esumfsup - Mathbox for Thierry Arnoux

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Theorem esumfsup 34071

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

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

**Assertion**
Ref	Expression
esumfsup	⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ ℕ(𝐹‘𝑘) = sup(ran seq1( +_𝑒 , 𝐹), ℝ^, < ))

Proof of Theorem esumfsup Dummy variables 𝑎 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1z 12647	. . . . . 6 ⊢ 1 ∈ ℤ
2		seqfn 14054	. . . . . 6 ⊢ (1 ∈ ℤ → seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1))
3	1, 2	ax-mp 5	. . . . 5 ⊢ seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1)
4		nnuz 12921	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
5	4	fneq2i 6666	. . . . 5 ⊢ (seq1( +_𝑒 , 𝐹) Fn ℕ ↔ seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1))
6	3, 5	mpbir 231	. . . 4 ⊢ seq1( +_𝑒 , 𝐹) Fn ℕ
7		iccssxr 13470	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
8		esumfsup.1	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
9	8	esumfzf 34070	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛))
10		ovex 7464	. . . . . . . 8 ⊢ (1...𝑛) ∈ V
11		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
12		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
13	8, 11, 12	nff 6732	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹:ℕ⟶(0[,]+∞)
14		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
15	13, 14	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ)
16		simpll 767	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
17		1nn 12277	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
18		fzssnn 13608	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℕ → (1...𝑛) ⊆ ℕ)
19	17, 18	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1...𝑛) ⊆ ℕ)
20		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
21	19, 20	sseldd 3984	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
22	16, 21	ffvelcdmd 7105	. . . . . . . . . 10 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
23	22	ex 412	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) → (𝐹‘𝑘) ∈ (0[,]+∞)))
24	15, 23	ralrimi 3257	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
25		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
26	25	esumcl 34031	. . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
27	10, 24, 26	sylancr 587	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
28	9, 27	eqeltrrd 2842	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ (0[,]+∞))
29	7, 28	sselid 3981	. . . . 5 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^*)
30	29	ralrimiva 3146	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^*)
31		fnfvrnss 7141	. . . 4 ⊢ ((seq1( +_𝑒 , 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^) → ran seq1( +_𝑒 , 𝐹) ⊆ ℝ^)
32	6, 30, 31	sylancr 587	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ran seq1( +_𝑒 , 𝐹) ⊆ ℝ^*)
33		nnex 12272	. . . . 5 ⊢ ℕ ∈ V
34		ffvelcdm 7101	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
35	34	ex 412	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ (0[,]+∞)))
36	13, 35	ralrimi 3257	. . . . 5 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ (0[,]+∞))
37	11	esumcl 34031	. . . . 5 ⊢ ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) ∈ (0[,]+∞))
38	33, 36, 37	sylancr 587	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) ∈ (0[,]+∞))
39	7, 38	sselid 3981	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ ℕ(𝐹‘𝑘) ∈ ℝ^)
40		fvelrnb 6969	. . . . . . . . 9 ⊢ (seq1( +_𝑒 , 𝐹) Fn ℕ → (𝑥 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
41	6, 40	mp1i 13	. . . . . . . 8 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑥 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
42		eqcom 2744	. . . . . . . . . 10 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑥 ↔ 𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
43	9	eqeq1d 2739	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑥 ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
44	42, 43	bitr3id 285	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (𝑥 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
45	44	rexbidva 3177	. . . . . . . 8 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (∃𝑛 ∈ ℕ 𝑥 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
46	41, 45	bitr4d 282	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑥 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ 𝑥 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
47	46	biimpa 476	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ran seq1( +_𝑒 , 𝐹)) → ∃𝑛 ∈ ℕ 𝑥 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
48	33	a1i 11	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → ℕ ∈ V)
49	34	adantlr 715	. . . . . . . . 9 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
50	17, 18	mp1i 13	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
51	15, 48, 49, 50	esummono 34055	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
52	51	ralrimiva 3146	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑛 ∈ ℕ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
53	52	adantr 480	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ran seq1( +_𝑒 , 𝐹)) → ∀𝑛 ∈ ℕ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
54	47, 53	jca 511	. . . . 5 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ran seq1( +_𝑒 , 𝐹)) → (∃𝑛 ∈ ℕ 𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∧ ∀𝑛 ∈ ℕ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)))
55		r19.29r 3116	. . . . 5 ⊢ ((∃𝑛 ∈ ℕ 𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∧ ∀𝑛 ∈ ℕ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑛 ∈ ℕ (𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∧ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘)))
56		breq1 5146	. . . . . . 7 ⊢ (𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) → (𝑥 ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘) ↔ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘)))
57	56	biimpar 477	. . . . . 6 ⊢ ((𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∧ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
58	57	rexlimivw 3151	. . . . 5 ⊢ (∃𝑛 ∈ ℕ (𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∧ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
59	54, 55, 58	3syl 18	. . . 4 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ran seq1( +_𝑒 , 𝐹)) → 𝑥 ≤ Σ^*𝑘 ∈ ℕ(𝐹‘𝑘))
60	59	ralrimiva 3146	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑥 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 ≤ Σ^*𝑘 ∈ ℕ(𝐹‘𝑘))
61		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ
62	13, 61	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ)
63		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑥
64		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 <
65	11	nfesum1 34041	. . . . . . . . . . 11 ⊢ Ⅎ𝑘Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)
66	63, 64, 65	nfbr 5190	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)
67	62, 66	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘))
68	33	a1i 11	. . . . . . . . 9 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) → ℕ ∈ V)
69		simplll 775	. . . . . . . . . 10 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(0[,]+∞))
70	69, 34	sylancom 588	. . . . . . . . 9 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
71		simplr 769	. . . . . . . . . 10 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ∈ ℝ)
72	71	rexrd 11311	. . . . . . . . 9 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ∈ ℝ^)
73		simpr 484	. . . . . . . . 9 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
74	67, 68, 70, 72, 73	esumlub 34061	. . . . . . . 8 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑎 ∈ (𝒫 ℕ ∩ Fin)𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘))
75		ssnnssfz 32789	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑛 ∈ ℕ 𝑎 ⊆ (1...𝑛))
76		r19.42v 3191	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ↔ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ ∃𝑛 ∈ ℕ 𝑎 ⊆ (1...𝑛)))
77		nfv 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑎 ⊆ (1...𝑛)
78	67, 77	nfan 1899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛))
79	10	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) → (1...𝑛) ∈ V)
80		simp-4l 783	. . . . . . . . . . . . . 14 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
81	17, 18	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ⊆ ℕ
82		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
83	81, 82	sselid 3981	. . . . . . . . . . . . . 14 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
84	80, 83	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
85		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) → 𝑎 ⊆ (1...𝑛))
86	78, 79, 84, 85	esummono 34055	. . . . . . . . . . . 12 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) → Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
87	86	reximi 3084	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) → ∃𝑛 ∈ ℕ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
88	76, 87	sylbir 235	. . . . . . . . . 10 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ ∃𝑛 ∈ ℕ 𝑎 ⊆ (1...𝑛)) → ∃𝑛 ∈ ℕ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
89	75, 88	sylan2 593	. . . . . . . . 9 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑛 ∈ ℕ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
90	89	ralrimiva 3146	. . . . . . . 8 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∀𝑎 ∈ (𝒫 ℕ ∩ Fin)∃𝑛 ∈ ℕ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
91		r19.29r 3116	. . . . . . . . 9 ⊢ ((∃𝑎 ∈ (𝒫 ℕ ∩ Fin)𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ ∀𝑎 ∈ (𝒫 ℕ ∩ Fin)∃𝑛 ∈ ℕ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → ∃𝑎 ∈ (𝒫 ℕ ∩ Fin)(𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ ∃𝑛 ∈ ℕ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
92		r19.42v 3191	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ℕ (𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) ↔ (𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ ∃𝑛 ∈ ℕ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
93	92	rexbii 3094	. . . . . . . . 9 ⊢ (∃𝑎 ∈ (𝒫 ℕ ∩ Fin)∃𝑛 ∈ ℕ (𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) ↔ ∃𝑎 ∈ (𝒫 ℕ ∩ Fin)(𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ ∃𝑛 ∈ ℕ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
94	91, 93	sylibr 234	. . . . . . . 8 ⊢ ((∃𝑎 ∈ (𝒫 ℕ ∩ Fin)𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ ∀𝑎 ∈ (𝒫 ℕ ∩ Fin)∃𝑛 ∈ ℕ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → ∃𝑎 ∈ (𝒫 ℕ ∩ Fin)∃𝑛 ∈ ℕ (𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
95	74, 90, 94	syl2anc 584	. . . . . . 7 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑎 ∈ (𝒫 ℕ ∩ Fin)∃𝑛 ∈ ℕ (𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
96		simp-4r 784	. . . . . . . . . . 11 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
97	96	rexrd 11311	. . . . . . . . . 10 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ^)
98		vex 3484	. . . . . . . . . . . 12 ⊢ 𝑎 ∈ V
99		nfcv 2905	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝑎
100	99	nfel1 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑎 ∈ (𝒫 ℕ ∩ Fin)
101	67, 100	nfan 1899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin))
102	101, 14	nfan 1899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ)
103		simp-5l 785	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑎) → 𝐹:ℕ⟶(0[,]+∞))
104		simpllr 776	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑎) → 𝑎 ∈ (𝒫 ℕ ∩ Fin))
105		inss1 4237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
106	105	sseli 3979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (𝒫 ℕ ∩ Fin) → 𝑎 ∈ 𝒫 ℕ)
107		elpwi 4607	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ 𝒫 ℕ → 𝑎 ⊆ ℕ)
108	104, 106, 107	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑎) → 𝑎 ⊆ ℕ)
109		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑎) → 𝑘 ∈ 𝑎)
110	108, 109	sseldd 3984	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑎) → 𝑘 ∈ ℕ)
111	103, 110	ffvelcdmd 7105	. . . . . . . . . . . . . 14 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑎) → (𝐹‘𝑘) ∈ (0[,]+∞))
112	111	ex 412	. . . . . . . . . . . . 13 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑎 → (𝐹‘𝑘) ∈ (0[,]+∞)))
113	102, 112	ralrimi 3257	. . . . . . . . . . . 12 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ 𝑎 (𝐹‘𝑘) ∈ (0[,]+∞))
114	99	esumcl 34031	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ V ∧ ∀𝑘 ∈ 𝑎 (𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝑎(𝐹‘𝑘) ∈ (0[,]+∞))
115	98, 113, 114	sylancr 587	. . . . . . . . . . 11 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∈ (0[,]+∞))
116	7, 115	sselid 3981	. . . . . . . . . 10 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∈ ℝ^*)
117		simp-5l 785	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
118		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
119	81, 118	sselid 3981	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
120	117, 119	ffvelcdmd 7105	. . . . . . . . . . . . . 14 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
121	120	ex 412	. . . . . . . . . . . . 13 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) → (𝐹‘𝑘) ∈ (0[,]+∞)))
122	102, 121	ralrimi 3257	. . . . . . . . . . . 12 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
123	10, 122, 26	sylancr 587	. . . . . . . . . . 11 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
124	7, 123	sselid 3981	. . . . . . . . . 10 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ ℝ^*)
125		xrltletr 13199	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∈ ℝ^ ∧ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ ℝ^) → ((𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → 𝑥 < Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
126	97, 116, 124, 125	syl3anc 1373	. . . . . . . . 9 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → ((𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → 𝑥 < Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
127	126	reximdva 3168	. . . . . . . 8 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) → (∃𝑛 ∈ ℕ (𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → ∃𝑛 ∈ ℕ 𝑥 < Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
128	127	rexlimdva 3155	. . . . . . 7 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → (∃𝑎 ∈ (𝒫 ℕ ∩ Fin)∃𝑛 ∈ ℕ (𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ∧ Σ^𝑘 ∈ 𝑎(𝐹‘𝑘) ≤ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → ∃𝑛 ∈ ℕ 𝑥 < Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
129	95, 128	mpd 15	. . . . . 6 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑛 ∈ ℕ 𝑥 < Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
130		fvelrnb 6969	. . . . . . . . . 10 ⊢ (seq1( +_𝑒 , 𝐹) Fn ℕ → (𝑦 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
131	6, 130	mp1i 13	. . . . . . . . 9 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑦 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
132		eqcom 2744	. . . . . . . . . . 11 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑦 ↔ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
133	9	eqeq1d 2739	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑦 ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
134	132, 133	bitr3id 285	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (𝑦 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
135	134	rexbidva 3177	. . . . . . . . 9 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (∃𝑛 ∈ ℕ 𝑦 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
136	131, 135	bitr4d 282	. . . . . . . 8 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑦 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ 𝑦 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
137		simpr 484	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
138	137	breq2d 5155	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → (𝑥 < 𝑦 ↔ 𝑥 < Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
139	27, 136, 138	rexxfr2d 5411	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦 ↔ ∃𝑛 ∈ ℕ 𝑥 < Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
140	139	ad2antrr 726	. . . . . 6 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → (∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦 ↔ ∃𝑛 ∈ ℕ 𝑥 < Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
141	129, 140	mpbird 257	. . . . 5 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦)
142	141	ex 412	. . . 4 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) → (𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦))
143	142	ralrimiva 3146	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑥 ∈ ℝ (𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦))
144		supxr2 13356	. . 3 ⊢ (((ran seq1( +_𝑒 , 𝐹) ⊆ ℝ^* ∧ Σ^𝑘 ∈ ℕ(𝐹‘𝑘) ∈ ℝ^) ∧ (∀𝑥 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘) ∧ ∀𝑥 ∈ ℝ (𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦))) → sup(ran seq1( +_𝑒 , 𝐹), ℝ^, < ) = Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
145	32, 39, 60, 143, 144	syl22anc 839	. 2 ⊢ (𝐹:ℕ⟶(0[,]+∞) → sup(ran seq1( +_𝑒 , 𝐹), ℝ^, < ) = Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
146	145	eqcomd 2743	1 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ ℕ(𝐹‘𝑘) = sup(ran seq1( +_𝑒 , 𝐹), ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 supcsup 9480 ℝcr 11154 0cc0 11155 1c1 11156 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 ℕcn 12266 ℤcz 12613 ℤ_≥cuz 12878 +_𝑒 cxad 13152 [,]cicc 13390 ...cfz 13547 seqcseq 14042 Σ^*cesum 34028

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-ordt 17546 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-ps 18611 df-tsr 18612 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-abv 20810 df-lmod 20860 df-scaf 20861 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tmd 24080 df-tgp 24081 df-tsms 24135 df-trg 24168 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 df-ii 24903 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-esum 34029

This theorem is referenced by: esumfsupre 34072 esumsup 34090

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