𝑛 + 1)}) = ∅)
72		simplr 769	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
73		fzssnn 13522	. . . . . . . . . . . 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 3919	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
77	72, 76	ffvelcdmd 7037	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
78		simplr 769	. . . . . . . . . 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 4583	. . . . . . . . . . . 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 767	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑛 ∈ ℕ)
83	82	peano2nnd 12191	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝑛 + 1) ∈ ℕ)
84	81, 83	eqeltrd 2836	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 ∈ ℕ)
85	78, 84	ffvelcdmd 7037	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝐹‘𝑘) ∈ (0[,]+∞))
86	61, 65, 66, 67, 69, 71, 77, 85	esumsplit 34197	. . . . . . . 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 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{(𝑛 + 1)}
88	25, 87, 66, 28, 31	cbvesum 34186	. . . . . . . . . 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 6844	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑥 = (𝑛 + 1)) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
91	49	peano2nnd 12191	. . . . . . . . . . 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 7037	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (𝐹‘(𝑛 + 1)) ∈ (0[,]+∞))
94	90, 91, 93	esumsn 34209	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑥 ∈ {(𝑛 + 1)} (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
95	88, 94	eqtrid 2783	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
96	95	oveq2d 7383	. . . . . . . 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 2776	. . . . . . 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 2778	. . . . 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 12192	. 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 1542 ∈ wcel 2114 Ⅎwnfc 2883 Vcvv 3429 ∪ cun 3887 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 {csn 4567 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 0cc0 11038 1c1 11039 + caddc 11041 +∞cpnf 11176 ℕcn 12174 ℤcz 12524 ℤ_≥cuz 12788 +_𝑒 cxad 13061 [,]cicc 13301 ...cfz 13461 seqcseq 13963 Σ^*cesum 34171

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-ordt 17465 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-ps 18532 df-tsr 18533 df-plusf 18607 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-subrng 20523 df-subrg 20547 df-abv 20786 df-lmod 20857 df-scaf 20858 df-sra 21168 df-rgmod 21169 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-tmd 24037 df-tgp 24038 df-tsms 24092 df-trg 24125 df-xms 24285 df-ms 24286 df-tms 24287 df-nm 24547 df-ngp 24548 df-nrg 24550 df-nlm 24551 df-ii 24844 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-esum 34172

This theorem is referenced by: esumfsup 34214 esumsup 34233

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