esumpcvgval - Mathbox for Thierry Arnoux

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

Theorem esumpcvgval 31339

Description: The value of the extended sum when the corresponding series sum is convergent. (Contributed by Thierry Arnoux, 31-Jul-2017.)

**Hypotheses**
Ref	Expression
esumpcvgval.1	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
esumpcvgval.2	⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵)
esumpcvgval.3	⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )

**Assertion**
Ref	Expression
esumpcvgval	⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Distinct variable groups: 𝑘,𝑙,𝑛 𝐴,𝑙,𝑛 𝐵,𝑘,𝑛 𝜑,𝑘,𝑛 Allowed substitution hints: 𝜑(𝑙) 𝐴(𝑘) 𝐵(𝑙)

Proof of Theorem esumpcvgval Dummy variables 𝑠 𝑥 𝑦 𝑧 𝑏 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltso 12537	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		nnuz 12284	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 12016	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
5		esumpcvgval.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6		esumpcvgval.2	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵)
7		eqcom 2830	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 ↔ 𝑙 = 𝑘)
8		eqcom 2830	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
9	6, 7, 8	3imtr3i 293	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → 𝐵 = 𝐴)
10	9	cbvmptv 5171	. . . . . . . . . 10 ⊢ (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴)
11	5, 10	fmptd 6880	. . . . . . . . 9 ⊢ (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
12	11	ffvelrnda 6853	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞))
13		elrege0 12845	. . . . . . . . 9 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥)))
14	13	simplbi 500	. . . . . . . 8 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ)
15	12, 14	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ)
16	3, 4, 15	serfre 13402	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ)
17	11	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
18		simpr 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
19	18	peano2nnd 11657	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
20	17, 19	ffvelrnd 6854	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞))
21		elrege0 12845	. . . . . . . . . 10 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
22	21	simprbi 499	. . . . . . . . 9 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))
23	20, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))
24	16	ffvelrnda 6853	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ)
25	21	simplbi 500	. . . . . . . . . 10 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ)
26	20, 25	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ)
27	24, 26	addge01d 11230	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))))
28	23, 27	mpbid 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
29	18, 3	eleqtrdi 2925	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
30		seqp1 13387	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘1) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
31	29, 30	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
32	28, 31	breqtrrd 5096	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)))
33		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
34	10	fvmpt2 6781	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
35	33, 5, 34	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
36		rge0ssre 12847	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
37	36, 5	sseldi 3967	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
38	16	feqmptd 6735	. . . . . . . . . 10 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
39		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
40		elfznn 12939	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
41	40	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
42	39, 41, 35	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
43	37	recnd 10671	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
44	39, 41, 43	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
45	42, 29, 44	fsumser 15089	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
46	45	eqcomd 2829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
47	46	mpteq2dva 5163	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
48	38, 47	eqtr2d 2859	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) = seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
49		esumpcvgval.3	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
50	48, 49	eqeltrrd 2916	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
51	3, 4, 35, 37, 50	isumrecl 15122	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
52		1zzd 12016	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
53		fzfid 13344	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
54		fzssuz 12951	. . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ (ℤ_≥‘1)
55	54, 3	sseqtrri 4006	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
57	35	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
58	37	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
59	5	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
60		elrege0 12845	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
61	60	simprbi 499	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (0[,)+∞) → 0 ≤ 𝐴)
62	59, 61	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
63	50	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
64	3, 52, 53, 56, 57, 58, 62, 63	isumless 15202	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴)
65	45, 64	eqbrtrrd 5092	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
66	65	ralrimiva 3184	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
67		brralrspcev 5128	. . . . . . 7 ⊢ ((Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴) → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
68	51, 66, 67	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
69	3, 4, 16, 32, 68	climsup 15028	. . . . 5 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
70	3, 4, 69, 24	climrecl 14942	. . . 4 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
71	70	rexrd 10693	. . 3 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^*)
72		eqid 2823	. . . . . . 7 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
73		sumex 15046	. . . . . . 7 ⊢ Σ𝑘 ∈ 𝑏 𝐴 ∈ V
74	72, 73	elrnmpti 5834	. . . . . 6 ⊢ (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
75		ssnnssfz 30512	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚))
76		fzfid 13344	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin)
77		elfznn 12939	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
78	77, 5	sylan2 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
79	60	simplbi 500	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (0[,)+∞) → 𝐴 ∈ ℝ)
80	78, 79	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
81	80	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
82	78, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
83	82	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
84		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚))
85	76, 81, 83, 84	fsumless 15153	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
86	85	ex 415	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
87	86	reximdv 3275	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
88	87	imp 409	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
89	75, 88	sylan2 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
90		breq1 5071	. . . . . . . . . 10 ⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
91	90	rexbidv 3299	. . . . . . . . 9 ⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
92	89, 91	syl5ibrcom 249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
93	92	rexlimdva 3286	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
94	93	imp 409	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
95	74, 94	sylan2b 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
96		simpr 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
97		inss2 4208	. . . . . . . . . . . . 13 ⊢ (𝒫 ℕ ∩ Fin) ⊆ Fin
98		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
99	97, 98	sseldi 3967	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ Fin)
100		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝜑)
101		inss1 4207	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
102		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
103	101, 102	sseldi 3967	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ 𝒫 ℕ)
104	103	elpwid 4552	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ⊆ ℕ)
105		simpr 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ 𝑏)
106	104, 105	sseldd 3970	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ ℕ)
107	100, 106, 5	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ (0[,)+∞))
108	107, 79	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ ℝ)
109	99, 108	fsumrecl 15093	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
110	109	adantr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
111	96, 110	eqeltrd 2915	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
112	111	r19.29an 3290	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
113	74, 112	sylan2b 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ∈ ℝ)
114	113	adantr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ)
115		fzfid 13344	. . . . . . . 8 ⊢ (𝜑 → (1...𝑚) ∈ Fin)
116	115, 80	fsumrecl 15093	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
117	116	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
118	70	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
119		simprr 771	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
120	16	frnd 6523	. . . . . . . 8 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
121	120	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
122		1nn 11651	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
123	122	ne0ii 4305	. . . . . . . . 9 ⊢ ℕ ≠ ∅
124		dm0rn0 5797	. . . . . . . . . . 11 ⊢ (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅)
125	16	fdmd 6525	. . . . . . . . . . . 12 ⊢ (𝜑 → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
126	125	eqeq1d 2825	. . . . . . . . . . 11 ⊢ (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
127	124, 126	syl5bbr 287	. . . . . . . . . 10 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
128	127	necon3bid 3062	. . . . . . . . 9 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ ≠ ∅))
129	123, 128	mpbiri 260	. . . . . . . 8 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
130	129	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
131		1z 12015	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
132		seqfn 13384	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
133	131, 132	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1)
134	3	fneq2i 6453	. . . . . . . . . . . . . . 15 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
135	133, 134	mpbir 233	. . . . . . . . . . . . . 14 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ
136		dffn5 6726	. . . . . . . . . . . . . 14 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
137	135, 136	mpbi 232	. . . . . . . . . . . . 13 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
138		fvex 6685	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ V
139	137, 138	elrnmpti 5834	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
140		r19.29 3256	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
141		breq1 5071	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧 ≤ 𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠))
142	141	biimparc 482	. . . . . . . . . . . . . 14 ⊢ (((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
143	142	rexlimivw 3284	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
144	140, 143	syl 17	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
145	139, 144	sylan2b 595	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → 𝑧 ≤ 𝑠)
146	145	ralrimiva 3184	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
147	146	reximi 3245	. . . . . . . . 9 ⊢ (∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
148	68, 147	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
149	148	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
150		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
151		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑)
152	77	adantl 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ)
153	151, 152, 35	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
154	150, 3	eleqtrdi 2925	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ_≥‘1))
155	151, 152, 5	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
156	155, 79	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
157	156	recnd 10671	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ)
158	153, 154, 157	fsumser 15089	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
159		fveq2 6672	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
160	159	rspceeqv 3640	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
161	150, 158, 160	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
162	137, 138	elrnmpti 5834	. . . . . . . . 9 ⊢ (Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
163	161, 162	sylibr 236	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
164	163	ad2ant2r 745	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
165		suprub 11604	. . . . . . 7 ⊢ (((ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) ∧ Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → Σ𝑘 ∈ (1...𝑚)𝐴 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
166	121, 130, 149, 164, 165	syl31anc 1369	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
167	114, 117, 118, 119, 166	letrd 10799	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
168	95, 167	rexlimddv 3293	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
169	70	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
170	113, 169	lenltd 10788	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → (𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥))
171	168, 170	mpbid 234	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥)
172		simpr1r 1227	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 = +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
173	172	3anassrs 1356	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
174	71	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^*)
175		pnfnlt 12526	. . . . . . . 8 ⊢ (sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^* → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
176	174, 175	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
177		breq1 5071	. . . . . . . . 9 ⊢ (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
178	177	notbid 320	. . . . . . . 8 ⊢ (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
179	178	adantl 484	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
180	176, 179	mpbird 259	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
181	173, 180	pm2.21dd 197	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
182		simplll 773	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝜑)
183		simpr1l 1226	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ^*)
184	183	3anassrs 1356	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ ℝ^*)
185		simplr 767	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 0 ≤ 𝑥)
186		simpr 487	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞)
187		0xr 10690	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
188		pnfxr 10697	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
189		elico1 12784	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^ ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)))
190	187, 188, 189	mp2an 690	. . . . . . 7 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞))
191	184, 185, 186, 190	syl3anbrc 1339	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ (0[,)+∞))
192		simpr1r 1227	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
193	192	3anassrs 1356	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
194	120	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
195	129	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
196	148	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
197	194, 195, 196	3jca 1124	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠))
198		simprl 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ (0[,)+∞))
199	36, 198	sseldi 3967	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ ℝ)
200		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
201		suprlub 11607	. . . . . . . . 9 ⊢ (((ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) ∧ 𝑥 ∈ ℝ) → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦))
202	201	biimpa 479	. . . . . . . 8 ⊢ ((((ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
203	197, 199, 200, 202	syl21anc 835	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
204	40	ssriv 3973	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑛) ⊆ ℕ
205		ovex 7191	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑛) ∈ V
206	205	elpw 4545	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑛) ∈ 𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ)
207	204, 206	mpbir 233	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ 𝒫 ℕ
208		fzfi 13343	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ Fin
209		elin 4171	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑛) ∈ 𝒫 ℕ ∧ (1...𝑛) ∈ Fin))
210	207, 208, 209	mpbir2an 709	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ∈ (𝒫 ℕ ∩ Fin)
211	210	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → (1...𝑛) ∈ (𝒫 ℕ ∩ Fin))
212		simpr 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
213	45	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
214	212, 213	eqtr4d 2861	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴)
215		sumeq1 15047	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (1...𝑛) → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
216	215	rspceeqv 3640	. . . . . . . . . . . . . 14 ⊢ (((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
217	211, 214, 216	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
218	217	ex 415	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
219	218	rexlimdva 3286	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
220	137, 138	elrnmpti 5834	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
221	72, 73	elrnmpti 5834	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
222	219, 220, 221	3imtr4g 298	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)))
223	222	ssrdv 3975	. . . . . . . . 9 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴))
224		ssrexv 4036	. . . . . . . . 9 ⊢ (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦))
225	223, 224	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦))
226	225	imp 409	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
227	203, 226	syldan 593	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
228	182, 191, 193, 227	syl12anc 834	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
229		simplrl 775	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ^*)
230		xrlelttric 30478	. . . . . . . 8 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
231	188, 230	mpan 688	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
232		xgepnf 12561	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ↔ 𝑥 = +∞))
233	232	orbi1d 913	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → ((+∞ ≤ 𝑥 ∨ 𝑥 < +∞) ↔ (𝑥 = +∞ ∨ 𝑥 < +∞)))
234	231, 233	mpbid 234	. . . . . 6 ⊢ (𝑥 ∈ ℝ^* → (𝑥 = +∞ ∨ 𝑥 < +∞))
235	229, 234	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → (𝑥 = +∞ ∨ 𝑥 < +∞))
236	181, 228, 235	mpjaodan 955	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
237		0elpw 5258	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 ℕ
238		0fin 8748	. . . . . . . . 9 ⊢ ∅ ∈ Fin
239		elin 4171	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ ∧ ∅ ∈ Fin))
240	237, 238, 239	mpbir2an 709	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 ℕ ∩ Fin)
241		sum0 15080	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0
242	241	eqcomi 2832	. . . . . . . 8 ⊢ 0 = Σ𝑘 ∈ ∅ 𝐴
243		sumeq1 15047	. . . . . . . . 9 ⊢ (𝑏 = ∅ → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴)
244	243	rspceeqv 3640	. . . . . . . 8 ⊢ ((∅ ∈ (𝒫 ℕ ∩ Fin) ∧ 0 = Σ𝑘 ∈ ∅ 𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
245	240, 242, 244	mp2an 690	. . . . . . 7 ⊢ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴
246	72, 73	elrnmpti 5834	. . . . . . 7 ⊢ (0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
247	245, 246	mpbir 233	. . . . . 6 ⊢ 0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
248		breq2 5072	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑥 < 𝑦 ↔ 𝑥 < 0))
249	248	rspcev 3625	. . . . . 6 ⊢ ((0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
250	247, 249	mpan 688	. . . . 5 ⊢ (𝑥 < 0 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
251	250	adantl 484	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
252		xrlelttric 30478	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
253	187, 252	mpan 688	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (0 ≤ 𝑥 ∨ 𝑥 < 0))
254	253	ad2antrl 726	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
255	236, 251, 254	mpjaodan 955	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
256	2, 71, 171, 255	eqsupd 8923	. 2 ⊢ (𝜑 → sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^*, < ) = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
257		nfv 1915	. . 3 ⊢ Ⅎ𝑘𝜑
258		nfcv 2979	. . 3 ⊢ Ⅎ𝑘ℕ
259		nnex 11646	. . . 4 ⊢ ℕ ∈ V
260	259	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
261		icossicc 12827	. . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
262	261, 5	sseldi 3967	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
263		elex 3514	. . . . . 6 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → 𝑏 ∈ V)
264	263	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ V)
265	107	fmpttd 6881	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞))
266		esumpfinvallem 31335	. . . . 5 ⊢ ((𝑏 ∈ V ∧ (𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)))
267	264, 265, 266	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)))
268	108	recnd 10671	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ ℂ)
269	99, 268	gsumfsum 20614	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴)
270	267, 269	eqtr3d 2860	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴)
271	257, 258, 260, 262, 270	esumval 31307	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^, < ))
272	3, 4, 35, 43, 69	isumclim 15114	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
273	256, 271, 272	3eqtr4d 2868	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 class class class wbr 5068 ↦ cmpt 5148 Or wor 5475 dom cdm 5557 ran crn 5558 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 supcsup 8906 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 +∞cpnf 10674 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 ℕcn 11640 ℤcz 11984 ℤ_≥cuz 12246 [,)cico 12743 [,]cicc 12744 ...cfz 12895 seqcseq 13372 ⇝ cli 14843 Σcsu 15044 ↾_s cress 16486 Σ_g cgsu 16716 ℝ^*_𝑠cxrs 16775 ℂ_fldccnfld 20547 Σ^*cesum 31288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xadd 12511 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-ordt 16776 df-xrs 16777 df-mre 16859 df-mrc 16860 df-acs 16862 df-ps 17812 df-tsr 17813 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-ntr 21630 df-nei 21708 df-cn 21837 df-haus 21925 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-tsms 22737 df-esum 31289

This theorem is referenced by: esumcvg 31347 esumcvgsum 31349

W3C validator