esumcvg - Mathbox for Thierry Arnoux

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Theorem esumcvg 34069

Description: The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 15669. (Contributed by Thierry Arnoux, 5-Sep-2017.)

**Hypotheses**
Ref	Expression
esumcvg.j	⊢ 𝐽 = (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
esumcvg.f	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
esumcvg.a	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
esumcvg.m	⊢ (𝑘 = 𝑚 → 𝐴 = 𝐵)

**Assertion**
Ref	Expression
esumcvg	⊢ (𝜑 → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)

Distinct variable groups: 𝑚,𝑛,𝐴 𝑘,𝑛,𝐵 𝑘,𝑚,𝐹,𝑛 𝑘,𝐽,𝑛 𝜑,𝑘,𝑚,𝑛 Allowed substitution hints: 𝐴(𝑘) 𝐵(𝑚) 𝐽(𝑚)

Proof of Theorem esumcvg Dummy variables 𝑙 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12812	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12540	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 1 ∈ ℤ)
3		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ )
4		rge0ssre 13393	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
5		ax-resscn 11101	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
6	4, 5	sstri 3953	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℂ
7		esumcvg.m	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → 𝐴 = 𝐵)
8	7	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝐴 ∈ (0[,)+∞) ↔ 𝐵 ∈ (0[,)+∞)))
9	8	cbvralvw 3213	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) ↔ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞))
10		rsp 3223	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) → (𝑘 ∈ ℕ → 𝐴 ∈ (0[,)+∞)))
11	9, 10	sylbir 235	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) → (𝑘 ∈ ℕ → 𝐴 ∈ (0[,)+∞)))
12	11	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → (𝑘 ∈ ℕ → 𝐴 ∈ (0[,)+∞)))
13	12	imp 406	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
14	6, 13	sselid 3941	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
15	14	adantlr 715	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
16		esumcvg.f	. . . . . . . . 9 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴)
17		fzfid 13914	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
18		elfznn 13490	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
19	18, 13	sylan2 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,)+∞))
20	19	adantlr 715	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,)+∞))
21	17, 20	esumpfinval 34058	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
22	21	mpteq2dva 5195	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
23	16, 22	eqtrid 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
24	6, 20	sselid 3941	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
25	17, 24	fsumcl 15675	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ∈ ℂ)
26	23, 25	fvmpt2d 6963	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
27	26	adantlr 715	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
28	1, 2, 3, 15, 27	isumclim3 15701	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ Σ𝑘 ∈ ℕ 𝐴)
29		esumcvg.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
30	17, 20	fsumrp0cl 33005	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,)+∞))
31	21, 30	eqeltrd 2828	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,)+∞))
32	31, 16	fmptd 7068	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹:ℕ⟶(0[,)+∞))
33	32	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹:ℕ⟶(0[,)+∞))
34		simplll 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝜑)
35		eqidd 2730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑚 ∈ ℕ ↦ 𝐵) = (𝑚 ∈ ℕ ↦ 𝐵))
36		eqcom 2736	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 ↔ 𝑚 = 𝑘)
37		eqcom 2736	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
38	7, 36, 37	3imtr3i 291	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → 𝐵 = 𝐴)
39	38	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 = 𝑘) → 𝐵 = 𝐴)
40		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
41		esumcvg.a	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
42	35, 39, 40, 41	fvmptd 6957	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
43	34, 42	sylancom 588	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
44	13	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
45		elrege0 13391	. . . . . . . . . 10 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
46	44, 45	sylib 218	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
47	46	simpld 494	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
48		ovex 7402	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ∈ V
49		simpll 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
50	18	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
51	49, 50, 41	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
52	51	ralrimiva 3125	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
53		nfcv 2891	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(1...𝑛)
54	53	esumcl 34013	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
55	48, 52, 54	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
56	55, 16	fmptd 7068	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶(0[,]+∞))
57	56	ffnd 6671	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
58	57	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 Fn ℕ)
59		1z 12539	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
60		seqfn 13954	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℤ → seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
61	59, 60	ax-mp 5	. . . . . . . . . . . . 13 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1)
62	1	fneq2i 6598	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
63	61, 62	mpbir 231	. . . . . . . . . . . 12 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn ℕ
64	63	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn ℕ)
65		simplll 774	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
66	18, 42	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑛)) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
67	65, 66	sylancom 588	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
68		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
69	68, 1	eleqtrdi 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
70	67, 69, 24	fsumser 15672	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵))‘𝑛))
71	26, 70	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵))‘𝑛))
72	58, 64, 71	eqfnfvd 6988	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 = seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)))
73	72	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 = seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)))
74	73, 3	eqeltrrd 2829	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
75	1, 2, 43, 47, 74	isumrecl 15707	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
76	46	simprd 495	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
77	1, 2, 43, 47, 74, 76	isumge0 15708	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 0 ≤ Σ𝑘 ∈ ℕ 𝐴)
78		elrege0 13391	. . . . . . 7 ⊢ (Σ𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) ↔ (Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ ℕ 𝐴))
79	75, 77, 78	sylanbrc 583	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → Σ𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞))
80		ssid 3966	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,)+∞)
81	29, 33, 79, 80	lmlimxrge0 33931	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝐹(⇝_𝑡‘𝐽)Σ𝑘 ∈ ℕ 𝐴 ↔ 𝐹 ⇝ Σ𝑘 ∈ ℕ 𝐴))
82	28, 81	mpbird 257	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)Σ𝑘 ∈ ℕ 𝐴)
83	16, 3	eqeltrrid 2833	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
84	22	eleq1d 2813	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ ↔ (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ ))
85	84	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ ↔ (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ ))
86	83, 85	mpbid 232	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
87	44, 7, 86	esumpcvgval 34061	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
88	82, 87	breqtrrd 5130	. . 3 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
89	32	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹:ℕ⟶(0[,)+∞))
90		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
91	90	nnzd 12532	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
92		uzid 12784	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
93		peano2uz 12836	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘𝑛) → (𝑛 + 1) ∈ (ℤ_≥‘𝑛))
94	91, 92, 93	3syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ (ℤ_≥‘𝑛))
95		simplll 774	. . . . . . . 8 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
96	95, 13	sylancom 588	. . . . . . 7 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
97	90, 94, 96	esumpmono 34062	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → Σ^𝑘 ∈ (1...𝑛)𝐴 ≤ Σ^𝑘 ∈ (1...(𝑛 + 1))𝐴)
98	26, 21	eqtr4d 2767	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ^*𝑘 ∈ (1...𝑛)𝐴)
99	98	adantlr 715	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ^*𝑘 ∈ (1...𝑛)𝐴)
100		oveq2 7377	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (1...𝑙) = (1...𝑛))
101		esumeq1 34017	. . . . . . . . . . 11 ⊢ ((1...𝑙) = (1...𝑛) → Σ^𝑘 ∈ (1...𝑙)𝐴 = Σ^𝑘 ∈ (1...𝑛)𝐴)
102	100, 101	syl 17	. . . . . . . . . 10 ⊢ (𝑙 = 𝑛 → Σ^𝑘 ∈ (1...𝑙)𝐴 = Σ^𝑘 ∈ (1...𝑛)𝐴)
103	102	cbvmptv 5206	. . . . . . . . 9 ⊢ (𝑙 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑙)𝐴) = (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)
104	16, 103	eqtr4i 2755	. . . . . . . 8 ⊢ 𝐹 = (𝑙 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑙)𝐴)
105	104	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → 𝐹 = (𝑙 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑙)𝐴))
106		simpr3 1197	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ (¬ 𝐹 ∈ dom ⇝ ∧ 𝑛 ∈ ℕ ∧ 𝑙 = (𝑛 + 1))) → 𝑙 = (𝑛 + 1))
107		oveq2 7377	. . . . . . . . 9 ⊢ (𝑙 = (𝑛 + 1) → (1...𝑙) = (1...(𝑛 + 1)))
108		esumeq1 34017	. . . . . . . . 9 ⊢ ((1...𝑙) = (1...(𝑛 + 1)) → Σ^𝑘 ∈ (1...𝑙)𝐴 = Σ^𝑘 ∈ (1...(𝑛 + 1))𝐴)
109	106, 107, 108	3syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ (¬ 𝐹 ∈ dom ⇝ ∧ 𝑛 ∈ ℕ ∧ 𝑙 = (𝑛 + 1))) → Σ^𝑘 ∈ (1...𝑙)𝐴 = Σ^𝑘 ∈ (1...(𝑛 + 1))𝐴)
110	109	3anassrs 1361	. . . . . . 7 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 = (𝑛 + 1)) → Σ^𝑘 ∈ (1...𝑙)𝐴 = Σ^𝑘 ∈ (1...(𝑛 + 1))𝐴)
111	90	peano2nnd 12179	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
112		ovex 7402	. . . . . . . 8 ⊢ (1...(𝑛 + 1)) ∈ V
113		simp-4l 782	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝜑)
114		elfznn 13490	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑛 + 1)) → 𝑘 ∈ ℕ)
115	114	adantl 481	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝑘 ∈ ℕ)
116	113, 115, 41	syl2anc 584	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝐴 ∈ (0[,]+∞))
117	116	ralrimiva 3125	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
118		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...(𝑛 + 1))
119	118	esumcl 34013	. . . . . . . 8 ⊢ (((1...(𝑛 + 1)) ∈ V ∧ ∀𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
120	112, 117, 119	sylancr 587	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
121	105, 110, 111, 120	fvmptd 6957	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘(𝑛 + 1)) = Σ^*𝑘 ∈ (1...(𝑛 + 1))𝐴)
122	97, 99, 121	3brtr4d 5134	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
123		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐹 ∈ dom ⇝ )
124	29, 89, 122, 123	lmdvglim 33937	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)+∞)
125		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞))
126		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑘ℕ
127		nnex 12168	. . . . . . . 8 ⊢ ℕ ∈ V
128	127	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → ℕ ∈ V)
129	41	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
130		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
131		simpll 766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
132		inss1 4196	. . . . . . . . . . . . . 14 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
133		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
134	132, 133	sselid 3941	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ∈ 𝒫 ℕ)
135	134	elpwid 4568	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ⊆ ℕ)
136		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑥)
137	135, 136	sseldd 3944	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ ℕ)
138	131, 137, 13	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ (0[,)+∞))
139	138	fmpttd 7069	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ 𝑥 ↦ 𝐴):𝑥⟶(0[,)+∞))
140		esumpfinvallem 34057	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 ℕ ∩ Fin) ∧ (𝑘 ∈ 𝑥 ↦ 𝐴):𝑥⟶(0[,)+∞)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑥 ↦ 𝐴)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 𝐴)))
141	130, 139, 140	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑥 ↦ 𝐴)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 𝐴)))
142		inss2 4197	. . . . . . . . . 10 ⊢ (𝒫 ℕ ∩ Fin) ⊆ Fin
143	142, 130	sselid 3941	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
144	131, 137, 14	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ ℂ)
145	143, 144	gsumfsum 21376	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑥 ↦ 𝐴)) = Σ𝑘 ∈ 𝑥 𝐴)
146	141, 145	eqtr3d 2766	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 𝐴)) = Σ𝑘 ∈ 𝑥 𝐴)
147	125, 126, 128, 129, 146	esumval 34029	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴), ℝ^, < ))
148	147	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴), ℝ^, < ))
149	89, 122, 123	lmdvg 33936	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ∀𝑦 ∈ ℝ ∃𝑙 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛))
150	149	r19.21bi 3227	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → ∃𝑙 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛))
151		nnz 12526	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ ℤ)
152		uzid 12784	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℤ → 𝑙 ∈ (ℤ_≥‘𝑙))
153	151, 152	syl 17	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ (ℤ_≥‘𝑙))
154		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 = 𝑙) → 𝑛 = 𝑙)
155	154	fveq2d 6844	. . . . . . . . . . . . 13 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 = 𝑙) → (𝐹‘𝑛) = (𝐹‘𝑙))
156	155	breq2d 5114	. . . . . . . . . . . 12 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 = 𝑙) → (𝑦 < (𝐹‘𝑛) ↔ 𝑦 < (𝐹‘𝑙)))
157	153, 156	rspcdv 3577	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ℕ → (∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛) → 𝑦 < (𝐹‘𝑙)))
158	157	reximia 3064	. . . . . . . . . 10 ⊢ (∃𝑙 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛) → ∃𝑙 ∈ ℕ 𝑦 < (𝐹‘𝑙))
159	150, 158	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → ∃𝑙 ∈ ℕ 𝑦 < (𝐹‘𝑙))
160		simplr 768	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝑦 ∈ ℝ)
161	89	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝐹:ℕ⟶(0[,)+∞))
162		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
163	161, 162	ffvelcdmd 7039	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) ∈ (0[,)+∞))
164	4, 163	sselid 3941	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) ∈ ℝ)
165		ltle 11238	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑙) ∈ ℝ) → (𝑦 < (𝐹‘𝑙) → 𝑦 ≤ (𝐹‘𝑙)))
166	160, 164, 165	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝑦 < (𝐹‘𝑙) → 𝑦 ≤ (𝐹‘𝑙)))
167		oveq2 7377	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙))
168		esumeq1 34017	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑛) = (1...𝑙) → Σ^𝑘 ∈ (1...𝑛)𝐴 = Σ^𝑘 ∈ (1...𝑙)𝐴)
169	167, 168	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑙 → Σ^𝑘 ∈ (1...𝑛)𝐴 = Σ^𝑘 ∈ (1...𝑙)𝐴)
170		esumex 34012	. . . . . . . . . . . . . . 15 ⊢ Σ^*𝑘 ∈ (1...𝑙)𝐴 ∈ V
171	170	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑙)𝐴 ∈ V)
172	16, 169, 162, 171	fvmptd3 6973	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) = Σ^*𝑘 ∈ (1...𝑙)𝐴)
173		fzfid 13914	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin)
174		simp-4l 782	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
175		elfznn 13490	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ)
176	175	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → 𝑘 ∈ ℕ)
177	174, 176, 13	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → 𝐴 ∈ (0[,)+∞))
178	173, 177	esumpfinval 34058	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑙)𝐴 = Σ𝑘 ∈ (1...𝑙)𝐴)
179	172, 178	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) = Σ𝑘 ∈ (1...𝑙)𝐴)
180	179	breq2d 5114	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
181	166, 180	sylibd 239	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝑦 < (𝐹‘𝑙) → 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
182	181	reximdva 3146	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → (∃𝑙 ∈ ℕ 𝑦 < (𝐹‘𝑙) → ∃𝑙 ∈ ℕ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
183	159, 182	mpd 15	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → ∃𝑙 ∈ ℕ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴)
184		fzssuz 13502	. . . . . . . . . . . . . 14 ⊢ (1...𝑙) ⊆ (ℤ_≥‘1)
185	184, 1	sseqtrri 3993	. . . . . . . . . . . . 13 ⊢ (1...𝑙) ⊆ ℕ
186		ovex 7402	. . . . . . . . . . . . . 14 ⊢ (1...𝑙) ∈ V
187	186	elpw 4563	. . . . . . . . . . . . 13 ⊢ ((1...𝑙) ∈ 𝒫 ℕ ↔ (1...𝑙) ⊆ ℕ)
188	185, 187	mpbir 231	. . . . . . . . . . . 12 ⊢ (1...𝑙) ∈ 𝒫 ℕ
189		fzfi 13913	. . . . . . . . . . . 12 ⊢ (1...𝑙) ∈ Fin
190		elin 3927	. . . . . . . . . . . 12 ⊢ ((1...𝑙) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑙) ∈ 𝒫 ℕ ∧ (1...𝑙) ∈ Fin))
191	188, 189, 190	mpbir2an 711	. . . . . . . . . . 11 ⊢ (1...𝑙) ∈ (𝒫 ℕ ∩ Fin)
192		sumex 15630	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (1...𝑙)𝐴 ∈ V
193		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) = (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)
194		sumeq1 15631	. . . . . . . . . . . 12 ⊢ (𝑥 = (1...𝑙) → Σ𝑘 ∈ 𝑥 𝐴 = Σ𝑘 ∈ (1...𝑙)𝐴)
195	193, 194	elrnmpt1s 5912	. . . . . . . . . . 11 ⊢ (((1...𝑙) ∈ (𝒫 ℕ ∩ Fin) ∧ Σ𝑘 ∈ (1...𝑙)𝐴 ∈ V) → Σ𝑘 ∈ (1...𝑙)𝐴 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴))
196	191, 192, 195	mp2an 692	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (1...𝑙)𝐴 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)
197		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴
198		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑧 = Σ𝑘 ∈ (1...𝑙)𝐴 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
199	197, 198	rspce 3574	. . . . . . . . . 10 ⊢ ((Σ𝑘 ∈ (1...𝑙)𝐴 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) ∧ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴) → ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
200	196, 199	mpan 690	. . . . . . . . 9 ⊢ (𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴 → ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
201	200	rexlimivw 3130	. . . . . . . 8 ⊢ (∃𝑙 ∈ ℕ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴 → ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
202	183, 201	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
203	202	ralrimiva 3125	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
204		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
205	142, 204	sselid 3941	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
206	138	adantllr 719	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ (0[,)+∞))
207	4, 206	sselid 3941	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ ℝ)
208	205, 207	fsumrecl 15676	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐴 ∈ ℝ)
209	208	rexrd 11200	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐴 ∈ ℝ^*)
210	209	fmpttd 7069	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴):(𝒫 ℕ ∩ Fin)⟶ℝ^*)
211		frn 6677	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴):(𝒫 ℕ ∩ Fin)⟶ℝ^* → ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) ⊆ ℝ^*)
212		supxrunb1 13255	. . . . . . 7 ⊢ (ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) ⊆ ℝ^* → (∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧 ↔ sup(ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴), ℝ^*, < ) = +∞))
213	210, 211, 212	3syl 18	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → (∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧 ↔ sup(ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴), ℝ^*, < ) = +∞))
214	203, 213	mpbid 232	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → sup(ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴), ℝ^*, < ) = +∞)
215	148, 214	eqtrd 2764	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → Σ^*𝑘 ∈ ℕ𝐴 = +∞)
216	124, 215	breqtrrd 5130	. . 3 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
217	88, 216	pm2.61dan 812	. 2 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
218	16	reseq1i 5935	. . . . . . . 8 ⊢ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘))
219		eleq1w 2811	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (𝑙 ∈ ℕ ↔ 𝑘 ∈ ℕ))
220	219	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((𝜑 ∧ 𝑙 ∈ ℕ) ↔ (𝜑 ∧ 𝑘 ∈ ℕ)))
221		sbequ12r 2253	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ([𝑙 / 𝑘]𝐴 = +∞ ↔ 𝐴 = +∞))
222	220, 221	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ↔ ((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞)))
223		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑘))
224	223	reseq2d 5939	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)))
225	223	xpeq1d 5660	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((ℤ_≥‘𝑙) × {+∞}) = ((ℤ_≥‘𝑘) × {+∞}))
226	224, 225	eqeq12d 2745	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((ℤ_≥‘𝑙) × {+∞}) ↔ ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})))
227	222, 226	imbi12d 344	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((ℤ_≥‘𝑙) × {+∞})) ↔ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))))
228		nfv 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑙 ∈ ℕ)
229		nfs1v 2157	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘[𝑙 / 𝑘]𝐴 = +∞
230	228, 229	nfan 1899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞)
231		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑛 ∈ (ℤ_≥‘𝑙)
232	230, 231	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙))
233		ovexd 7404	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → (1...𝑛) ∈ V)
234		simp-4l 782	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
235	18	adantl 481	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
236	234, 235, 41	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
237		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ ℕ)
238		elnnuz 12813	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ ℕ ↔ 𝑙 ∈ (ℤ_≥‘1))
239		eluzfz 13456	. . . . . . . . . . . . . . 15 ⊢ ((𝑙 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ (1...𝑛))
240	238, 239	sylanb 581	. . . . . . . . . . . . . 14 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ (1...𝑛))
241	237, 240	sylancom 588	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ (1...𝑛))
242		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → [𝑙 / 𝑘]𝐴 = +∞)
243		sbequ12 2252	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → (𝐴 = +∞ ↔ [𝑙 / 𝑘]𝐴 = +∞))
244	229, 243	rspce 3574	. . . . . . . . . . . . 13 ⊢ ((𝑙 ∈ (1...𝑛) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ∃𝑘 ∈ (1...𝑛)𝐴 = +∞)
245	241, 242, 244	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → ∃𝑘 ∈ (1...𝑛)𝐴 = +∞)
246	232, 233, 236, 245	esumpinfval 34056	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞)
247	246	ralrimiva 3125	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞)
248		eqidd 2730	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑙))
249		mpteq12 5190	. . . . . . . . . . . 12 ⊢ (((ℤ_≥‘𝑙) = (ℤ_≥‘𝑙) ∧ ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^𝑘 ∈ (1...𝑛)𝐴 = +∞) → (𝑛 ∈ (ℤ_≥‘𝑙) ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ +∞))
250	248, 249	sylan 580	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^𝑘 ∈ (1...𝑛)𝐴 = +∞) → (𝑛 ∈ (ℤ_≥‘𝑙) ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ +∞))
251		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → 𝑙 ∈ ℕ)
252		uznnssnn 12830	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℕ → (ℤ_≥‘𝑙) ⊆ ℕ)
253		resmpt 5997	. . . . . . . . . . . . 13 ⊢ ((ℤ_≥‘𝑙) ⊆ ℕ → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ Σ^𝑘 ∈ (1...𝑛)𝐴))
254	251, 252, 253	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ Σ^𝑘 ∈ (1...𝑛)𝐴))
255	254	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^𝑘 ∈ (1...𝑛)𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
256		fconstmpt 5693	. . . . . . . . . . . 12 ⊢ ((ℤ_≥‘𝑙) × {+∞}) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ +∞)
257	256	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞) → ((ℤ_≥‘𝑙) × {+∞}) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ +∞))
258	250, 255, 257	3eqtr4d 2774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^𝑘 ∈ (1...𝑛)𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((ℤ_≥‘𝑙) × {+∞}))
259	247, 258	mpdan 687	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((ℤ_≥‘𝑙) × {+∞}))
260	227, 259	chvarvv 1989	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
261	218, 260	eqtrid 2776	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞) → (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
262	261	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 = +∞ → (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})))
263	262	reximdva 3146	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ℕ 𝐴 = +∞ → ∃𝑘 ∈ ℕ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})))
264	263	imp 406	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → ∃𝑘 ∈ ℕ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
265		xrge0topn 33926	. . . . . . . . . . 11 ⊢ (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
266	29, 265	eqtri 2752	. . . . . . . . . 10 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
267		letopon 23125	. . . . . . . . . . 11 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
268		iccssxr 13367	. . . . . . . . . . 11 ⊢ (0[,]+∞) ⊆ ℝ^*
269		resttopon 23081	. . . . . . . . . . 11 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ (0[,]+∞) ⊆ ℝ^) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)))
270	267, 268, 269	mp2an 692	. . . . . . . . . 10 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))
271	266, 270	eqeltri 2824	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
272	271	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ (TopOn‘(0[,]+∞)))
273		0xr 11197	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
274		pnfxr 11204	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
275		0lepnf 13069	. . . . . . . . . 10 ⊢ 0 ≤ +∞
276		ubicc2 13402	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
277	273, 274, 275, 276	mp3an 1463	. . . . . . . . 9 ⊢ +∞ ∈ (0[,]+∞)
278	277	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → +∞ ∈ (0[,]+∞))
279	40	nnzd 12532	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
280		eqid 2729	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑘) = (ℤ_≥‘𝑘)
281	280	lmconst 23181	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘(0[,]+∞)) ∧ +∞ ∈ (0[,]+∞) ∧ 𝑘 ∈ ℤ) → ((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞)
282	272, 278, 279, 281	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞)
283		breq1 5105	. . . . . . . 8 ⊢ ((𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}) → ((𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞ ↔ ((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞))
284	283	biimprd 248	. . . . . . 7 ⊢ ((𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}) → (((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞ → (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞))
285	282, 284	mpan9 506	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})) → (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞)
286		ovexd 7404	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0[,]+∞) ∈ V)
287		cnex 11125	. . . . . . . . . 10 ⊢ ℂ ∈ V
288	287	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ℂ ∈ V)
289	56	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(0[,]+∞))
290		nnsscn 12167	. . . . . . . . . 10 ⊢ ℕ ⊆ ℂ
291	290	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ℕ ⊆ ℂ)
292		elpm2r 8795	. . . . . . . . 9 ⊢ ((((0[,]+∞) ∈ V ∧ ℂ ∈ V) ∧ (𝐹:ℕ⟶(0[,]+∞) ∧ ℕ ⊆ ℂ)) → 𝐹 ∈ ((0[,]+∞) ↑_pm ℂ))
293	286, 288, 289, 291, 292	syl22anc 838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹 ∈ ((0[,]+∞) ↑_pm ℂ))
294	272, 293, 279	lmres 23220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹(⇝_𝑡‘𝐽)+∞ ↔ (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞))
295	294	biimpar 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞) → 𝐹(⇝_𝑡‘𝐽)+∞)
296	285, 295	syldan 591	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})) → 𝐹(⇝_𝑡‘𝐽)+∞)
297	296	r19.29an 3137	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})) → 𝐹(⇝_𝑡‘𝐽)+∞)
298	264, 297	syldan 591	. . 3 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → 𝐹(⇝_𝑡‘𝐽)+∞)
299		nfv 1914	. . . . 5 ⊢ Ⅎ𝑘𝜑
300		nfre1 3260	. . . . 5 ⊢ Ⅎ𝑘∃𝑘 ∈ ℕ 𝐴 = +∞
301	299, 300	nfan 1899	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞)
302	127	a1i 11	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → ℕ ∈ V)
303	41	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
304		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → ∃𝑘 ∈ ℕ 𝐴 = +∞)
305	301, 302, 303, 304	esumpinfval 34056	. . 3 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → Σ^*𝑘 ∈ ℕ𝐴 = +∞)
306	298, 305	breqtrrd 5130	. 2 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
307		eleq1w 2811	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ ℕ ↔ 𝑚 ∈ ℕ))
308	307	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝜑 ∧ 𝑘 ∈ ℕ) ↔ (𝜑 ∧ 𝑚 ∈ ℕ)))
309	7	eleq1d 2813	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝐴 ∈ (0[,]+∞) ↔ 𝐵 ∈ (0[,]+∞)))
310	308, 309	imbi12d 344	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ (0[,]+∞))))
311	310, 41	chvarvv 1989	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ (0[,]+∞))
312		eliccelico 32750	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞)))
313	273, 274, 275, 312	mp3an 1463	. . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
314	311, 313	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
315	314	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
316		r19.30 3100	. . . 4 ⊢ (∀𝑚 ∈ ℕ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞) → (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑚 ∈ ℕ 𝐵 = +∞))
317	315, 316	syl 17	. . 3 ⊢ (𝜑 → (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑚 ∈ ℕ 𝐵 = +∞))
318	7	eqeq1d 2731	. . . . 5 ⊢ (𝑘 = 𝑚 → (𝐴 = +∞ ↔ 𝐵 = +∞))
319	318	cbvrexvw 3214	. . . 4 ⊢ (∃𝑘 ∈ ℕ 𝐴 = +∞ ↔ ∃𝑚 ∈ ℕ 𝐵 = +∞)
320	319	orbi2i 912	. . 3 ⊢ ((∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑘 ∈ ℕ 𝐴 = +∞) ↔ (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑚 ∈ ℕ 𝐵 = +∞))
321	317, 320	sylibr 234	. 2 ⊢ (𝜑 → (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑘 ∈ ℕ 𝐴 = +∞))
322	217, 306, 321	mpjaodan 960	1 ⊢ (𝜑 → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 [wsb 2065 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 Vcvv 3444 ∩ cin 3910 ⊆ wss 3911 𝒫 cpw 4559 {csn 4585 class class class wbr 5102 ↦ cmpt 5183 × cxp 5629 dom cdm 5631 ran crn 5632 ↾ cres 5633 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ↑_pm cpm 8777 Fincfn 8895 supcsup 9367 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 +∞cpnf 11181 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 ℕcn 12162 ℤcz 12505 ℤ_≥cuz 12769 [,)cico 13284 [,]cicc 13285 ...cfz 13444 seqcseq 13942 ⇝ cli 15426 Σcsu 15628 ↾_s cress 17176 ↾_t crest 17359 TopOpenctopn 17360 Σ_g cgsu 17379 ordTopcordt 17438 ℝ^*_𝑠cxrs 17439 ℂ_fldccnfld 21296 TopOnctopon 22830 ⇝_𝑡clm 23146 Σ^*cesum 34010

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-ordt 17440 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-ps 18507 df-tsr 18508 df-plusf 18548 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-subrng 20466 df-subrg 20490 df-abv 20729 df-lmod 20800 df-scaf 20801 df-sra 21112 df-rgmod 21113 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-lm 23149 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-tmd 23992 df-tgp 23993 df-tsms 24047 df-trg 24080 df-xms 24241 df-ms 24242 df-tms 24243 df-nm 24503 df-ngp 24504 df-nrg 24506 df-nlm 24507 df-ii 24803 df-cncf 24804 df-limc 25800 df-dv 25801 df-log 26498 df-esum 34011

This theorem is referenced by: esumcvg2 34070

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