esumfsup - Mathbox for Thierry Arnoux

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Theorem esumfsup 34469

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 12630	. . . . . 6 ⊢ 1 ∈ ℤ
2		seqfn 14056	. . . . . 6 ⊢ (1 ∈ ℤ → seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1))
3	1, 2	ax-mp 5	. . . . 5 ⊢ seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1)
4		nnuz 12907	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
5	4	fneq2i 6633	. . . . 5 ⊢ (seq1( +_𝑒 , 𝐹) Fn ℕ ↔ seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1))
6	3, 5	mpbir 234	. . . 4 ⊢ seq1( +_𝑒 , 𝐹) Fn ℕ
7		iccssxr 13463	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
8		esumfsup.1	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
9	8	esumfzf 34468	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛))
10		ovex 7445	. . . . . . . 8 ⊢ (1...𝑛) ∈ V
11		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
12		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
13	8, 11, 12	nff 6701	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹:ℕ⟶(0[,]+∞)
14		nfv 1943	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
15	13, 14	nfan 1928	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ)
16		simpll 778	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
17		1nn 12250	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
18		fzssnn 13603	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℕ → (1...𝑛) ⊆ ℕ)
19	17, 18	mp1i 14	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1...𝑛) ⊆ ℕ)
20		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
21	19, 20	sseldd 3937	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
22	16, 21	ffvelcdmd 7080	. . . . . . . . . 10 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
23	22	ex 417	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) → (𝐹‘𝑘) ∈ (0[,]+∞)))
24	15, 23	ralrimi 3262	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
25		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
26	25	esumcl 34429	. . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
27	10, 24, 26	sylancr 598	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
28	9, 27	eqeltrrd 2863	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ (0[,]+∞))
29	7, 28	sselid 3934	. . . . 5 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^*)
30	29	ralrimiva 3156	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^*)
31		fnfvrnss 7116	. . . 4 ⊢ ((seq1( +_𝑒 , 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^) → ran seq1( +_𝑒 , 𝐹) ⊆ ℝ^)
32	6, 30, 31	sylancr 598	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ran seq1( +_𝑒 , 𝐹) ⊆ ℝ^*)
33		nnex 12245	. . . . 5 ⊢ ℕ ∈ V
34		ffvelcdm 7076	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
35	34	ex 417	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ (0[,]+∞)))
36	13, 35	ralrimi 3262	. . . . 5 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ (0[,]+∞))
37	11	esumcl 34429	. . . . 5 ⊢ ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) ∈ (0[,]+∞))
38	33, 36, 37	sylancr 598	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) ∈ (0[,]+∞))
39	7, 38	sselid 3934	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ ℕ(𝐹‘𝑘) ∈ ℝ^)
40		fvelrnb 6941	. . . . . . . . 9 ⊢ (seq1( +_𝑒 , 𝐹) Fn ℕ → (𝑥 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
41	6, 40	mp1i 14	. . . . . . . 8 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑥 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
42		eqcom 2769	. . . . . . . . . 10 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑥 ↔ 𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
43	9	eqeq1d 2764	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑥 ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
44	42, 43	bitr3id 288	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (𝑥 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
45	44	rexbidva 3186	. . . . . . . 8 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (∃𝑛 ∈ ℕ 𝑥 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
46	41, 45	bitr4d 285	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑥 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ 𝑥 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
47	46	biimpa 481	. . . . . 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 727	. . . . . . . . 9 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
50	17, 18	mp1i 14	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
51	15, 48, 49, 50	esummono 34453	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
52	51	ralrimiva 3156	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑛 ∈ ℕ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
53	52	adantr 485	. . . . . 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 520	. . . . 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 3128	. . . . 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 5111	. . . . . . 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 482	. . . . . 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 3161	. . . . 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 19	. . . 4 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ran seq1( +_𝑒 , 𝐹)) → 𝑥 ≤ Σ^*𝑘 ∈ ℕ(𝐹‘𝑘))
60	59	ralrimiva 3156	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑥 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 ≤ Σ^*𝑘 ∈ ℕ(𝐹‘𝑘))
61		nfv 1943	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ
62	13, 61	nfan 1928	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ)
63		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑥
64		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 <
65	11	nfesum1 34439	. . . . . . . . . . 11 ⊢ Ⅎ𝑘Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)
66	63, 64, 65	nfbr 5157	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)
67	62, 66	nfan 1928	. . . . . . . . 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 786	. . . . . . . . . 10 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(0[,]+∞))
70	69, 34	sylancom 599	. . . . . . . . 9 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
71		simplr 780	. . . . . . . . . 10 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ∈ ℝ)
72	71	rexrd 11265	. . . . . . . . 9 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ∈ ℝ^)
73		simpr 489	. . . . . . . . 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 34459	. . . . . . . 8 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑎 ∈ (𝒫 ℕ ∩ Fin)𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘))
75		ssnnssfz 33143	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑛 ∈ ℕ 𝑎 ⊆ (1...𝑛))
76		r19.42v 3196	. . . . . . . . . . 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 1943	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑎 ⊆ (1...𝑛)
78	67, 77	nfan 1928	. . . . . . . . . . . . 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 794	. . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . 14 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
84	80, 83	ffvelcdmd 7080	. . . . . . . . . . . . 13 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
85		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) → 𝑎 ⊆ (1...𝑛))
86	78, 79, 84, 85	esummono 34453	. . . . . . . . . . . 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 3102	. . . . . . . . . . 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 238	. . . . . . . . . 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 604	. . . . . . . . 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 3156	. . . . . . . 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 3128	. . . . . . . . 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 3196	. . . . . . . . . 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 3111	. . . . . . . . 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 237	. . . . . . . 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 595	. . . . . . 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 795	. . . . . . . . . . 11 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
97	96	rexrd 11265	. . . . . . . . . 10 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ^)
98		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑎 ∈ V
99		nfcv 2924	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝑎
100	99	nfel1 2940	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑎 ∈ (𝒫 ℕ ∩ Fin)
101	67, 100	nfan 1928	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin))
102	101, 14	nfan 1928	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ)
103		simp-5l 796	. . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . 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 4188	. . . . . . . . . . . . . . . . . 18 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
106	105	sseli 3932	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (𝒫 ℕ ∩ Fin) → 𝑎 ∈ 𝒫 ℕ)
107		elpwi 4568	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ 𝒫 ℕ → 𝑎 ⊆ ℕ)
108	104, 106, 107	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑎) → 𝑎 ⊆ ℕ)
109		simpr 489	. . . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . 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 7080	. . . . . . . . . . . . . 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 417	. . . . . . . . . . . . 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 3262	. . . . . . . . . . . 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 34429	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ V ∧ ∀𝑘 ∈ 𝑎 (𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝑎(𝐹‘𝑘) ∈ (0[,]+∞))
115	98, 113, 114	sylancr 598	. . . . . . . . . . 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 3934	. . . . . . . . . 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 796	. . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . 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 7080	. . . . . . . . . . . . . 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 417	. . . . . . . . . . . . 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 3262	. . . . . . . . . . . 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 598	. . . . . . . . . . 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 3934	. . . . . . . . . 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 13188	. . . . . . . . . 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 1397	. . . . . . . . 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 3177	. . . . . . . 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 3165	. . . . . . 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 16	. . . . . 6 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑛 ∈ ℕ 𝑥 < Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
130		fvelrnb 6941	. . . . . . . . . 10 ⊢ (seq1( +_𝑒 , 𝐹) Fn ℕ → (𝑦 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
131	6, 130	mp1i 14	. . . . . . . . 9 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑦 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
132		eqcom 2769	. . . . . . . . . . 11 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑦 ↔ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
133	9	eqeq1d 2764	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑦 ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
134	132, 133	bitr3id 288	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (𝑦 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
135	134	rexbidva 3186	. . . . . . . . 9 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (∃𝑛 ∈ ℕ 𝑦 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ↔ ∃𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
136	131, 135	bitr4d 285	. . . . . . . 8 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑦 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ 𝑦 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
137		simpr 489	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
138	137	breq2d 5120	. . . . . . . 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 5381	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦 ↔ ∃𝑛 ∈ ℕ 𝑥 < Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
140	139	ad2antrr 738	. . . . . 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 260	. . . . 5 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦)
142	141	ex 417	. . . 4 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) → (𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦))
143	142	ralrimiva 3156	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑥 ∈ ℝ (𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦))
144		supxr2 13346	. . 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 851	. 2 ⊢ (𝐹:ℕ⟶(0[,]+∞) → sup(ran seq1( +_𝑒 , 𝐹), ℝ^, < ) = Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
146	145	eqcomd 2768	1 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ ℕ(𝐹‘𝑘) = sup(ran seq1( +_𝑒 , 𝐹), ℝ^, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 Ⅎwnfc 2909 ∀wral 3078 ∃wrex 3088 Vcvv 3454 ∩ cin 3903 ⊆ wss 3904 𝒫 cpw 4561 class class class wbr 5108 ran crn 5661 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 Fincfn 8941 supcsup 9398 ℝcr 11105 0cc0 11106 1c1 11107 +∞cpnf 11246 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 ℕcn 12239 ℤcz 12597 ℤ_≥cuz 12868 +_𝑒 cxad 13141 [,]cicc 13381 ...cfz 13541 seqcseq 14044 Σ^*cesum 34426

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-ordt 17561 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-ps 18628 df-tsr 18629 df-plusf 18703 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-submnd 18848 df-grp 19009 df-minusg 19010 df-sbg 19011 df-mulg 19140 df-subg 19195 df-cntz 19393 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-subrng 20656 df-subrg 20680 df-abv 20923 df-lmod 20994 df-scaf 20995 df-sra 21305 df-rgmod 21306 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-tmd 24240 df-tgp 24241 df-tsms 24295 df-trg 24328 df-xms 24488 df-ms 24489 df-tms 24490 df-nm 24750 df-ngp 24751 df-nrg 24753 df-nlm 24754 df-ii 25047 df-cncf 25048 df-limc 26036 df-dv 26037 df-log 26732 df-esum 34427

This theorem is used by: esumfsupre 34470 esumsup 34488

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