esumfsup - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > esumfsup		Structured version Visualization version GIF version

Theorem esumfsup 34369

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 12603	. . . . . 6 ⊢ 1 ∈ ℤ
2		seqfn 14028	. . . . . 6 ⊢ (1 ∈ ℤ → seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1))
3	1, 2	ax-mp 5	. . . . 5 ⊢ seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1)
4		nnuz 12880	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
5	4	fneq2i 6621	. . . . 5 ⊢ (seq1( +_𝑒 , 𝐹) Fn ℕ ↔ seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1))
6	3, 5	mpbir 233	. . . 4 ⊢ seq1( +_𝑒 , 𝐹) Fn ℕ
7		iccssxr 13436	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
8		esumfsup.1	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
9	8	esumfzf 34368	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛))
10		ovex 7431	. . . . . . . 8 ⊢ (1...𝑛) ∈ V
11		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
12		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
13	8, 11, 12	nff 6689	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹:ℕ⟶(0[,]+∞)
14		nfv 1936	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
15	13, 14	nfan 1921	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ)
16		simpll 776	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
17		1nn 12223	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
18		fzssnn 13575	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
21	19, 20	sseldd 3939	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
22	16, 21	ffvelcdmd 7068	. . . . . . . . . 10 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
23	22	ex 416	. . . . . . . . 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 2926	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
26	25	esumcl 34329	. . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
27	10, 24, 26	sylancr 596	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
28	9, 27	eqeltrrd 2865	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ (0[,]+∞))
29	7, 28	sselid 3936	. . . . 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 7104	. . . 4 ⊢ ((seq1( +_𝑒 , 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^) → ran seq1( +_𝑒 , 𝐹) ⊆ ℝ^)
32	6, 30, 31	sylancr 596	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ran seq1( +_𝑒 , 𝐹) ⊆ ℝ^*)
33		nnex 12218	. . . . 5 ⊢ ℕ ∈ V
34		ffvelcdm 7064	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
35	34	ex 416	. . . . . 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 34329	. . . . 5 ⊢ ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) ∈ (0[,]+∞))
38	33, 36, 37	sylancr 596	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) ∈ (0[,]+∞))
39	7, 38	sselid 3936	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^𝑘 ∈ ℕ(𝐹‘𝑘) ∈ ℝ^)
40		fvelrnb 6929	. . . . . . . . 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 2771	. . . . . . . . . 10 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑥 ↔ 𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
43	9	eqeq1d 2766	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑥 ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑥))
44	42, 43	bitr3id 287	. . . . . . . . 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 284	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑥 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ 𝑥 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
47	46	biimpa 480	. . . . . 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 725	. . . . . . . . 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 34353	. . . . . . . 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 484	. . . . . 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 519	. . . . 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 5105	. . . . . . 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 481	. . . . . 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 18	. . . 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 1936	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ
62	13, 61	nfan 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ)
63		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑥
64		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 <
65	11	nfesum1 34339	. . . . . . . . . . 11 ⊢ Ⅎ𝑘Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)
66	63, 64, 65	nfbr 5149	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)
67	62, 66	nfan 1921	. . . . . . . . 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 784	. . . . . . . . . 10 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(0[,]+∞))
70	69, 34	sylancom 597	. . . . . . . . 9 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
71		simplr 778	. . . . . . . . . 10 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ∈ ℝ)
72	71	rexrd 11234	. . . . . . . . 9 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ∈ ℝ^)
73		simpr 488	. . . . . . . . 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 34359	. . . . . . . 8 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑎 ∈ (𝒫 ℕ ∩ Fin)𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘))
75		ssnnssfz 32991	. . . . . . . . . 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 1936	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑎 ⊆ (1...𝑛)
78	67, 77	nfan 1921	. . . . . . . . . . . . 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 792	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . 14 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
84	80, 83	ffvelcdmd 7068	. . . . . . . . . . . . 13 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
85		simpr 488	. . . . . . . . . . . . 13 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) → 𝑎 ⊆ (1...𝑛))
86	78, 79, 84, 85	esummono 34353	. . . . . . . . . . . 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 237	. . . . . . . . . 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 602	. . . . . . . . 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 236	. . . . . . . 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 593	. . . . . . 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 793	. . . . . . . . . . 11 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
97	96	rexrd 11234	. . . . . . . . . 10 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ^)
98		vex 3460	. . . . . . . . . . . 12 ⊢ 𝑎 ∈ V
99		nfcv 2926	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝑎
100	99	nfel1 2942	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑎 ∈ (𝒫 ℕ ∩ Fin)
101	67, 100	nfan 1921	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin))
102	101, 14	nfan 1921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ)
103		simp-5l 794	. . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . 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 4190	. . . . . . . . . . . . . . . . . 18 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
106	105	sseli 3934	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (𝒫 ℕ ∩ Fin) → 𝑎 ∈ 𝒫 ℕ)
107		elpwi 4564	. . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . 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 7068	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 34329	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ V ∧ ∀𝑘 ∈ 𝑎 (𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝑎(𝐹‘𝑘) ∈ (0[,]+∞))
115	98, 113, 114	sylancr 596	. . . . . . . . . . 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 3936	. . . . . . . . . 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 794	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . 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 7068	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 596	. . . . . . . . . . 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 3936	. . . . . . . . . 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 13161	. . . . . . . . . 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 1392	. . . . . . . . 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 15	. . . . . 6 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑛 ∈ ℕ 𝑥 < Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
130		fvelrnb 6929	. . . . . . . . . 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 2771	. . . . . . . . . . 11 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑦 ↔ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
133	9	eqeq1d 2766	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑦 ↔ (seq1( +_𝑒 , 𝐹)‘𝑛) = 𝑦))
134	132, 133	bitr3id 287	. . . . . . . . . 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 284	. . . . . . . 8 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑦 ∈ ran seq1( +_𝑒 , 𝐹) ↔ ∃𝑛 ∈ ℕ 𝑦 = Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
137		simpr 488	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
138	137	breq2d 5114	. . . . . . . 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 5370	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦 ↔ ∃𝑛 ∈ ℕ 𝑥 < Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
140	139	ad2antrr 736	. . . . . 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 259	. . . . 5 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦)
142	141	ex 416	. . . 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 13319	. . 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 849	. 2 ⊢ (𝐹:ℕ⟶(0[,]+∞) → sup(ran seq1( +_𝑒 , 𝐹), ℝ^, < ) = Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
146	145	eqcomd 2770	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 208 ∧ wa 399 = wceq 1562 ∈ wcel 2144 Ⅎwnfc 2911 ∀wral 3078 ∃wrex 3088 Vcvv 3456 ∩ cin 3905 ⊆ wss 3906 𝒫 cpw 4557 class class class wbr 5102 ran crn 5650 Fn wfn 6518 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 Fincfn 8929 supcsup 9388 ℝcr 11074 0cc0 11075 1c1 11076 +∞cpnf 11215 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 ℕcn 12212 ℤcz 12570 ℤ_≥cuz 12841 +_𝑒 cxad 13114 [,]cicc 13354 ...cfz 13514 seqcseq 14016 Σ^*cesum 34326

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-fl 13804 df-mod 13882 df-seq 14017 df-exp 14077 df-fac 14289 df-bc 14318 df-hash 14346 df-shft 15082 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-limsup 15500 df-clim 15517 df-rlim 15518 df-sum 15716 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-ordt 17533 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-ps 18600 df-tsr 18601 df-plusf 18675 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-mhm 18819 df-submnd 18820 df-grp 18980 df-minusg 18981 df-sbg 18982 df-mulg 19112 df-subg 19167 df-cntz 19359 df-cmn 19824 df-abl 19825 df-mgp 20189 df-rng 20201 df-ur 20234 df-ring 20287 df-cring 20288 df-subrng 20598 df-subrg 20622 df-abv 20860 df-lmod 20931 df-scaf 20932 df-sra 21242 df-rgmod 21243 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-lp 23198 df-perf 23199 df-cn 23289 df-cnp 23290 df-haus 23377 df-tx 23624 df-hmeo 23817 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-tmd 24134 df-tgp 24135 df-tsms 24189 df-trg 24222 df-xms 24382 df-ms 24383 df-tms 24384 df-nm 24644 df-ngp 24645 df-nrg 24647 df-nlm 24648 df-ii 24941 df-cncf 24942 df-limc 25930 df-dv 25931 df-log 26623 df-esum 34327

This theorem is referenced by: esumfsupre 34370 esumsup 34388

W3C validator