esumcvg - Mathbox for Thierry Arnoux

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Theorem esumcvg 30495

Description: The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 14700. (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 11960	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 11693	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 1 ∈ ℤ)
3		simpr 473	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ )
4		rge0ssre 12519	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
5		ax-resscn 10287	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
6	4, 5	sstri 3818	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℂ
7		esumcvg.m	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → 𝐴 = 𝐵)
8	7	eleq1d 2881	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝐴 ∈ (0[,)+∞) ↔ 𝐵 ∈ (0[,)+∞)))
9	8	cbvralv 3371	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) ↔ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞))
10		rsp 3128	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) → (𝑘 ∈ ℕ → 𝐴 ∈ (0[,)+∞)))
11	9, 10	sylbir 226	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) → (𝑘 ∈ ℕ → 𝐴 ∈ (0[,)+∞)))
12	11	adantl 469	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → (𝑘 ∈ ℕ → 𝐴 ∈ (0[,)+∞)))
13	12	imp 395	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
14	6, 13	sseldi 3807	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
15	14	adantlr 697	. . . . . 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 13015	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
18		elfznn 12612	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
19	18, 13	sylan2 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,)+∞))
20	19	adantlr 697	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,)+∞))
21	17, 20	esumpfinval 30484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
22	21	mpteq2dva 4949	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
23	16, 22	syl5eq 2863	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
24	6, 20	sseldi 3807	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
25	17, 24	fsumcl 14706	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ∈ ℂ)
26	23, 25	fvmpt2d 6523	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
27	26	adantlr 697	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
28	1, 2, 3, 15, 27	isumclim3 14732	. . . . 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 30042	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,)+∞))
31	21, 30	eqeltrd 2896	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,)+∞))
32	31, 16	fmptd 6615	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹:ℕ⟶(0[,)+∞))
33	32	adantr 468	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹:ℕ⟶(0[,)+∞))
34		simplll 782	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝜑)
35		eqidd 2818	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑚 ∈ ℕ ↦ 𝐵) = (𝑚 ∈ ℕ ↦ 𝐵))
36		eqcom 2824	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 ↔ 𝑚 = 𝑘)
37		eqcom 2824	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
38	7, 36, 37	3imtr3i 282	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → 𝐵 = 𝐴)
39	38	adantl 469	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 = 𝑘) → 𝐵 = 𝐴)
40		simpr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
41		esumcvg.a	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
42	35, 39, 40, 41	fvmptd 6518	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
43	34, 42	sylancom 578	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
44	13	adantlr 697	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
45		elrege0 12517	. . . . . . . . . 10 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
46	44, 45	sylib 209	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
47	46	simpld 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
48		ovex 6915	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ∈ V
49		simpll 774	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
50	18	adantl 469	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
51	49, 50, 41	syl2anc 575	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
52	51	ralrimiva 3165	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
53		nfcv 2959	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(1...𝑛)
54	53	esumcl 30439	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
55	48, 52, 54	sylancr 577	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
56	55, 16	fmptd 6615	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶(0[,]+∞))
57	56	ffnd 6266	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
58	57	adantr 468	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 Fn ℕ)
59		1z 11692	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
60		seqfn 13055	. . . . . . . . . . . . . 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 6206	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
63	61, 62	mpbir 222	. . . . . . . . . . . 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 782	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
66	18, 42	sylan2 582	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑛)) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
67	65, 66	sylancom 578	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
68		simpr 473	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
69	68, 1	syl6eleq 2906	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
70	67, 69, 24	fsumser 14703	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵))‘𝑛))
71	26, 70	eqtrd 2851	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵))‘𝑛))
72	58, 64, 71	eqfnfvd 6545	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 = seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)))
73	72	adantr 468	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 = seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)))
74	73, 3	eqeltrrd 2897	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
75	1, 2, 43, 47, 74	isumrecl 14738	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
76	46	simprd 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
77	1, 2, 43, 47, 74, 76	isumge0 14739	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 0 ≤ Σ𝑘 ∈ ℕ 𝐴)
78		elrege0 12517	. . . . . . 7 ⊢ (Σ𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) ↔ (Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ ℕ 𝐴))
79	75, 77, 78	sylanbrc 574	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → Σ𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞))
80		ssid 3831	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,)+∞)
81	29, 33, 79, 80	lmlimxrge0 30341	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝐹(⇝_𝑡‘𝐽)Σ𝑘 ∈ ℕ 𝐴 ↔ 𝐹 ⇝ Σ𝑘 ∈ ℕ 𝐴))
82	28, 81	mpbird 248	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)Σ𝑘 ∈ ℕ 𝐴)
83	16, 3	syl5eqelr 2901	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
84	22	eleq1d 2881	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ ↔ (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ ))
85	84	adantr 468	. . . . . 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 223	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
87	44, 7, 86	esumpcvgval 30487	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
88	82, 87	breqtrrd 4883	. . 3 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
89	32	adantr 468	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹:ℕ⟶(0[,)+∞))
90		simpr 473	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
91	90	nnzd 11766	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
92		uzid 11938	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
93		peano2uz 11978	. . . . . . . 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 782	. . . . . . . 8 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
96	95, 13	sylancom 578	. . . . . . 7 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
97	90, 94, 96	esumpmono 30488	. . . . . 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 2854	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ^*𝑘 ∈ (1...𝑛)𝐴)
99	98	adantlr 697	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ^*𝑘 ∈ (1...𝑛)𝐴)
100		oveq2 6891	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (1...𝑙) = (1...𝑛))
101		esumeq1 30443	. . . . . . . . . . 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 4955	. . . . . . . . 9 ⊢ (𝑙 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑙)𝐴) = (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)
104	16, 103	eqtr4i 2842	. . . . . . . 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 1245	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ (¬ 𝐹 ∈ dom ⇝ ∧ 𝑛 ∈ ℕ ∧ 𝑙 = (𝑛 + 1))) → 𝑙 = (𝑛 + 1))
107		oveq2 6891	. . . . . . . . 9 ⊢ (𝑙 = (𝑛 + 1) → (1...𝑙) = (1...(𝑛 + 1)))
108		esumeq1 30443	. . . . . . . . 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 1462	. . . . . . 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 11333	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
112		ovex 6915	. . . . . . . 8 ⊢ (1...(𝑛 + 1)) ∈ V
113		simp-4l 792	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝜑)
114		elfznn 12612	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑛 + 1)) → 𝑘 ∈ ℕ)
115	114	adantl 469	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝑘 ∈ ℕ)
116	113, 115, 41	syl2anc 575	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝐴 ∈ (0[,]+∞))
117	116	ralrimiva 3165	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
118		nfcv 2959	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...(𝑛 + 1))
119	118	esumcl 30439	. . . . . . . 8 ⊢ (((1...(𝑛 + 1)) ∈ V ∧ ∀𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
120	112, 117, 119	sylancr 577	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
121	105, 110, 111, 120	fvmptd 6518	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘(𝑛 + 1)) = Σ^*𝑘 ∈ (1...(𝑛 + 1))𝐴)
122	97, 99, 121	3brtr4d 4887	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
123		simpr 473	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐹 ∈ dom ⇝ )
124	29, 89, 122, 123	lmdvglim 30347	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)+∞)
125		nfv 2005	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞))
126		nfcv 2959	. . . . . . 7 ⊢ Ⅎ𝑘ℕ
127		nnex 11321	. . . . . . . 8 ⊢ ℕ ∈ V
128	127	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → ℕ ∈ V)
129	41	adantlr 697	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
130		simpr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
131		simpll 774	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
132		inss1 4040	. . . . . . . . . . . . . 14 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
133		simplr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
134	132, 133	sseldi 3807	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ∈ 𝒫 ℕ)
135	134	elpwid 4374	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ⊆ ℕ)
136		simpr 473	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑥)
137	135, 136	sseldd 3810	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ ℕ)
138	131, 137, 13	syl2anc 575	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ (0[,)+∞))
139	138	fmpttd 6616	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ 𝑥 ↦ 𝐴):𝑥⟶(0[,)+∞))
140		esumpfinvallem 30483	. . . . . . . . 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 575	. . . . . . . 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 4041	. . . . . . . . . 10 ⊢ (𝒫 ℕ ∩ Fin) ⊆ Fin
143	142, 130	sseldi 3807	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
144	131, 137, 14	syl2anc 575	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ ℂ)
145	143, 144	gsumfsum 20040	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑥 ↦ 𝐴)) = Σ𝑘 ∈ 𝑥 𝐴)
146	141, 145	eqtr3d 2853	. . . . . . 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 30455	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴), ℝ^, < ))
148	147	adantr 468	. . . . 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 30346	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ∀𝑦 ∈ ℝ ∃𝑙 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛))
150	149	r19.21bi 3131	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → ∃𝑙 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛))
151		nnz 11684	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ ℤ)
152		uzid 11938	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℤ → 𝑙 ∈ (ℤ_≥‘𝑙))
153	151, 152	syl 17	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ (ℤ_≥‘𝑙))
154		simpr 473	. . . . . . . . . . . . . 14 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 = 𝑙) → 𝑛 = 𝑙)
155	154	fveq2d 6421	. . . . . . . . . . . . 13 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 = 𝑙) → (𝐹‘𝑛) = (𝐹‘𝑙))
156	155	breq2d 4867	. . . . . . . . . . . 12 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 = 𝑙) → (𝑦 < (𝐹‘𝑛) ↔ 𝑦 < (𝐹‘𝑙)))
157	153, 156	rspcdv 3516	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ℕ → (∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛) → 𝑦 < (𝐹‘𝑙)))
158	157	reximia 3207	. . . . . . . . . 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 776	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝑦 ∈ ℝ)
161	89	ad2antrr 708	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝐹:ℕ⟶(0[,)+∞))
162		simpr 473	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
163	161, 162	ffvelrnd 6591	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) ∈ (0[,)+∞))
164	4, 163	sseldi 3807	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) ∈ ℝ)
165		ltle 10420	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑙) ∈ ℝ) → (𝑦 < (𝐹‘𝑙) → 𝑦 ≤ (𝐹‘𝑙)))
166	160, 164, 165	syl2anc 575	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝑦 < (𝐹‘𝑙) → 𝑦 ≤ (𝐹‘𝑙)))
167	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝐹 = (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴))
168		oveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙))
169		esumeq1 30443	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑛) = (1...𝑙) → Σ^𝑘 ∈ (1...𝑛)𝐴 = Σ^𝑘 ∈ (1...𝑙)𝐴)
170	168, 169	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑙 → Σ^𝑘 ∈ (1...𝑛)𝐴 = Σ^𝑘 ∈ (1...𝑙)𝐴)
171	170	adantl 469	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑛 = 𝑙) → Σ^𝑘 ∈ (1...𝑛)𝐴 = Σ^𝑘 ∈ (1...𝑙)𝐴)
172		esumex 30438	. . . . . . . . . . . . . . 15 ⊢ Σ^*𝑘 ∈ (1...𝑙)𝐴 ∈ V
173	172	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑙)𝐴 ∈ V)
174	167, 171, 162, 173	fvmptd 6518	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) = Σ^*𝑘 ∈ (1...𝑙)𝐴)
175		fzfid 13015	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin)
176		simp-4l 792	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
177		elfznn 12612	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ)
178	177	adantl 469	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → 𝑘 ∈ ℕ)
179	176, 178, 13	syl2anc 575	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → 𝐴 ∈ (0[,)+∞))
180	175, 179	esumpfinval 30484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑙)𝐴 = Σ𝑘 ∈ (1...𝑙)𝐴)
181	174, 180	eqtrd 2851	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) = Σ𝑘 ∈ (1...𝑙)𝐴)
182	181	breq2d 4867	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
183	166, 182	sylibd 230	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝑦 < (𝐹‘𝑙) → 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
184	183	reximdva 3215	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → (∃𝑙 ∈ ℕ 𝑦 < (𝐹‘𝑙) → ∃𝑙 ∈ ℕ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
185	159, 184	mpd 15	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → ∃𝑙 ∈ ℕ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴)
186		fzssuz 12624	. . . . . . . . . . . . . 14 ⊢ (1...𝑙) ⊆ (ℤ_≥‘1)
187	186, 1	sseqtr4i 3846	. . . . . . . . . . . . 13 ⊢ (1...𝑙) ⊆ ℕ
188		ovex 6915	. . . . . . . . . . . . . 14 ⊢ (1...𝑙) ∈ V
189	188	elpw 4368	. . . . . . . . . . . . 13 ⊢ ((1...𝑙) ∈ 𝒫 ℕ ↔ (1...𝑙) ⊆ ℕ)
190	187, 189	mpbir 222	. . . . . . . . . . . 12 ⊢ (1...𝑙) ∈ 𝒫 ℕ
191		fzfi 13014	. . . . . . . . . . . 12 ⊢ (1...𝑙) ∈ Fin
192		elin 4006	. . . . . . . . . . . 12 ⊢ ((1...𝑙) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑙) ∈ 𝒫 ℕ ∧ (1...𝑙) ∈ Fin))
193	190, 191, 192	mpbir2an 693	. . . . . . . . . . 11 ⊢ (1...𝑙) ∈ (𝒫 ℕ ∩ Fin)
194		sumex 14660	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (1...𝑙)𝐴 ∈ V
195		eqid 2817	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) = (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)
196		sumeq1 14661	. . . . . . . . . . . 12 ⊢ (𝑥 = (1...𝑙) → Σ𝑘 ∈ 𝑥 𝐴 = Σ𝑘 ∈ (1...𝑙)𝐴)
197	195, 196	elrnmpt1s 5587	. . . . . . . . . . 11 ⊢ (((1...𝑙) ∈ (𝒫 ℕ ∩ Fin) ∧ Σ𝑘 ∈ (1...𝑙)𝐴 ∈ V) → Σ𝑘 ∈ (1...𝑙)𝐴 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴))
198	193, 194, 197	mp2an 675	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (1...𝑙)𝐴 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)
199		nfv 2005	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴
200		breq2 4859	. . . . . . . . . . 11 ⊢ (𝑧 = Σ𝑘 ∈ (1...𝑙)𝐴 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
201	199, 200	rspce 3508	. . . . . . . . . 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)
202	198, 201	mpan 673	. . . . . . . . 9 ⊢ (𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴 → ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
203	202	rexlimivw 3228	. . . . . . . 8 ⊢ (∃𝑙 ∈ ℕ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴 → ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
204	185, 203	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
205	204	ralrimiva 3165	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
206		simpr 473	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
207	142, 206	sseldi 3807	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
208	138	adantllr 701	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ (0[,)+∞))
209	4, 208	sseldi 3807	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ ℝ)
210	207, 209	fsumrecl 14707	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐴 ∈ ℝ)
211	210	rexrd 10383	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐴 ∈ ℝ^*)
212	211	fmpttd 6616	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴):(𝒫 ℕ ∩ Fin)⟶ℝ^*)
213		frn 6271	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴):(𝒫 ℕ ∩ Fin)⟶ℝ^* → ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) ⊆ ℝ^*)
214		supxrunb1 12386	. . . . . . 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^*, < ) = +∞))
215	212, 213, 214	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^*, < ) = +∞))
216	205, 215	mpbid 223	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → sup(ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴), ℝ^*, < ) = +∞)
217	148, 216	eqtrd 2851	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → Σ^*𝑘 ∈ ℕ𝐴 = +∞)
218	124, 217	breqtrrd 4883	. . 3 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
219	88, 218	pm2.61dan 838	. 2 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
220	16	reseq1i 5606	. . . . . . . 8 ⊢ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘))
221		eleq1w 2879	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (𝑙 ∈ ℕ ↔ 𝑘 ∈ ℕ))
222	221	anbi2d 616	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((𝜑 ∧ 𝑙 ∈ ℕ) ↔ (𝜑 ∧ 𝑘 ∈ ℕ)))
223		sbequ12r 2281	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ([𝑙 / 𝑘]𝐴 = +∞ ↔ 𝐴 = +∞))
224	222, 223	anbi12d 618	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ↔ ((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞)))
225		fveq2 6417	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑘))
226	225	reseq2d 5610	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)))
227	225	xpeq1d 5352	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((ℤ_≥‘𝑙) × {+∞}) = ((ℤ_≥‘𝑘) × {+∞}))
228	226, 227	eqeq12d 2832	. . . . . . . . . 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) × {+∞})))
229	224, 228	imbi12d 335	. . . . . . . . 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) × {+∞}))))
230		nfv 2005	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑙 ∈ ℕ)
231		nfs1v 2288	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘[𝑙 / 𝑘]𝐴 = +∞
232	230, 231	nfan 1990	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞)
233		nfv 2005	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑛 ∈ (ℤ_≥‘𝑙)
234	232, 233	nfan 1990	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙))
235		ovexd 6917	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → (1...𝑛) ∈ V)
236		simp-4l 792	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
237	18	adantl 469	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
238	236, 237, 41	syl2anc 575	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
239		simpllr 784	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ ℕ)
240		elnnuz 11961	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ ℕ ↔ 𝑙 ∈ (ℤ_≥‘1))
241		eluzfz 12579	. . . . . . . . . . . . . . 15 ⊢ ((𝑙 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ (1...𝑛))
242	240, 241	sylanb 572	. . . . . . . . . . . . . 14 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ (1...𝑛))
243	239, 242	sylancom 578	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ (1...𝑛))
244		simplr 776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → [𝑙 / 𝑘]𝐴 = +∞)
245		sbequ12 2280	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → (𝐴 = +∞ ↔ [𝑙 / 𝑘]𝐴 = +∞))
246	231, 245	rspce 3508	. . . . . . . . . . . . 13 ⊢ ((𝑙 ∈ (1...𝑛) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ∃𝑘 ∈ (1...𝑛)𝐴 = +∞)
247	243, 244, 246	syl2anc 575	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → ∃𝑘 ∈ (1...𝑛)𝐴 = +∞)
248	234, 235, 238, 247	esumpinfval 30482	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞)
249	248	ralrimiva 3165	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞)
250		eqidd 2818	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑙))
251		mpteq12 4941	. . . . . . . . . . . 12 ⊢ (((ℤ_≥‘𝑙) = (ℤ_≥‘𝑙) ∧ ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^𝑘 ∈ (1...𝑛)𝐴 = +∞) → (𝑛 ∈ (ℤ_≥‘𝑙) ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ +∞))
252	250, 251	sylan 571	. . . . . . . . . . 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) ↦ +∞))
253		simplr 776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → 𝑙 ∈ ℕ)
254		uznnssnn 11972	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℕ → (ℤ_≥‘𝑙) ⊆ ℕ)
255		resmpt 5667	. . . . . . . . . . . . 13 ⊢ ((ℤ_≥‘𝑙) ⊆ ℕ → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ Σ^𝑘 ∈ (1...𝑛)𝐴))
256	253, 254, 255	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))
257	256	adantr 468	. . . . . . . . . . 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))
258		fconstmpt 5376	. . . . . . . . . . . 12 ⊢ ((ℤ_≥‘𝑙) × {+∞}) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ +∞)
259	258	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞) → ((ℤ_≥‘𝑙) × {+∞}) = (𝑛 ∈ (ℤ_≥‘𝑙) ↦ +∞))
260	252, 257, 259	3eqtr4d 2861	. . . . . . . . . 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) × {+∞}))
261	249, 260	mpdan 670	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((ℤ_≥‘𝑙) × {+∞}))
262	229, 261	chvarv 2439	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
263	220, 262	syl5eq 2863	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞) → (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
264	263	ex 399	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 = +∞ → (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})))
265	264	reximdva 3215	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ℕ 𝐴 = +∞ → ∃𝑘 ∈ ℕ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})))
266	265	imp 395	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → ∃𝑘 ∈ ℕ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
267		xrge0topn 30336	. . . . . . . . . . 11 ⊢ (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
268	29, 267	eqtri 2839	. . . . . . . . . 10 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
269		letopon 21243	. . . . . . . . . . 11 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
270		iccssxr 12493	. . . . . . . . . . 11 ⊢ (0[,]+∞) ⊆ ℝ^*
271		resttopon 21199	. . . . . . . . . . 11 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ (0[,]+∞) ⊆ ℝ^) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)))
272	269, 270, 271	mp2an 675	. . . . . . . . . 10 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))
273	268, 272	eqeltri 2892	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
274	273	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ (TopOn‘(0[,]+∞)))
275		0xr 10380	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
276		pnfxr 10386	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
277		0lepnf 12201	. . . . . . . . . 10 ⊢ 0 ≤ +∞
278		ubicc2 12528	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
279	275, 276, 277, 278	mp3an 1578	. . . . . . . . 9 ⊢ +∞ ∈ (0[,]+∞)
280	279	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → +∞ ∈ (0[,]+∞))
281	40	nnzd 11766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
282		eqid 2817	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑘) = (ℤ_≥‘𝑘)
283	282	lmconst 21299	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘(0[,]+∞)) ∧ +∞ ∈ (0[,]+∞) ∧ 𝑘 ∈ ℤ) → ((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞)
284	274, 280, 281, 283	syl3anc 1483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞)
285		breq1 4858	. . . . . . . 8 ⊢ ((𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}) → ((𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞ ↔ ((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞))
286	285	biimprd 239	. . . . . . 7 ⊢ ((𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}) → (((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞ → (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞))
287	284, 286	mpan9 498	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})) → (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞)
288		ovexd 6917	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0[,]+∞) ∈ V)
289		cnex 10311	. . . . . . . . . 10 ⊢ ℂ ∈ V
290	289	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ℂ ∈ V)
291	56	adantr 468	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(0[,]+∞))
292		nnsscn 11319	. . . . . . . . . 10 ⊢ ℕ ⊆ ℂ
293	292	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ℕ ⊆ ℂ)
294		elpm2r 8119	. . . . . . . . 9 ⊢ ((((0[,]+∞) ∈ V ∧ ℂ ∈ V) ∧ (𝐹:ℕ⟶(0[,]+∞) ∧ ℕ ⊆ ℂ)) → 𝐹 ∈ ((0[,]+∞) ↑_pm ℂ))
295	288, 290, 291, 293, 294	syl22anc 858	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹 ∈ ((0[,]+∞) ↑_pm ℂ))
296	274, 295, 281	lmres 21338	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹(⇝_𝑡‘𝐽)+∞ ↔ (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞))
297	296	biimpar 465	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞) → 𝐹(⇝_𝑡‘𝐽)+∞)
298	287, 297	syldan 581	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})) → 𝐹(⇝_𝑡‘𝐽)+∞)
299	298	r19.29an 3276	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})) → 𝐹(⇝_𝑡‘𝐽)+∞)
300	266, 299	syldan 581	. . 3 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → 𝐹(⇝_𝑡‘𝐽)+∞)
301		nfv 2005	. . . . 5 ⊢ Ⅎ𝑘𝜑
302		nfre1 3203	. . . . 5 ⊢ Ⅎ𝑘∃𝑘 ∈ ℕ 𝐴 = +∞
303	301, 302	nfan 1990	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞)
304	127	a1i 11	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → ℕ ∈ V)
305	41	adantlr 697	. . . 4 ⊢ (((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
306		simpr 473	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → ∃𝑘 ∈ ℕ 𝐴 = +∞)
307	303, 304, 305, 306	esumpinfval 30482	. . 3 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → Σ^*𝑘 ∈ ℕ𝐴 = +∞)
308	300, 307	breqtrrd 4883	. 2 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
309		eleq1w 2879	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ ℕ ↔ 𝑚 ∈ ℕ))
310	309	anbi2d 616	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝜑 ∧ 𝑘 ∈ ℕ) ↔ (𝜑 ∧ 𝑚 ∈ ℕ)))
311	7	eleq1d 2881	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝐴 ∈ (0[,]+∞) ↔ 𝐵 ∈ (0[,]+∞)))
312	310, 311	imbi12d 335	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ (0[,]+∞))))
313	312, 41	chvarv 2439	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ (0[,]+∞))
314		eliccelico 29888	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞)))
315	275, 276, 277, 314	mp3an 1578	. . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
316	313, 315	sylib 209	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
317	316	ralrimiva 3165	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
318		r19.30 3281	. . . 4 ⊢ (∀𝑚 ∈ ℕ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞) → (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑚 ∈ ℕ 𝐵 = +∞))
319	317, 318	syl 17	. . 3 ⊢ (𝜑 → (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑚 ∈ ℕ 𝐵 = +∞))
320	7	eqeq1d 2819	. . . . 5 ⊢ (𝑘 = 𝑚 → (𝐴 = +∞ ↔ 𝐵 = +∞))
321	320	cbvrexv 3372	. . . 4 ⊢ (∃𝑘 ∈ ℕ 𝐴 = +∞ ↔ ∃𝑚 ∈ ℕ 𝐵 = +∞)
322	321	orbi2i 927	. . 3 ⊢ ((∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑘 ∈ ℕ 𝐴 = +∞) ↔ (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑚 ∈ ℕ 𝐵 = +∞))
323	319, 322	sylibr 225	. 2 ⊢ (𝜑 → (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑘 ∈ ℕ 𝐴 = +∞))
324	219, 308, 323	mpjaodan 972	1 ⊢ (𝜑 → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 197 ∧ wa 384 ∨ wo 865 ∧ w3a 1100 = wceq 1637 [wsb 2061 ∈ wcel 2157 ∀wral 3107 ∃wrex 3108 Vcvv 3402 ∩ cin 3779 ⊆ wss 3780 𝒫 cpw 4362 {csn 4381 class class class wbr 4855 ↦ cmpt 4934 × cxp 5322 dom cdm 5324 ran crn 5325 ↾ cres 5326 Fn wfn 6105 ⟶wf 6106 ‘cfv 6110 (class class class)co 6883 ↑_pm cpm 8102 Fincfn 8201 supcsup 8594 ℂcc 10228 ℝcr 10229 0cc0 10230 1c1 10231 + caddc 10233 +∞cpnf 10365 ℝ^*cxr 10367 < clt 10368 ≤ cle 10369 ℕcn 11314 ℤcz 11662 ℤ_≥cuz 11923 [,)cico 12414 [,]cicc 12415 ...cfz 12568 seqcseq 13043 ⇝ cli 14457 Σcsu 14658 ↾_s cress 16088 ↾_t crest 16305 TopOpenctopn 16306 Σ_g cgsu 16325 ordTopcordt 16383 ℝ^*_𝑠cxrs 16384 ℂ_fldccnfld 19973 TopOnctopon 20948 ⇝_𝑡clm 21264 Σ^*cesum 30436

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2069 ax-7 2105 ax-8 2159 ax-9 2166 ax-10 2186 ax-11 2202 ax-12 2215 ax-13 2422 ax-ext 2795 ax-rep 4977 ax-sep 4988 ax-nul 4996 ax-pow 5048 ax-pr 5109 ax-un 7188 ax-inf2 8794 ax-cnex 10286 ax-resscn 10287 ax-1cn 10288 ax-icn 10289 ax-addcl 10290 ax-addrcl 10291 ax-mulcl 10292 ax-mulrcl 10293 ax-mulcom 10294 ax-addass 10295 ax-mulass 10296 ax-distr 10297 ax-i2m1 10298 ax-1ne0 10299 ax-1rid 10300 ax-rnegex 10301 ax-rrecex 10302 ax-cnre 10303 ax-pre-lttri 10304 ax-pre-lttrn 10305 ax-pre-ltadd 10306 ax-pre-mulgt0 10307 ax-pre-sup 10308 ax-addf 10309 ax-mulf 10310

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-fal 1651 df-ex 1860 df-nf 1864 df-sb 2062 df-mo 2635 df-eu 2642 df-clab 2804 df-cleq 2810 df-clel 2813 df-nfc 2948 df-ne 2990 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3404 df-sbc 3645 df-csb 3740 df-dif 3783 df-un 3785 df-in 3787 df-ss 3794 df-pss 3796 df-nul 4128 df-if 4291 df-pw 4364 df-sn 4382 df-pr 4384 df-tp 4386 df-op 4388 df-uni 4642 df-int 4681 df-iun 4725 df-iin 4726 df-br 4856 df-opab 4918 df-mpt 4935 df-tr 4958 df-id 5232 df-eprel 5237 df-po 5245 df-so 5246 df-fr 5283 df-se 5284 df-we 5285 df-xp 5330 df-rel 5331 df-cnv 5332 df-co 5333 df-dm 5334 df-rn 5335 df-res 5336 df-ima 5337 df-pred 5906 df-ord 5952 df-on 5953 df-lim 5954 df-suc 5955 df-iota 6073 df-fun 6112 df-fn 6113 df-f 6114 df-f1 6115 df-fo 6116 df-f1o 6117 df-fv 6118 df-isom 6119 df-riota 6844 df-ov 6886 df-oprab 6887 df-mpt2 6888 df-of 7136 df-om 7305 df-1st 7407 df-2nd 7408 df-supp 7539 df-wrecs 7651 df-recs 7713 df-rdg 7751 df-1o 7805 df-2o 7806 df-oadd 7809 df-er 7988 df-map 8103 df-pm 8104 df-ixp 8155 df-en 8202 df-dom 8203 df-sdom 8204 df-fin 8205 df-fsupp 8524 df-fi 8565 df-sup 8596 df-inf 8597 df-oi 8663 df-card 9057 df-cda 9284 df-pnf 10370 df-mnf 10371 df-xr 10372 df-ltxr 10373 df-le 10374 df-sub 10562 df-neg 10563 df-div 10979 df-nn 11315 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 df-9 11382 df-n0 11579 df-xnn0 11649 df-z 11663 df-dec 11779 df-uz 11924 df-q 12027 df-rp 12066 df-xneg 12181 df-xadd 12182 df-xmul 12183 df-ioo 12416 df-ioc 12417 df-ico 12418 df-icc 12419 df-fz 12569 df-fzo 12709 df-fl 12836 df-mod 12912 df-seq 13044 df-exp 13103 df-fac 13300 df-bc 13329 df-hash 13357 df-shft 14049 df-cj 14081 df-re 14082 df-im 14083 df-sqrt 14217 df-abs 14218 df-limsup 14444 df-clim 14461 df-rlim 14462 df-sum 14659 df-ef 15037 df-sin 15039 df-cos 15040 df-pi 15042 df-struct 16089 df-ndx 16090 df-slot 16091 df-base 16093 df-sets 16094 df-ress 16095 df-plusg 16185 df-mulr 16186 df-starv 16187 df-sca 16188 df-vsca 16189 df-ip 16190 df-tset 16191 df-ple 16192 df-ds 16194 df-unif 16195 df-hom 16196 df-cco 16197 df-rest 16307 df-topn 16308 df-0g 16326 df-gsum 16327 df-topgen 16328 df-pt 16329 df-prds 16332 df-ordt 16385 df-xrs 16386 df-qtop 16391 df-imas 16392 df-xps 16394 df-mre 16470 df-mrc 16471 df-acs 16473 df-ps 17424 df-tsr 17425 df-plusf 17465 df-mgm 17466 df-sgrp 17508 df-mnd 17519 df-mhm 17559 df-submnd 17560 df-grp 17649 df-minusg 17650 df-sbg 17651 df-mulg 17765 df-subg 17812 df-cntz 17970 df-cmn 18415 df-abl 18416 df-mgp 18711 df-ur 18723 df-ring 18770 df-cring 18771 df-subrg 19001 df-abv 19040 df-lmod 19088 df-scaf 19089 df-sra 19400 df-rgmod 19401 df-psmet 19965 df-xmet 19966 df-met 19967 df-bl 19968 df-mopn 19969 df-fbas 19970 df-fg 19971 df-cnfld 19974 df-top 20932 df-topon 20949 df-topsp 20971 df-bases 20984 df-cld 21057 df-ntr 21058 df-cls 21059 df-nei 21136 df-lp 21174 df-perf 21175 df-cn 21265 df-cnp 21266 df-lm 21267 df-haus 21353 df-tx 21599 df-hmeo 21792 df-fil 21883 df-fm 21975 df-flim 21976 df-flf 21977 df-tmd 22109 df-tgp 22110 df-tsms 22163 df-trg 22196 df-xms 22358 df-ms 22359 df-tms 22360 df-nm 22620 df-ngp 22621 df-nrg 22623 df-nlm 22624 df-ii 22913 df-cncf 22914 df-limc 23866 df-dv 23867 df-log 24539 df-esum 30437

This theorem is referenced by: esumcvg2 30496

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