esumfzf - Mathbox for Thierry Arnoux

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Theorem esumfzf 31703

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 1922	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 1
2		oveq2 7199	. . . . . 6 ⊢ (𝑖 = 1 → (1...𝑖) = (1...1))
3	1, 2	esumeq1d 31669	. . . . 5 ⊢ (𝑖 = 1 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...1)(𝐹‘𝑘))
4		fveq2 6695	. . . . 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 344	. . 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 1922	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑛
8		oveq2 7199	. . . . . 6 ⊢ (𝑖 = 𝑛 → (1...𝑖) = (1...𝑛))
9	7, 8	esumeq1d 31669	. . . . 5 ⊢ (𝑖 = 𝑛 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
10		fveq2 6695	. . . . 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 344	. . 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 1922	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = (𝑛 + 1)
14		oveq2 7199	. . . . . 6 ⊢ (𝑖 = (𝑛 + 1) → (1...𝑖) = (1...(𝑛 + 1)))
15	13, 14	esumeq1d 31669	. . . . 5 ⊢ (𝑖 = (𝑛 + 1) → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘))
16		fveq2 6695	. . . . 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 344	. . 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 1922	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑁
20		oveq2 7199	. . . . . 6 ⊢ (𝑖 = 𝑁 → (1...𝑖) = (1...𝑁))
21	19, 20	esumeq1d 31669	. . . . 5 ⊢ (𝑖 = 𝑁 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑁)(𝐹‘𝑘))
22		fveq2 6695	. . . . 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 344	. . 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 6695	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
26		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑥{1}
27		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑘{1}
28		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑘)
29		esumfzf.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐹
30		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
31	29, 30	nffv 6705	. . . . . 6 ⊢ Ⅎ𝑘(𝐹‘𝑥)
32	25, 26, 27, 28, 31	cbvesum 31676	. . . . 5 ⊢ Σ^𝑘 ∈ {1} (𝐹‘𝑘) = Σ^𝑥 ∈ {1} (𝐹‘𝑥)
33		simpr 488	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 = 1) → 𝑥 = 1)
34	33	fveq2d 6699	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 = 1) → (𝐹‘𝑥) = (𝐹‘1))
35		1z 12172	. . . . . . 7 ⊢ 1 ∈ ℤ
36	35	a1i 11	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → 1 ∈ ℤ)
37		1nn 11806	. . . . . . 7 ⊢ 1 ∈ ℕ
38		ffvelrn 6880	. . . . . . 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 31699	. . . . 5 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑥 ∈ {1} (𝐹‘𝑥) = (𝐹‘1))
41	32, 40	syl5eq 2783	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ {1} (𝐹‘𝑘) = (𝐹‘1))
42		fzsn 13119	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
43	35, 42	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
44		esumeq1 31668	. . . . 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 13552	. . . . 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 486	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝑛 ∈ ℕ)
50		nnuz 12442	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
51	49, 50	eleqtrdi 2841	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝑛 ∈ (ℤ_≥‘1))
52		seqp1 13554	. . . . . . . 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 484	. . . . . 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 488	. . . . . . 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 7206	. . . . . 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 1922	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
58	57	nfci 2880	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
59		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
60	29, 58, 59	nff 6519	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹:ℕ⟶(0[,]+∞)
61	57, 60	nfan 1907	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞))
62		fzsuc 13124	. . . . . . . . . 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 31669	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = Σ^𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝐹‘𝑘))
65		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
66		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑘{(𝑛 + 1)}
67		ovexd 7226	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (1...𝑛) ∈ V)
68		snex 5309	. . . . . . . . . 10 ⊢ {(𝑛 + 1)} ∈ V
69	68	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → {(𝑛 + 1)} ∈ V)
70		fzp1disj 13136	. . . . . . . . . 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 13121	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → (1...𝑛) ⊆ ℕ)
74	37, 73	ax-mp 5	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
75		simpr 488	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
76	74, 75	sseldi 3885	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
77	72, 76	ffvelrnd 6883	. . . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 ∈ {(𝑛 + 1)})
80		velsn 4543	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {(𝑛 + 1)} ↔ 𝑘 = (𝑛 + 1))
81	79, 80	sylib 221	. . . . . . . . . . 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 11812	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝑛 + 1) ∈ ℕ)
84	81, 83	eqeltrd 2831	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 ∈ ℕ)
85	78, 84	ffvelrnd 6883	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝐹‘𝑘) ∈ (0[,]+∞))
86	61, 65, 66, 67, 69, 71, 77, 85	esumsplit 31687	. . . . . . . 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 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{(𝑛 + 1)}
88	25, 87, 66, 28, 31	cbvesum 31676	. . . . . . . . . 10 ⊢ Σ^𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘) = Σ^𝑥 ∈ {(𝑛 + 1)} (𝐹‘𝑥)
89		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑥 = (𝑛 + 1)) → 𝑥 = (𝑛 + 1))
90	89	fveq2d 6699	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑥 = (𝑛 + 1)) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
91	49	peano2nnd 11812	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (𝑛 + 1) ∈ ℕ)
92		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝐹:ℕ⟶(0[,]+∞))
93	92, 91	ffvelrnd 6883	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (𝐹‘(𝑛 + 1)) ∈ (0[,]+∞))
94	90, 91, 93	esumsn 31699	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑥 ∈ {(𝑛 + 1)} (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
95	88, 94	syl5eq 2783	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
96	95	oveq2d 7207	. . . . . . . 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 484	. . . . . 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 423	. . . 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 11813	. 2 ⊢ (𝑁 ∈ ℕ → (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ (1...𝑁)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑁)))
103	102	impcom 411	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 399 = wceq 1543 ∈ wcel 2112 Ⅎwnfc 2877 Vcvv 3398 ∪ cun 3851 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 {csn 4527 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 0cc0 10694 1c1 10695 + caddc 10697 +∞cpnf 10829 ℕcn 11795 ℤcz 12141 ℤ_≥cuz 12403 +_𝑒 cxad 12667 [,]cicc 12903 ...cfz 13060 seqcseq 13539 Σ^*cesum 31661

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-fac 13805 df-bc 13834 df-hash 13862 df-shft 14595 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-ef 15592 df-sin 15594 df-cos 15595 df-pi 15597 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-ordt 16960 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-ps 18026 df-tsr 18027 df-plusf 18067 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-subrg 19752 df-abv 19807 df-lmod 19855 df-scaf 19856 df-sra 20163 df-rgmod 20164 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-lp 21987 df-perf 21988 df-cn 22078 df-cnp 22079 df-haus 22166 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-tmd 22923 df-tgp 22924 df-tsms 22978 df-trg 23011 df-xms 23172 df-ms 23173 df-tms 23174 df-nm 23434 df-ngp 23435 df-nrg 23437 df-nlm 23438 df-ii 23728 df-cncf 23729 df-limc 24717 df-dv 24718 df-log 25399 df-esum 31662

This theorem is referenced by: esumfsup 31704 esumsup 31723

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