esumcvg - Mathbox for Thierry Arnoux

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Theorem esumcvg 34066

Description: The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 15759. (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 12918	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12645	. . . . . 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 13492	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
5		ax-resscn 11209	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
6	4, 5	sstri 4004	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℂ
7		esumcvg.m	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → 𝐴 = 𝐵)
8	7	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝐴 ∈ (0[,)+∞) ↔ 𝐵 ∈ (0[,)+∞)))
9	8	cbvralvw 3234	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) ↔ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞))
10		rsp 3244	. . . . . . . . . . 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 3992	. . . . . . 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 14010	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
18		elfznn 13589	. . . . . . . . . . . . 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 34055	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
22	21	mpteq2dva 5247	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
23	16, 22	eqtrid 2786	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
24	6, 20	sselid 3992	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
25	17, 24	fsumcl 15765	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ∈ ℂ)
26	23, 25	fvmpt2d 7028	. . . . . . 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 15791	. . . . 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 33008	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,)+∞))
31	21, 30	eqeltrd 2838	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,)+∞))
32	31, 16	fmptd 7133	. . . . . . 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 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝜑)
35		eqidd 2735	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑚 ∈ ℕ ↦ 𝐵) = (𝑚 ∈ ℕ ↦ 𝐵))
36		eqcom 2741	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 ↔ 𝑚 = 𝑘)
37		eqcom 2741	. . . . . . . . . . . 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 7022	. . . . . . . . 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 13490	. . . . . . . . . 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 7463	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ∈ V
49		simpll 767	. . . . . . . . . . . . . . . . 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 3143	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
53		nfcv 2902	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(1...𝑛)
54	53	esumcl 34010	. . . . . . . . . . . . . . 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 7133	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶(0[,]+∞))
57	56	ffnd 6737	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
58	57	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 Fn ℕ)
59		1z 12644	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
60		seqfn 14050	. . . . . . . . . . . . . 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 6666	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
70	67, 69, 24	fsumser 15762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵))‘𝑛))
71	26, 70	eqtrd 2774	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵))‘𝑛))
72	58, 64, 71	eqfnfvd 7053	. . . . . . . . . 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 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
75	1, 2, 43, 47, 74	isumrecl 15797	. . . . . . 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 15798	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 0 ≤ Σ𝑘 ∈ ℕ 𝐴)
78		elrege0 13490	. . . . . . 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 4017	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,)+∞)
81	29, 33, 79, 80	lmlimxrge0 33908	. . . . 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 2843	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
84	22	eleq1d 2823	. . . . . . 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 34058	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
88	82, 87	breqtrrd 5175	. . 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 12637	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
92		uzid 12890	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
93		peano2uz 12940	. . . . . . . 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 775	. . . . . . . 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 34059	. . . . . 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 2777	. . . . . . 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 7438	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (1...𝑙) = (1...𝑛))
101		esumeq1 34014	. . . . . . . . . . 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 5260	. . . . . . . . 9 ⊢ (𝑙 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑙)𝐴) = (𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴)
104	16, 103	eqtr4i 2765	. . . . . . . 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 1195	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ (¬ 𝐹 ∈ dom ⇝ ∧ 𝑛 ∈ ℕ ∧ 𝑙 = (𝑛 + 1))) → 𝑙 = (𝑛 + 1))
107		oveq2 7438	. . . . . . . . 9 ⊢ (𝑙 = (𝑛 + 1) → (1...𝑙) = (1...(𝑛 + 1)))
108		esumeq1 34014	. . . . . . . . 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 1359	. . . . . . 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 12280	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
112		ovex 7463	. . . . . . . 8 ⊢ (1...(𝑛 + 1)) ∈ V
113		simp-4l 783	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝜑)
114		elfznn 13589	. . . . . . . . . . 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 3143	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
118		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...(𝑛 + 1))
119	118	esumcl 34010	. . . . . . . 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 7022	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘(𝑛 + 1)) = Σ^*𝑘 ∈ (1...(𝑛 + 1))𝐴)
122	97, 99, 121	3brtr4d 5179	. . . . 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 33914	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)+∞)
125		nfv 1911	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞))
126		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑘ℕ
127		nnex 12269	. . . . . . . 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 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
132		inss1 4244	. . . . . . . . . . . . . 14 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
133		simplr 769	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
134	132, 133	sselid 3992	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ∈ 𝒫 ℕ)
135	134	elpwid 4613	. . . . . . . . . . . 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 3995	. . . . . . . . . . 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 7134	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ 𝑥 ↦ 𝐴):𝑥⟶(0[,)+∞))
140		esumpfinvallem 34054	. . . . . . . . 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 4245	. . . . . . . . . 10 ⊢ (𝒫 ℕ ∩ Fin) ⊆ Fin
143	142, 130	sselid 3992	. . . . . . . . 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 21469	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑥 ↦ 𝐴)) = Σ𝑘 ∈ 𝑥 𝐴)
146	141, 145	eqtr3d 2776	. . . . . . 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 34026	. . . . . 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 33913	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ∀𝑦 ∈ ℝ ∃𝑙 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛))
150	149	r19.21bi 3248	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → ∃𝑙 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛))
151		nnz 12631	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ ℤ)
152		uzid 12890	. . . . . . . . . . . . 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 6910	. . . . . . . . . . . . 13 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 = 𝑙) → (𝐹‘𝑛) = (𝐹‘𝑙))
156	155	breq2d 5159	. . . . . . . . . . . 12 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 = 𝑙) → (𝑦 < (𝐹‘𝑛) ↔ 𝑦 < (𝐹‘𝑙)))
157	153, 156	rspcdv 3613	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ℕ → (∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛) → 𝑦 < (𝐹‘𝑙)))
158	157	reximia 3078	. . . . . . . . . 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 769	. . . . . . . . . . . 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 7104	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) ∈ (0[,)+∞))
164	4, 163	sselid 3992	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) ∈ ℝ)
165		ltle 11346	. . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙))
168		esumeq1 34014	. . . . . . . . . . . . . . 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 34009	. . . . . . . . . . . . . . 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 7038	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) = Σ^*𝑘 ∈ (1...𝑙)𝐴)
173		fzfid 14010	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin)
174		simp-4l 783	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
175		elfznn 13589	. . . . . . . . . . . . . . . 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 34055	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) = Σ𝑘 ∈ (1...𝑙)𝐴)
180	179	breq2d 5159	. . . . . . . . . . 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 3165	. . . . . . . . 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 13601	. . . . . . . . . . . . . 14 ⊢ (1...𝑙) ⊆ (ℤ_≥‘1)
185	184, 1	sseqtrri 4032	. . . . . . . . . . . . 13 ⊢ (1...𝑙) ⊆ ℕ
186		ovex 7463	. . . . . . . . . . . . . 14 ⊢ (1...𝑙) ∈ V
187	186	elpw 4608	. . . . . . . . . . . . 13 ⊢ ((1...𝑙) ∈ 𝒫 ℕ ↔ (1...𝑙) ⊆ ℕ)
188	185, 187	mpbir 231	. . . . . . . . . . . 12 ⊢ (1...𝑙) ∈ 𝒫 ℕ
189		fzfi 14009	. . . . . . . . . . . 12 ⊢ (1...𝑙) ∈ Fin
190		elin 3978	. . . . . . . . . . . 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 15720	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (1...𝑙)𝐴 ∈ V
193		eqid 2734	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) = (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)
194		sumeq1 15721	. . . . . . . . . . . 12 ⊢ (𝑥 = (1...𝑙) → Σ𝑘 ∈ 𝑥 𝐴 = Σ𝑘 ∈ (1...𝑙)𝐴)
195	193, 194	elrnmpt1s 5972	. . . . . . . . . . 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 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴
198		breq2 5151	. . . . . . . . . . 11 ⊢ (𝑧 = Σ𝑘 ∈ (1...𝑙)𝐴 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
199	197, 198	rspce 3610	. . . . . . . . . 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 3148	. . . . . . . 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 3143	. . . . . 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 3992	. . . . . . . . . 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 3992	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ ℝ)
208	205, 207	fsumrecl 15766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐴 ∈ ℝ)
209	208	rexrd 11308	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐴 ∈ ℝ^*)
210	209	fmpttd 7134	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴):(𝒫 ℕ ∩ Fin)⟶ℝ^*)
211		frn 6743	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴):(𝒫 ℕ ∩ Fin)⟶ℝ^* → ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) ⊆ ℝ^*)
212		supxrunb1 13357	. . . . . . 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 2774	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → Σ^*𝑘 ∈ ℕ𝐴 = +∞)
216	124, 215	breqtrrd 5175	. . 3 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
217	88, 216	pm2.61dan 813	. 2 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
218	16	reseq1i 5995	. . . . . . . 8 ⊢ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘))
219		eleq1w 2821	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (𝑙 ∈ ℕ ↔ 𝑘 ∈ ℕ))
220	219	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((𝜑 ∧ 𝑙 ∈ ℕ) ↔ (𝜑 ∧ 𝑘 ∈ ℕ)))
221		sbequ12r 2249	. . . . . . . . . . 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 6906	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑘))
224	223	reseq2d 5999	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)))
225	223	xpeq1d 5717	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((ℤ_≥‘𝑙) × {+∞}) = ((ℤ_≥‘𝑘) × {+∞}))
226	224, 225	eqeq12d 2750	. . . . . . . . . 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 1911	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑙 ∈ ℕ)
229		nfs1v 2153	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘[𝑙 / 𝑘]𝐴 = +∞
230	228, 229	nfan 1896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞)
231		nfv 1911	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑛 ∈ (ℤ_≥‘𝑙)
232	230, 231	nfan 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙))
233		ovexd 7465	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → (1...𝑛) ∈ V)
234		simp-4l 783	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ ℕ)
238		elnnuz 12919	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ ℕ ↔ 𝑙 ∈ (ℤ_≥‘1))
239		eluzfz 13555	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → [𝑙 / 𝑘]𝐴 = +∞)
243		sbequ12 2248	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → (𝐴 = +∞ ↔ [𝑙 / 𝑘]𝐴 = +∞))
244	229, 243	rspce 3610	. . . . . . . . . . . . 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 34053	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞)
247	246	ralrimiva 3143	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞)
248		eqidd 2735	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑙))
249		mpteq12 5239	. . . . . . . . . . . 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 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → 𝑙 ∈ ℕ)
252		uznnssnn 12934	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℕ → (ℤ_≥‘𝑙) ⊆ ℕ)
253		resmpt 6056	. . . . . . . . . . . . 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 5750	. . . . . . . . . . . 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 2784	. . . . . . . . . 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 1995	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
261	218, 260	eqtrid 2786	. . . . . . 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 3165	. . . . 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 33903	. . . . . . . . . . 11 ⊢ (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
266	29, 265	eqtri 2762	. . . . . . . . . 10 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
267		letopon 23228	. . . . . . . . . . 11 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
268		iccssxr 13466	. . . . . . . . . . 11 ⊢ (0[,]+∞) ⊆ ℝ^*
269		resttopon 23184	. . . . . . . . . . 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 2834	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
272	271	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ (TopOn‘(0[,]+∞)))
273		0xr 11305	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
274		pnfxr 11312	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
275		0lepnf 13171	. . . . . . . . . 10 ⊢ 0 ≤ +∞
276		ubicc2 13501	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
277	273, 274, 275, 276	mp3an 1460	. . . . . . . . 9 ⊢ +∞ ∈ (0[,]+∞)
278	277	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → +∞ ∈ (0[,]+∞))
279	40	nnzd 12637	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
280		eqid 2734	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑘) = (ℤ_≥‘𝑘)
281	280	lmconst 23284	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘(0[,]+∞)) ∧ +∞ ∈ (0[,]+∞) ∧ 𝑘 ∈ ℤ) → ((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞)
282	272, 278, 279, 281	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞)
283		breq1 5150	. . . . . . . 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 7465	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0[,]+∞) ∈ V)
287		cnex 11233	. . . . . . . . . 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 12268	. . . . . . . . . 10 ⊢ ℕ ⊆ ℂ
291	290	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ℕ ⊆ ℂ)
292		elpm2r 8883	. . . . . . . . 9 ⊢ ((((0[,]+∞) ∈ V ∧ ℂ ∈ V) ∧ (𝐹:ℕ⟶(0[,]+∞) ∧ ℕ ⊆ ℂ)) → 𝐹 ∈ ((0[,]+∞) ↑_pm ℂ))
293	286, 288, 289, 291, 292	syl22anc 839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹 ∈ ((0[,]+∞) ↑_pm ℂ))
294	272, 293, 279	lmres 23323	. . . . . . 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 3155	. . . 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 1911	. . . . 5 ⊢ Ⅎ𝑘𝜑
300		nfre1 3282	. . . . 5 ⊢ Ⅎ𝑘∃𝑘 ∈ ℕ 𝐴 = +∞
301	299, 300	nfan 1896	. . . 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 34053	. . 3 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → Σ^*𝑘 ∈ ℕ𝐴 = +∞)
306	298, 305	breqtrrd 5175	. 2 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
307		eleq1w 2821	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ ℕ ↔ 𝑚 ∈ ℕ))
308	307	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝜑 ∧ 𝑘 ∈ ℕ) ↔ (𝜑 ∧ 𝑚 ∈ ℕ)))
309	7	eleq1d 2823	. . . . . . . 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 1995	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ (0[,]+∞))
312		eliccelico 32785	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞)))
313	273, 274, 275, 312	mp3an 1460	. . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
314	311, 313	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
315	314	ralrimiva 3143	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
316		r19.30 3117	. . . 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 2736	. . . . 5 ⊢ (𝑘 = 𝑚 → (𝐴 = +∞ ↔ 𝐵 = +∞))
319	318	cbvrexvw 3235	. . . 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 1536 [wsb 2061 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 𝒫 cpw 4604 {csn 4630 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 dom cdm 5688 ran crn 5689 ↾ cres 5690 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ↑_pm cpm 8865 Fincfn 8983 supcsup 9477 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 +∞cpnf 11289 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 ℕcn 12263 ℤcz 12610 ℤ_≥cuz 12875 [,)cico 13385 [,]cicc 13386 ...cfz 13543 seqcseq 14038 ⇝ cli 15516 Σcsu 15718 ↾_s cress 17273 ↾_t crest 17466 TopOpenctopn 17467 Σ_g cgsu 17486 ordTopcordt 17545 ℝ^*_𝑠cxrs 17546 ℂ_fldccnfld 21381 TopOnctopon 22931 ⇝_𝑡clm 23249 Σ^*cesum 34007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-ordt 17547 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-ps 18623 df-tsr 18624 df-plusf 18664 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-subrng 20562 df-subrg 20586 df-abv 20826 df-lmod 20876 df-scaf 20877 df-sra 21189 df-rgmod 21190 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-lm 23252 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-tmd 24095 df-tgp 24096 df-tsms 24150 df-trg 24183 df-xms 24345 df-ms 24346 df-tms 24347 df-nm 24610 df-ngp 24611 df-nrg 24613 df-nlm 24614 df-ii 24916 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612 df-esum 34008

This theorem is referenced by: esumcvg2 34067

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