esumfsup - Mathbox for Thierry Arnoux

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Theorem esumfsup 34613

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 12673	. . . . . 6 ⊢ 1 ∈ ℤ
2		seqfn 14102	. . . . . 6 ⊢ (1 ∈ ℤ → seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1))
3	1, 2	ax-mp 5	. . . . 5 ⊢ seq1( +_𝑒 , 𝐹) Fn (ℤ_≥‘1)
4		nnuz 12951	. . . . . 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 13508	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
8		esumfsup.1	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
9	8	esumfzf 34612	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = (seq1( +_𝑒 , 𝐹)‘𝑛))
10		ovex 7449	. . . . . . . 8 ⊢ (1...𝑛) ∈ V
11		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑘ℕ
12		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
13	8, 11, 12	nff 6701	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹:ℕ⟶(0[,]+∞)
14		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
15	13, 14	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ)
16		simpll 779	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐹:ℕ⟶(0[,]+∞))
17		1nn 12293	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
18		fzssnn 13648	. . . . . . . . . . . . 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 490	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...𝑛))
21	19, 20	sseldd 3932	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
22	16, 21	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
23	22	ex 418	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) → (𝐹‘𝑘) ∈ (0[,]+∞)))
24	15, 23	ralrimi 3260	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
25		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...𝑛)
26	25	esumcl 34573	. . . . . . . 8 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
27	10, 24, 26	sylancr 599	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ∈ (0[,]+∞))
28	9, 27	eqeltrrd 2861	. . . . . 6 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ (0[,]+∞))
29	7, 28	sselid 3929	. . . . 5 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^*)
30	29	ralrimiva 3154	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^*)
31		fnfvrnss 7117	. . . 4 ⊢ ((seq1( +_𝑒 , 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (seq1( +_𝑒 , 𝐹)‘𝑛) ∈ ℝ^) → ran seq1( +_𝑒 , 𝐹) ⊆ ℝ^)
32	6, 30, 31	sylancr 599	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ran seq1( +_𝑒 , 𝐹) ⊆ ℝ^*)
33		nnex 12288	. . . . 5 ⊢ ℕ ∈ V
34		ffvelcdm 7077	. . . . . . 7 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
35	34	ex 418	. . . . . 6 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (𝑘 ∈ ℕ → (𝐹‘𝑘) ∈ (0[,]+∞)))
36	13, 35	ralrimi 3260	. . . . 5 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ (0[,]+∞))
37	11	esumcl 34573	. . . . 5 ⊢ ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) ∈ (0[,]+∞))
38	33, 36, 37	sylancr 599	. . . 4 ⊢ (𝐹:ℕ⟶(0[,]+∞) → Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) ∈ (0[,]+∞))
39	7, 38	sselid 3929	. . 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 2767	. . . . . . . . . 10 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑥 ↔ 𝑥 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
43	9	eqeq1d 2762	. . . . . . . . . 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 3184	. . . . . . . 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 482	. . . . . 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 728	. . . . . . . . 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 34597	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
52	51	ralrimiva 3154	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑛 ∈ ℕ Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) ≤ Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
53	52	adantr 486	. . . . . 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 521	. . . . 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 3126	. . . . 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 5106	. . . . . . 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 483	. . . . . 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 3159	. . . . 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 3154	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑥 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 ≤ Σ^*𝑘 ∈ ℕ(𝐹‘𝑘))
61		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ
62	13, 61	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ)
63		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑥
64		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 <
65	11	nfesum1 34583	. . . . . . . . . . 11 ⊢ Ⅎ𝑘Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)
66	63, 64, 65	nfbr 5152	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)
67	62, 66	nfan 1932	. . . . . . . . 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 787	. . . . . . . . . 10 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(0[,]+∞))
70	69, 34	sylancom 600	. . . . . . . . 9 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (0[,]+∞))
71		simplr 781	. . . . . . . . . 10 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ∈ ℝ)
72	71	rexrd 11308	. . . . . . . . 9 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → 𝑥 ∈ ℝ^)
73		simpr 490	. . . . . . . . 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 34603	. . . . . . . 8 ⊢ (((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) → ∃𝑎 ∈ (𝒫 ℕ ∩ Fin)𝑥 < Σ^𝑘 ∈ 𝑎(𝐹‘𝑘))
75		ssnnssfz 33290	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑛 ∈ ℕ 𝑎 ⊆ (1...𝑛))
76		r19.42v 3194	. . . . . . . . . . 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 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑎 ⊆ (1...𝑛)
78	67, 77	nfan 1932	. . . . . . . . . . . . 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 795	. . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . 14 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
84	80, 83	ffvelcdmd 7081	. . . . . . . . . . . . 13 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) ∧ 𝑘 ∈ (1...𝑛)) → (𝐹‘𝑘) ∈ (0[,]+∞))
85		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ⊆ (1...𝑛)) → 𝑎 ⊆ (1...𝑛))
86	78, 79, 84, 85	esummono 34597	. . . . . . . . . . . 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 3100	. . . . . . . . . . 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 605	. . . . . . . . 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 3154	. . . . . . . 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 3126	. . . . . . . . 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 3194	. . . . . . . . . 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 3109	. . . . . . . . 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 596	. . . . . . 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 796	. . . . . . . . . . 11 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
97	96	rexrd 11308	. . . . . . . . . 10 ⊢ (((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ^)
98		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑎 ∈ V
99		nfcv 2922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝑎
100	99	nfel1 2938	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑎 ∈ (𝒫 ℕ ∩ Fin)
101	67, 100	nfan 1932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin))
102	101, 14	nfan 1932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘)) ∧ 𝑎 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑛 ∈ ℕ)
103		simp-5l 797	. . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . 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 4182	. . . . . . . . . . . . . . . . . 18 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
106	105	sseli 3927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (𝒫 ℕ ∩ Fin) → 𝑎 ∈ 𝒫 ℕ)
107		elpwi 4564	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . 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 7081	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 3260	. . . . . . . . . . . 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 34573	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ V ∧ ∀𝑘 ∈ 𝑎 (𝐹‘𝑘) ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝑎(𝐹‘𝑘) ∈ (0[,]+∞))
115	98, 113, 114	sylancr 599	. . . . . . . . . . 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 3929	. . . . . . . . . 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 797	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . 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 7081	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 3260	. . . . . . . . . . . 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 599	. . . . . . . . . . 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 3929	. . . . . . . . . 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 13233	. . . . . . . . . 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 1398	. . . . . . . . 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 3175	. . . . . . . 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 3163	. . . . . . 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 2767	. . . . . . . . . . 11 ⊢ (Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘) = 𝑦 ↔ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
133	9	eqeq1d 2762	. . . . . . . . . . 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 3184	. . . . . . . . 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 490	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘)) → 𝑦 = Σ^𝑘 ∈ (1...𝑛)(𝐹‘𝑘))
138	137	breq2d 5115	. . . . . . . 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 5376	. . . . . . 7 ⊢ (𝐹:ℕ⟶(0[,]+∞) → (∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦 ↔ ∃𝑛 ∈ ℕ 𝑥 < Σ^*𝑘 ∈ (1...𝑛)(𝐹‘𝑘)))
140	139	ad2antrr 739	. . . . . 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 418	. . . 4 ⊢ ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑥 ∈ ℝ) → (𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦))
143	142	ralrimiva 3154	. . 3 ⊢ (𝐹:ℕ⟶(0[,]+∞) → ∀𝑥 ∈ ℝ (𝑥 < Σ^*𝑘 ∈ ℕ(𝐹‘𝑘) → ∃𝑦 ∈ ran seq1( +_𝑒 , 𝐹)𝑥 < 𝑦))
144		supxr2 13391	. . 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 852	. 2 ⊢ (𝐹:ℕ⟶(0[,]+∞) → sup(ran seq1( +_𝑒 , 𝐹), ℝ^, < ) = Σ^𝑘 ∈ ℕ(𝐹‘𝑘))
146	145	eqcomd 2766	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 401 = wceq 1570 ∈ wcel 2145 Ⅎwnfc 2907 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4557 class class class wbr 5103 ran crn 5656 Fn wfn 6530 ⟶wf 6531 ‘cfv 6535 (class class class)co 7416 Fincfn 8959 supcsup 9417 ℝcr 11148 0cc0 11149 1c1 11150 +∞cpnf 11289 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 ℕcn 12282 ℤcz 12640 ℤ_≥cuz 12912 +_𝑒 cxad 13186 [,]cicc 13426 ...cfz 13586 seqcseq 14090 Σ^*cesum 34570

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 ax-mulf 11229

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 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 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-q 13023 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-ioo 13427 df-ioc 13428 df-ico 13429 df-icc 13430 df-fz 13587 df-fzo 13735 df-fl 13878 df-mod 13956 df-seq 14091 df-exp 14151 df-fac 14363 df-bc 14392 df-hash 14420 df-shft 15165 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-limsup 15583 df-clim 15600 df-rlim 15601 df-sum 15799 df-ef 16178 df-sin 16180 df-cos 16181 df-pi 16183 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-rest 17532 df-topn 17533 df-0g 17551 df-gsum 17552 df-topgen 17553 df-pt 17554 df-prds 17557 df-ordt 17612 df-xrs 17613 df-qtop 17618 df-imas 17619 df-xps 17621 df-mre 17695 df-mrc 17696 df-acs 17698 df-ps 18679 df-tsr 18680 df-plusf 18754 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-mhm 18917 df-submnd 18918 df-grp 19086 df-minusg 19087 df-sbg 19088 df-mulg 19217 df-subg 19272 df-cntz 19470 df-cmn 19935 df-abl 19936 df-mgp 20300 df-rng 20314 df-ur 20347 df-ring 20400 df-cring 20401 df-subrng 20737 df-subrg 20761 df-abv 21005 df-lmod 21076 df-scaf 21077 df-sra 21387 df-rgmod 21388 df-psmet 21609 df-xmet 21610 df-met 21611 df-bl 21612 df-mopn 21613 df-fbas 21614 df-fg 21615 df-cnfld 21618 df-top 23151 df-topon 23168 df-topsp 23190 df-bases 23203 df-cld 23276 df-ntr 23277 df-cls 23278 df-nei 23355 df-lp 23393 df-perf 23394 df-cn 23484 df-cnp 23485 df-haus 23572 df-tx 23820 df-hmeo 24013 df-fil 24104 df-fm 24196 df-flim 24197 df-flf 24198 df-tmd 24330 df-tgp 24331 df-tsms 24385 df-trg 24418 df-xms 24578 df-ms 24579 df-tms 24580 df-nm 24840 df-ngp 24841 df-nrg 24843 df-nlm 24844 df-ii 25137 df-cncf 25138 df-limc 26125 df-dv 26126 df-log 26825 df-esum 34571

This theorem is used by: esumfsupre 34614 esumsup 34632

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