esumfzf - Mathbox for Thierry Arnoux

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Theorem esumfzf 33098

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 1918	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 1
2		oveq2 7417	. . . . . 6 ⊢ (𝑖 = 1 → (1...𝑖) = (1...1))
3	1, 2	esumeq1d 33064	. . . . 5 ⊢ (𝑖 = 1 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...1)(𝐹‘𝑘))
4		fveq2 6892	. . . . 5 ⊢ (𝑖 = 1 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘1))
5	3, 4	eqeq12d 2749	. . . 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 341	. . 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 1918	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑛
8		oveq2 7417	. . . . . 6 ⊢ (𝑖 = 𝑛 → (1...𝑖) = (1...𝑛))
9	7, 8	esumeq1d 33064	. . . . 5 ⊢ (𝑖 = 𝑛 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
10		fveq2 6892	. . . . 5 ⊢ (𝑖 = 𝑛 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘𝑛))
11	9, 10	eqeq12d 2749	. . . 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 341	. . 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 1918	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = (𝑛 + 1)
14		oveq2 7417	. . . . . 6 ⊢ (𝑖 = (𝑛 + 1) → (1...𝑖) = (1...(𝑛 + 1)))
15	13, 14	esumeq1d 33064	. . . . 5 ⊢ (𝑖 = (𝑛 + 1) → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘))
16		fveq2 6892	. . . . 5 ⊢ (𝑖 = (𝑛 + 1) → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)))
17	15, 16	eqeq12d 2749	. . . 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 341	. . 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 1918	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑁
20		oveq2 7417	. . . . . 6 ⊢ (𝑖 = 𝑁 → (1...𝑖) = (1...𝑁))
21	19, 20	esumeq1d 33064	. . . . 5 ⊢ (𝑖 = 𝑁 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑁)(𝐹‘𝑘))
22		fveq2 6892	. . . . 5 ⊢ (𝑖 = 𝑁 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘𝑁))
23	21, 22	eqeq12d 2749	. . . 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 341	. . 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 6892	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
26		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑥{1}
27		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑘{1}
28		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑘)
29		esumfzf.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐹
30		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
31	29, 30	nffv 6902	. . . . . 6 ⊢ Ⅎ𝑘(𝐹‘𝑥)
32	25, 26, 27, 28, 31	cbvesum 33071	. . . . 5 ⊢ Σ^𝑘 ∈ {1} (𝐹‘𝑘) = Σ^𝑥 ∈ {1} (𝐹‘𝑥)
33		simpr 486	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 = 1) → 𝑥 = 1)
34	33	fveq2d 6896	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 = 1) → (𝐹‘𝑥) = (𝐹‘1))
35		1z 12592	. . . . . . 7 ⊢ 1 ∈ ℤ
36	35	a1i 11	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → 1 ∈ ℤ)
37		1nn 12223	. . . . . . 7 ⊢ 1 ∈ ℕ
38		ffvelcdm 7084	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ (0[,]+∞))
39	37, 38	mpan2 690	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝐹‘1) ∈ (0[,]+∞))
40	34, 36, 39	esumsn 33094	. . . . 5 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑥 ∈ {1} (𝐹‘𝑥) = (𝐹‘1))
41	32, 40	eqtrid 2785	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ {1} (𝐹‘𝑘) = (𝐹‘1))
42		fzsn 13543	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
43	35, 42	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
44		esumeq1 33063	. . . . 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 13979	. . . . 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 2798	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ (1...1)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘1))
49		simpl 484	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝑛 ∈ ℕ)
50		nnuz 12865	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
51	49, 50	eleqtrdi 2844	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝑛 ∈ (ℤ_≥‘1))
52		seqp1 13981	. . . . . . . 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 482	. . . . . 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 486	. . . . . . 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 7424	. . . . . 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 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
58	57	nfci 2887	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
59		nfcv 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
60	29, 58, 59	nff 6714	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹:ℕ⟶(0[,]+∞)
61	57, 60	nfan 1903	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞))
62		fzsuc 13548	. . . . . . . . . 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 33064	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = Σ^𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝐹‘𝑘))
65		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
66		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑘{(𝑛 + 1)}
67		ovexd 7444	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (1...𝑛) ∈ V)
68		snex 5432	. . . . . . . . . 10 ⊢ {(𝑛 + 1)} ∈ V
69	68	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → {(𝑛 + 1)} ∈ V)
70		fzp1disj 13560	. . . . . . . . . 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 13545	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → (1...𝑛) ⊆ ℕ)
74	37, 73	ax-mp 5	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
75		simpr 486	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
76	74, 75	sselid 3981	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
77	72, 76	ffvelcdmd 7088	. . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 ∈ {(𝑛 + 1)})
80		velsn 4645	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {(𝑛 + 1)} ↔ 𝑘 = (𝑛 + 1))
81	79, 80	sylib 217	. . . . . . . . . . 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 12229	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝑛 + 1) ∈ ℕ)
84	81, 83	eqeltrd 2834	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 ∈ ℕ)
85	78, 84	ffvelcdmd 7088	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝐹‘𝑘) ∈ (0[,]+∞))
86	61, 65, 66, 67, 69, 71, 77, 85	esumsplit 33082	. . . . . . . 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 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{(𝑛 + 1)}
88	25, 87, 66, 28, 31	cbvesum 33071	. . . . . . . . . 10 ⊢ Σ^𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘) = Σ^𝑥 ∈ {(𝑛 + 1)} (𝐹‘𝑥)
89		simpr 486	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑥 = (𝑛 + 1)) → 𝑥 = (𝑛 + 1))
90	89	fveq2d 6896	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑥 = (𝑛 + 1)) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
91	49	peano2nnd 12229	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (𝑛 + 1) ∈ ℕ)
92		simpr 486	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝐹:ℕ⟶(0[,]+∞))
93	92, 91	ffvelcdmd 7088	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (𝐹‘(𝑛 + 1)) ∈ (0[,]+∞))
94	90, 91, 93	esumsn 33094	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑥 ∈ {(𝑛 + 1)} (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
95	88, 94	eqtrid 2785	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
96	95	oveq2d 7425	. . . . . . . 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 2778	. . . . . . 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 482	. . . . . 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 2780	. . . . 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 421	. . . 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 12230	. 2 ⊢ (𝑁 ∈ ℕ → (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ (1...𝑁)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑁)))
103	102	impcom 409	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 397 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2884 Vcvv 3475 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 {csn 4629 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 +∞cpnf 11245 ℕcn 12212 ℤcz 12558 ℤ_≥cuz 12822 +_𝑒 cxad 13090 [,]cicc 13327 ...cfz 13484 seqcseq 13966 Σ^*cesum 33056

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-ordt 17447 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-ps 18519 df-tsr 18520 df-plusf 18560 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-subrg 20317 df-abv 20425 df-lmod 20473 df-scaf 20474 df-sra 20785 df-rgmod 20786 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-tmd 23576 df-tgp 23577 df-tsms 23631 df-trg 23664 df-xms 23826 df-ms 23827 df-tms 23828 df-nm 24091 df-ngp 24092 df-nrg 24094 df-nlm 24095 df-ii 24393 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-esum 33057

This theorem is referenced by: esumfsup 33099 esumsup 33118

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