esumfsup - Mathbox for Thierry Arnoux

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Theorem esumfsup 34060

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 12563	. . . . . 6 ⊢ 1 ∈ ℤ
2		seqfn 13978	. . . . . 6 ⊢ (1 ∈ ℤ → seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1))
3	1, 2	ax-mp 5	. . . . 5 ⊢ seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1)
4		nnuz 12836	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
5	4	fneq2i 6616	. . . . 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 13391	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
8		esumfsup.1	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
9	8	esumfzf 34059	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛))
10		ovex 7420	. . . . . . . 8 ⊢ (1...𝑛) ∈ V
11		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
12		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
13	8, 11, 12	nff 6684	. . . . . . . . . 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 766	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
17		1nn 12197	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
18		fzssnn 13529	. . . . . . . . . . . . 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 3947	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
22	16, 21	ffvelcdmd 7057	. . . . . . . . . 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 3235	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
25		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
26	25	esumcl 34020	. . . . . . . 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 2829	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ (0[,]+∞))
29	7, 28	sselid 3944	. . . . 5 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^*)
30	29	ralrimiva 3125	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^*)
31		fnfvrnss 7093	. . . 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 12192	. . . . 5 ⊢ ℕ ∈ V
34		ffvelcdm 7053	. . . . . . 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 3235	. . . . 5 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ (0[,]+∞))
37	11	esumcl 34020	. . . . 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 3944	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ ℕ(𝐹‘𝑘) ∈ ℝ^)
40		fvelrnb 6921	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑥 ↔ 𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
43	9	eqeq1d 2731	. . . . . . . . . 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 3155	. . . . . . . 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 34044	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
52	51	ralrimiva 3125	. . . . . . 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 3096	. . . . 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 5110	. . . . . . 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 3130	. . . . 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 3125	. . 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 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑥
64		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 <
65	11	nfesum1 34030	. . . . . . . . . . 11 ⊢ Ⅎ𝑘Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)
66	63, 64, 65	nfbr 5154	. . . . . . . . . 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 774	. . . . . . . . . 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 768	. . . . . . . . . 10 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ∈ ℝ)
72	71	rexrd 11224	. . . . . . . . 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 34050	. . . . . . . 8 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑎 ∈ (𝒫 ℕ ∩ Fin)𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘))
75		ssnnssfz 32710	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑛 ∈ ℕ 𝑎 ⊆ (1...𝑛))
76		r19.42v 3169	. . . . . . . . . . 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 782	. . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . 14 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
84	80, 83	ffvelcdmd 7057	. . . . . . . . . . . . 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 34044	. . . . . . . . . . . 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 3067	. . . . . . . . . . 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 3125	. . . . . . . 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 3096	. . . . . . . . 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 3169	. . . . . . . . . 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 3076	. . . . . . . . 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 783	. . . . . . . . . . 11 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
97	96	rexrd 11224	. . . . . . . . . 10 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ^)
98		vex 3451	. . . . . . . . . . . 12 ⊢ 𝑎 ∈ V
99		nfcv 2891	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝑎
100	99	nfel1 2908	. . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . 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 4200	. . . . . . . . . . . . . . . . . 18 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
106	105	sseli 3942	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (𝒫 ℕ ∩ Fin) → 𝑎 ∈ 𝒫 ℕ)
107		elpwi 4570	. . . . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . 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 7057	. . . . . . . . . . . . . 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 3235	. . . . . . . . . . . 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 34020	. . . . . . . . . . . 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 3944	. . . . . . . . . 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 784	. . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . 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 7057	. . . . . . . . . . . . . 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 3235	. . . . . . . . . . . 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 3944	. . . . . . . . . 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 13117	. . . . . . . . . 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 3146	. . . . . . . 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 3134	. . . . . . 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 6921	. . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑦 ↔ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
133	9	eqeq1d 2731	. . . . . . . . . . 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 3155	. . . . . . . . 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 5119	. . . . . . . 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 5366	. . . . . . 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 3125	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑥 ∈ ℝ (𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦))
144		supxr2 13274	. . 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 838	. 2 ⊢ (𝐹:ℕ⟶(0[,]+∞) → sup(ran seq1( +_𝑒 , 𝐹), ℝ^, < ) = Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
146	145	eqcomd 2735	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 2109 Ⅎwnfc 2876 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 𝒫 cpw 4563 class class class wbr 5107 ran crn 5639 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 supcsup 9391 ℝcr 11067 0cc0 11068 1c1 11069 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 ℕcn 12186 ℤcz 12529 ℤ_≥cuz 12793 +_𝑒 cxad 13070 [,]cicc 13309 ...cfz 13468 seqcseq 13966 Σ^*cesum 34017

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-ordt 17464 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-ps 18525 df-tsr 18526 df-plusf 18566 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-subrng 20455 df-subrg 20479 df-abv 20718 df-lmod 20768 df-scaf 20769 df-sra 21080 df-rgmod 21081 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-tmd 23959 df-tgp 23960 df-tsms 24014 df-trg 24047 df-xms 24208 df-ms 24209 df-tms 24210 df-nm 24470 df-ngp 24471 df-nrg 24473 df-nlm 24474 df-ii 24770 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-esum 34018

This theorem is referenced by: esumfsupre 34061 esumsup 34079

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