esumfzf - Mathbox for Thierry Arnoux

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Theorem esumfzf 31004

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 1874	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 1
2		oveq2 6990	. . . . . 6 ⊢ (𝑖 = 1 → (1...𝑖) = (1...1))
3	1, 2	esumeq1d 30970	. . . . 5 ⊢ (𝑖 = 1 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...1)(𝐹‘𝑘))
4		fveq2 6504	. . . . 5 ⊢ (𝑖 = 1 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘1))
5	3, 4	eqeq12d 2795	. . . 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 333	. . 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 1874	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑛
8		oveq2 6990	. . . . . 6 ⊢ (𝑖 = 𝑛 → (1...𝑖) = (1...𝑛))
9	7, 8	esumeq1d 30970	. . . . 5 ⊢ (𝑖 = 𝑛 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
10		fveq2 6504	. . . . 5 ⊢ (𝑖 = 𝑛 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘𝑛))
11	9, 10	eqeq12d 2795	. . . 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 333	. . 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 1874	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = (𝑛 + 1)
14		oveq2 6990	. . . . . 6 ⊢ (𝑖 = (𝑛 + 1) → (1...𝑖) = (1...(𝑛 + 1)))
15	13, 14	esumeq1d 30970	. . . . 5 ⊢ (𝑖 = (𝑛 + 1) → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘))
16		fveq2 6504	. . . . 5 ⊢ (𝑖 = (𝑛 + 1) → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘(𝑛 + 1)))
17	15, 16	eqeq12d 2795	. . . 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 333	. . 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 1874	. . . . . 6 ⊢ Ⅎ𝑘 𝑖 = 𝑁
20		oveq2 6990	. . . . . 6 ⊢ (𝑖 = 𝑁 → (1...𝑖) = (1...𝑁))
21	19, 20	esumeq1d 30970	. . . . 5 ⊢ (𝑖 = 𝑁 → Σ^𝑘 ∈ (1...𝑖)(𝐹‘𝑘) = Σ^𝑘 ∈ (1...𝑁)(𝐹‘𝑘))
22		fveq2 6504	. . . . 5 ⊢ (𝑖 = 𝑁 → (seq1( +_𝑒 , 𝐹)‘𝑖) = (seq1( +_𝑒 , 𝐹)‘𝑁))
23	21, 22	eqeq12d 2795	. . . 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 333	. . 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 6504	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
26		nfcv 2934	. . . . . 6 ⊢ Ⅎ𝑥{1}
27		nfcv 2934	. . . . . 6 ⊢ Ⅎ𝑘{1}
28		nfcv 2934	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑘)
29		esumfzf.1	. . . . . . 7 ⊢ Ⅎ𝑘𝐹
30		nfcv 2934	. . . . . . 7 ⊢ Ⅎ𝑘𝑥
31	29, 30	nffv 6514	. . . . . 6 ⊢ Ⅎ𝑘(𝐹‘𝑥)
32	25, 26, 27, 28, 31	cbvesum 30977	. . . . 5 ⊢ Σ^𝑘 ∈ {1} (𝐹‘𝑘) = Σ^𝑥 ∈ {1} (𝐹‘𝑥)
33		simpr 477	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 = 1) → 𝑥 = 1)
34	33	fveq2d 6508	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 = 1) → (𝐹‘𝑥) = (𝐹‘1))
35		1z 11831	. . . . . . 7 ⊢ 1 ∈ ℤ
36	35	a1i 11	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → 1 ∈ ℤ)
37		1nn 11458	. . . . . . 7 ⊢ 1 ∈ ℕ
38		ffvelrn 6680	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ (0[,]+∞))
39	37, 38	mpan2 679	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝐹‘1) ∈ (0[,]+∞))
40	34, 36, 39	esumsn 31000	. . . . 5 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑥 ∈ {1} (𝐹‘𝑥) = (𝐹‘1))
41	32, 40	syl5eq 2828	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ {1} (𝐹‘𝑘) = (𝐹‘1))
42		fzsn 12771	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
43	35, 42	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
44		esumeq1 30969	. . . . 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 13203	. . . . 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 2841	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ (1...1)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘1))
49		simpl 475	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝑛 ∈ ℕ)
50		nnuz 12101	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
51	49, 50	syl6eleq 2878	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝑛 ∈ (ℤ_≥‘1))
52		seqp1 13205	. . . . . . . 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 473	. . . . . 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 477	. . . . . . 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 6997	. . . . . 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 1874	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
58	57	nfci 2921	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
59		nfcv 2934	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
60	29, 58, 59	nff 6345	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹:ℕ⟶(0[,]+∞)
61	57, 60	nfan 1863	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞))
62		fzsuc 12776	. . . . . . . . . 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 30970	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^𝑘 ∈ (1...(𝑛 + 1))(𝐹‘𝑘) = Σ^𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝐹‘𝑘))
65		nfcv 2934	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
66		nfcv 2934	. . . . . . . . 9 ⊢ Ⅎ𝑘{(𝑛 + 1)}
67		ovexd 7016	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (1...𝑛) ∈ V)
68		snex 5192	. . . . . . . . . 10 ⊢ {(𝑛 + 1)} ∈ V
69	68	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → {(𝑛 + 1)} ∈ V)
70		fzp1disj 12788	. . . . . . . . . 10 ⊢ ((1...𝑛) ∩ {(𝑛 + 1)}) = ∅
71	70	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → ((1...𝑛) ∩ {(𝑛 + 1)}) = ∅)
72		simplr 757	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
73		fzssnn 12773	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → (1...𝑛) ⊆ ℕ)
74	37, 73	ax-mp 5	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
75		simpr 477	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
76	74, 75	sseldi 3858	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
77	72, 76	ffvelrnd 6683	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
78		simplr 757	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝐹:ℕ⟶(0[,]+∞))
79		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 ∈ {(𝑛 + 1)})
80		velsn 4460	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {(𝑛 + 1)} ↔ 𝑘 = (𝑛 + 1))
81	79, 80	sylib 210	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 = (𝑛 + 1))
82		simpll 755	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑛 ∈ ℕ)
83	82	peano2nnd 11464	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝑛 + 1) ∈ ℕ)
84	81, 83	eqeltrd 2868	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → 𝑘 ∈ ℕ)
85	78, 84	ffvelrnd 6683	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑘 ∈ {(𝑛 + 1)}) → (𝐹‘𝑘) ∈ (0[,]+∞))
86	61, 65, 66, 67, 69, 71, 77, 85	esumsplit 30988	. . . . . . . 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 2934	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{(𝑛 + 1)}
88	25, 87, 66, 28, 31	cbvesum 30977	. . . . . . . . . 10 ⊢ Σ^𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘) = Σ^𝑥 ∈ {(𝑛 + 1)} (𝐹‘𝑥)
89		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑥 = (𝑛 + 1)) → 𝑥 = (𝑛 + 1))
90	89	fveq2d 6508	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) ∧ 𝑥 = (𝑛 + 1)) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
91	49	peano2nnd 11464	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (𝑛 + 1) ∈ ℕ)
92		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → 𝐹:ℕ⟶(0[,]+∞))
93	92, 91	ffvelrnd 6683	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → (𝐹‘(𝑛 + 1)) ∈ (0[,]+∞))
94	90, 91, 93	esumsn 31000	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑥 ∈ {(𝑛 + 1)} (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
95	88, 94	syl5eq 2828	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐹:ℕ⟶(0[,]+∞)) → Σ^*𝑘 ∈ {(𝑛 + 1)} (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
96	95	oveq2d 6998	. . . . . . . 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 2821	. . . . . . 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 473	. . . . . 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 2823	. . . . 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 412	. . . 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 11465	. 2 ⊢ (𝑁 ∈ ℕ → (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ (1...𝑁)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑁)))
103	102	impcom 399	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 387 = wceq 1508 ∈ wcel 2051 Ⅎwnfc 2918 Vcvv 3417 ∪ cun 3829 ∩ cin 3830 ⊆ wss 3831 ∅c0 4181 {csn 4444 ⟶wf 6189 ‘cfv 6193 (class class class)co 6982 0cc0 10341 1c1 10342 + caddc 10344 +∞cpnf 10477 ℕcn 11445 ℤcz 11799 ℤ_≥cuz 12064 +_𝑒 cxad 12328 [,]cicc 12563 ...cfz 12714 seqcseq 13190 Σ^*cesum 30962

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-inf2 8904 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 ax-pre-sup 10419 ax-addf 10420 ax-mulf 10421

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-iin 4800 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-se 5371 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-isom 6202 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-of 7233 df-om 7403 df-1st 7507 df-2nd 7508 df-supp 7640 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-2o 7912 df-oadd 7915 df-er 8095 df-map 8214 df-pm 8215 df-ixp 8266 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-fsupp 8635 df-fi 8676 df-sup 8707 df-inf 8708 df-oi 8775 df-card 9168 df-cda 9394 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-nn 11446 df-2 11509 df-3 11510 df-4 11511 df-5 11512 df-6 11513 df-7 11514 df-8 11515 df-9 11516 df-n0 11714 df-z 11800 df-dec 11918 df-uz 12065 df-q 12169 df-rp 12211 df-xneg 12330 df-xadd 12331 df-xmul 12332 df-ioo 12564 df-ioc 12565 df-ico 12566 df-icc 12567 df-fz 12715 df-fzo 12856 df-fl 12983 df-mod 13059 df-seq 13191 df-exp 13251 df-fac 13455 df-bc 13484 df-hash 13512 df-shft 14293 df-cj 14325 df-re 14326 df-im 14327 df-sqrt 14461 df-abs 14462 df-limsup 14695 df-clim 14712 df-rlim 14713 df-sum 14910 df-ef 15287 df-sin 15289 df-cos 15290 df-pi 15292 df-struct 16347 df-ndx 16348 df-slot 16349 df-base 16351 df-sets 16352 df-ress 16353 df-plusg 16440 df-mulr 16441 df-starv 16442 df-sca 16443 df-vsca 16444 df-ip 16445 df-tset 16446 df-ple 16447 df-ds 16449 df-unif 16450 df-hom 16451 df-cco 16452 df-rest 16558 df-topn 16559 df-0g 16577 df-gsum 16578 df-topgen 16579 df-pt 16580 df-prds 16583 df-ordt 16636 df-xrs 16637 df-qtop 16642 df-imas 16643 df-xps 16645 df-mre 16727 df-mrc 16728 df-acs 16730 df-ps 17680 df-tsr 17681 df-plusf 17721 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mulg 18024 df-subg 18072 df-cntz 18230 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-cring 19035 df-subrg 19268 df-abv 19322 df-lmod 19370 df-scaf 19371 df-sra 19678 df-rgmod 19679 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-fbas 20259 df-fg 20260 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-cls 21348 df-nei 21425 df-lp 21463 df-perf 21464 df-cn 21554 df-cnp 21555 df-haus 21642 df-tx 21889 df-hmeo 22082 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-tmd 22399 df-tgp 22400 df-tsms 22453 df-trg 22486 df-xms 22648 df-ms 22649 df-tms 22650 df-nm 22910 df-ngp 22911 df-nrg 22913 df-nlm 22914 df-ii 23203 df-cncf 23204 df-limc 24182 df-dv 24183 df-log 24856 df-esum 30963

This theorem is referenced by: esumfsup 31005 esumsup 31024

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