esumcvg - Mathbox for Thierry Arnoux

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Theorem esumcvg 33382

Description: The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 15677. (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 12869	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12597	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 1 ∈ ℤ)
3		simpr 483	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ )
4		rge0ssre 13437	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
5		ax-resscn 11169	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
6	4, 5	sstri 3990	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℂ
7		esumcvg.m	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → 𝐴 = 𝐵)
8	7	eleq1d 2816	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝐴 ∈ (0[,)+∞) ↔ 𝐵 ∈ (0[,)+∞)))
9	8	cbvralvw 3232	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) ↔ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞))
10		rsp 3242	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) → (𝑘 ∈ ℕ → 𝐴 ∈ (0[,)+∞)))
11	9, 10	sylbir 234	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) → (𝑘 ∈ ℕ → 𝐴 ∈ (0[,)+∞)))
12	11	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → (𝑘 ∈ ℕ → 𝐴 ∈ (0[,)+∞)))
13	12	imp 405	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
14	6, 13	sselid 3979	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
15	14	adantlr 711	. . . . . 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 13942	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
18		elfznn 13534	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
19	18, 13	sylan2 591	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,)+∞))
20	19	adantlr 711	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,)+∞))
21	17, 20	esumpfinval 33371	. . . . . . . . . 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 2782	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
24	6, 20	sselid 3979	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
25	17, 24	fsumcl 15683	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ∈ ℂ)
26	23, 25	fvmpt2d 7010	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
27	26	adantlr 711	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
28	1, 2, 3, 15, 27	isumclim3 15709	. . . . 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 32463	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,)+∞))
31	21, 30	eqeltrd 2831	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,)+∞))
32	31, 16	fmptd 7114	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹:ℕ⟶(0[,)+∞))
33	32	adantr 479	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹:ℕ⟶(0[,)+∞))
34		simplll 771	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝜑)
35		eqidd 2731	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑚 ∈ ℕ ↦ 𝐵) = (𝑚 ∈ ℕ ↦ 𝐵))
36		eqcom 2737	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 ↔ 𝑚 = 𝑘)
37		eqcom 2737	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
38	7, 36, 37	3imtr3i 290	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → 𝐵 = 𝐴)
39	38	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 = 𝑘) → 𝐵 = 𝐴)
40		simpr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
41		esumcvg.a	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
42	35, 39, 40, 41	fvmptd 7004	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
43	34, 42	sylancom 586	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
44	13	adantlr 711	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
45		elrege0 13435	. . . . . . . . . 10 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
46	44, 45	sylib 217	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
47	46	simpld 493	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
48		ovex 7444	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ∈ V
49		simpll 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
50	18	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
51	49, 50, 41	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
52	51	ralrimiva 3144	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
53		nfcv 2901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(1...𝑛)
54	53	esumcl 33326	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑛) ∈ V ∧ ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
55	48, 52, 54	sylancr 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)𝐴 ∈ (0[,]+∞))
56	55, 16	fmptd 7114	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶(0[,]+∞))
57	56	ffnd 6717	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
58	57	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 Fn ℕ)
59		1z 12596	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
60		seqfn 13982	. . . . . . . . . . . . . 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 6646	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
63	61, 62	mpbir 230	. . . . . . . . . . . 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 771	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
66	18, 42	sylan2 591	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑛)) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
67	65, 66	sylancom 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑚 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
68		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
69	68, 1	eleqtrdi 2841	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
70	67, 69, 24	fsumser 15680	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵))‘𝑛))
71	26, 70	eqtrd 2770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (seq1( + , (𝑚 ∈ ℕ ↦ 𝐵))‘𝑛))
72	58, 64, 71	eqfnfvd 7034	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹 = seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)))
73	72	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 = seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)))
74	73, 3	eqeltrrd 2832	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → seq1( + , (𝑚 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
75	1, 2, 43, 47, 74	isumrecl 15715	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
76	46	simprd 494	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
77	1, 2, 43, 47, 74, 76	isumge0 15716	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 0 ≤ Σ𝑘 ∈ ℕ 𝐴)
78		elrege0 13435	. . . . . . 7 ⊢ (Σ𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞) ↔ (Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ ℕ 𝐴))
79	75, 77, 78	sylanbrc 581	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → Σ𝑘 ∈ ℕ 𝐴 ∈ (0[,)+∞))
80		ssid 4003	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,)+∞)
81	29, 33, 79, 80	lmlimxrge0 33226	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝐹(⇝_𝑡‘𝐽)Σ𝑘 ∈ ℕ 𝐴 ↔ 𝐹 ⇝ Σ𝑘 ∈ ℕ 𝐴))
82	28, 81	mpbird 256	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)Σ𝑘 ∈ ℕ 𝐴)
83	16, 3	eqeltrrid 2836	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
84	22	eleq1d 2816	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ ↔ (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ ))
85	84	adantr 479	. . . . . 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 231	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝐹 ∈ dom ⇝ ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
87	44, 7, 86	esumpcvgval 33374	. . . 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 479	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹:ℕ⟶(0[,)+∞))
90		simpr 483	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
91	90	nnzd 12589	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
92		uzid 12841	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
93		peano2uz 12889	. . . . . . . 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 771	. . . . . . . 8 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
96	95, 13	sylancom 586	. . . . . . 7 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
97	90, 94, 96	esumpmono 33375	. . . . . 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 2773	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ^*𝑘 ∈ (1...𝑛)𝐴)
99	98	adantlr 711	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = Σ^*𝑘 ∈ (1...𝑛)𝐴)
100		oveq2 7419	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑛 → (1...𝑙) = (1...𝑛))
101		esumeq1 33330	. . . . . . . . . . 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 2761	. . . . . . . 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 1194	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ (¬ 𝐹 ∈ dom ⇝ ∧ 𝑛 ∈ ℕ ∧ 𝑙 = (𝑛 + 1))) → 𝑙 = (𝑛 + 1))
107		oveq2 7419	. . . . . . . . 9 ⊢ (𝑙 = (𝑛 + 1) → (1...𝑙) = (1...(𝑛 + 1)))
108		esumeq1 33330	. . . . . . . . 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 1358	. . . . . . 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 12233	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
112		ovex 7444	. . . . . . . 8 ⊢ (1...(𝑛 + 1)) ∈ V
113		simp-4l 779	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝜑)
114		elfznn 13534	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑛 + 1)) → 𝑘 ∈ ℕ)
115	114	adantl 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝑘 ∈ ℕ)
116	113, 115, 41	syl2anc 582	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑛 + 1))) → 𝐴 ∈ (0[,]+∞))
117	116	ralrimiva 3144	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
118		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑘(1...(𝑛 + 1))
119	118	esumcl 33326	. . . . . . . 8 ⊢ (((1...(𝑛 + 1)) ∈ V ∧ ∀𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
120	112, 117, 119	sylancr 585	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...(𝑛 + 1))𝐴 ∈ (0[,]+∞))
121	105, 110, 111, 120	fvmptd 7004	. . . . . 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 483	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐹 ∈ dom ⇝ )
124	29, 89, 122, 123	lmdvglim 33232	. . . 4 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → 𝐹(⇝_𝑡‘𝐽)+∞)
125		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞))
126		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑘ℕ
127		nnex 12222	. . . . . . . 8 ⊢ ℕ ∈ V
128	127	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → ℕ ∈ V)
129	41	adantlr 711	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
130		simpr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
131		simpll 763	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
132		inss1 4227	. . . . . . . . . . . . . 14 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
133		simplr 765	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
134	132, 133	sselid 3979	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ∈ 𝒫 ℕ)
135	134	elpwid 4610	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑥 ⊆ ℕ)
136		simpr 483	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑥)
137	135, 136	sseldd 3982	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ ℕ)
138	131, 137, 13	syl2anc 582	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ (0[,)+∞))
139	138	fmpttd 7115	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ 𝑥 ↦ 𝐴):𝑥⟶(0[,)+∞))
140		esumpfinvallem 33370	. . . . . . . . 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 582	. . . . . . . 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 4228	. . . . . . . . . 10 ⊢ (𝒫 ℕ ∩ Fin) ⊆ Fin
143	142, 130	sselid 3979	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
144	131, 137, 14	syl2anc 582	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ ℂ)
145	143, 144	gsumfsum 21212	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑥 ↦ 𝐴)) = Σ𝑘 ∈ 𝑥 𝐴)
146	141, 145	eqtr3d 2772	. . . . . . 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 33342	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴), ℝ^, < ))
148	147	adantr 479	. . . . 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 33231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ∀𝑦 ∈ ℝ ∃𝑙 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛))
150	149	r19.21bi 3246	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) → ∃𝑙 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛))
151		nnz 12583	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ ℤ)
152		uzid 12841	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℤ → 𝑙 ∈ (ℤ_≥‘𝑙))
153	151, 152	syl 17	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ (ℤ_≥‘𝑙))
154		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 = 𝑙) → 𝑛 = 𝑙)
155	154	fveq2d 6894	. . . . . . . . . . . . 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 3603	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ℕ → (∀𝑛 ∈ (ℤ_≥‘𝑙)𝑦 < (𝐹‘𝑛) → 𝑦 < (𝐹‘𝑙)))
158	157	reximia 3079	. . . . . . . . . 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 765	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝑦 ∈ ℝ)
161	89	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝐹:ℕ⟶(0[,)+∞))
162		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
163	161, 162	ffvelcdmd 7086	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) ∈ (0[,)+∞))
164	4, 163	sselid 3979	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) ∈ ℝ)
165		ltle 11306	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑙) ∈ ℝ) → (𝑦 < (𝐹‘𝑙) → 𝑦 ≤ (𝐹‘𝑙)))
166	160, 164, 165	syl2anc 582	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝑦 < (𝐹‘𝑙) → 𝑦 ≤ (𝐹‘𝑙)))
167		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙))
168		esumeq1 33330	. . . . . . . . . . . . . . 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 33325	. . . . . . . . . . . . . . 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 7020	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝐹‘𝑙) = Σ^*𝑘 ∈ (1...𝑙)𝐴)
173		fzfid 13942	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin)
174		simp-4l 779	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)))
175		elfznn 13534	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ)
176	175	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → 𝑘 ∈ ℕ)
177	174, 176, 13	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → 𝐴 ∈ (0[,)+∞))
178	173, 177	esumpfinval 33371	. . . . . . . . . . . . 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 2770	. . . . . . . . . . . 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 238	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑦 ∈ ℝ) ∧ 𝑙 ∈ ℕ) → (𝑦 < (𝐹‘𝑙) → 𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴))
182	181	reximdva 3166	. . . . . . . . 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 13546	. . . . . . . . . . . . . 14 ⊢ (1...𝑙) ⊆ (ℤ_≥‘1)
185	184, 1	sseqtrri 4018	. . . . . . . . . . . . 13 ⊢ (1...𝑙) ⊆ ℕ
186		ovex 7444	. . . . . . . . . . . . . 14 ⊢ (1...𝑙) ∈ V
187	186	elpw 4605	. . . . . . . . . . . . 13 ⊢ ((1...𝑙) ∈ 𝒫 ℕ ↔ (1...𝑙) ⊆ ℕ)
188	185, 187	mpbir 230	. . . . . . . . . . . 12 ⊢ (1...𝑙) ∈ 𝒫 ℕ
189		fzfi 13941	. . . . . . . . . . . 12 ⊢ (1...𝑙) ∈ Fin
190		elin 3963	. . . . . . . . . . . 12 ⊢ ((1...𝑙) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑙) ∈ 𝒫 ℕ ∧ (1...𝑙) ∈ Fin))
191	188, 189, 190	mpbir2an 707	. . . . . . . . . . 11 ⊢ (1...𝑙) ∈ (𝒫 ℕ ∩ Fin)
192		sumex 15638	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (1...𝑙)𝐴 ∈ V
193		eqid 2730	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) = (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)
194		sumeq1 15639	. . . . . . . . . . . 12 ⊢ (𝑥 = (1...𝑙) → Σ𝑘 ∈ 𝑥 𝐴 = Σ𝑘 ∈ (1...𝑙)𝐴)
195	193, 194	elrnmpt1s 5955	. . . . . . . . . . 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 688	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (1...𝑙)𝐴 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)
197		nfv 1915	. . . . . . . . . . 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 3600	. . . . . . . . . 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 686	. . . . . . . . 9 ⊢ (𝑦 ≤ Σ𝑘 ∈ (1...𝑙)𝐴 → ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
201	200	rexlimivw 3149	. . . . . . . 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 3144	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴)𝑦 ≤ 𝑧)
204		simpr 483	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ (𝒫 ℕ ∩ Fin))
205	142, 204	sselid 3979	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
206	138	adantllr 715	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ (0[,)+∞))
207	4, 206	sselid 3979	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐴 ∈ ℝ)
208	205, 207	fsumrecl 15684	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐴 ∈ ℝ)
209	208	rexrd 11268	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐴 ∈ ℝ^*)
210	209	fmpttd 7115	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴):(𝒫 ℕ ∩ Fin)⟶ℝ^*)
211		frn 6723	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴):(𝒫 ℕ ∩ Fin)⟶ℝ^* → ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴) ⊆ ℝ^*)
212		supxrunb1 13302	. . . . . . 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 231	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) ∧ ¬ 𝐹 ∈ dom ⇝ ) → sup(ran (𝑥 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐴), ℝ^*, < ) = +∞)
215	148, 214	eqtrd 2770	. . . 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 809	. 2 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞)) → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)
218	16	reseq1i 5976	. . . . . . . 8 ⊢ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘))
219		eleq1w 2814	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (𝑙 ∈ ℕ ↔ 𝑘 ∈ ℕ))
220	219	anbi2d 627	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((𝜑 ∧ 𝑙 ∈ ℕ) ↔ (𝜑 ∧ 𝑘 ∈ ℕ)))
221		sbequ12r 2242	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ([𝑙 / 𝑘]𝐴 = +∞ ↔ 𝐴 = +∞))
222	220, 221	anbi12d 629	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ↔ ((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞)))
223		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑘))
224	223	reseq2d 5980	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((𝑛 ∈ ℕ ↦ Σ^𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)))
225	223	xpeq1d 5704	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → ((ℤ_≥‘𝑙) × {+∞}) = ((ℤ_≥‘𝑘) × {+∞}))
226	224, 225	eqeq12d 2746	. . . . . . . . . 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 343	. . . . . . . . 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 1915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑙 ∈ ℕ)
229		nfs1v 2151	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘[𝑙 / 𝑘]𝐴 = +∞
230	228, 229	nfan 1900	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞)
231		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑛 ∈ (ℤ_≥‘𝑙)
232	230, 231	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙))
233		ovexd 7446	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → (1...𝑛) ∈ V)
234		simp-4l 779	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
235	18	adantl 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
236	234, 235, 41	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (0[,]+∞))
237		simpllr 772	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ ℕ)
238		elnnuz 12870	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ ℕ ↔ 𝑙 ∈ (ℤ_≥‘1))
239		eluzfz 13500	. . . . . . . . . . . . . . 15 ⊢ ((𝑙 ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ (1...𝑛))
240	238, 239	sylanb 579	. . . . . . . . . . . . . 14 ⊢ ((𝑙 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ (1...𝑛))
241	237, 240	sylancom 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → 𝑙 ∈ (1...𝑛))
242		simplr 765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → [𝑙 / 𝑘]𝐴 = +∞)
243		sbequ12 2241	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → (𝐴 = +∞ ↔ [𝑙 / 𝑘]𝐴 = +∞))
244	229, 243	rspce 3600	. . . . . . . . . . . . 13 ⊢ ((𝑙 ∈ (1...𝑛) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ∃𝑘 ∈ (1...𝑛)𝐴 = +∞)
245	241, 242, 244	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → ∃𝑘 ∈ (1...𝑛)𝐴 = +∞)
246	232, 233, 236, 245	esumpinfval 33369	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) ∧ 𝑛 ∈ (ℤ_≥‘𝑙)) → Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞)
247	246	ralrimiva 3144	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ∀𝑛 ∈ (ℤ_≥‘𝑙)Σ^*𝑘 ∈ (1...𝑛)𝐴 = +∞)
248		eqidd 2731	. . . . . . . . . . . 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 578	. . . . . . . . . . 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 765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → 𝑙 ∈ ℕ)
252		uznnssnn 12883	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ℕ → (ℤ_≥‘𝑙) ⊆ ℕ)
253		resmpt 6036	. . . . . . . . . . . . 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 479	. . . . . . . . . . 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 5737	. . . . . . . . . . . 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 2780	. . . . . . . . . 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 683	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑙 ∈ ℕ) ∧ [𝑙 / 𝑘]𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑙)) = ((ℤ_≥‘𝑙) × {+∞}))
260	227, 259	chvarvv 2000	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞) → ((𝑛 ∈ ℕ ↦ Σ^*𝑘 ∈ (1...𝑛)𝐴) ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
261	218, 260	eqtrid 2782	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝐴 = +∞) → (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
262	261	ex 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 = +∞ → (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})))
263	262	reximdva 3166	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ℕ 𝐴 = +∞ → ∃𝑘 ∈ ℕ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})))
264	263	imp 405	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → ∃𝑘 ∈ ℕ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}))
265		xrge0topn 33221	. . . . . . . . . . 11 ⊢ (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
266	29, 265	eqtri 2758	. . . . . . . . . 10 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
267		letopon 22929	. . . . . . . . . . 11 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
268		iccssxr 13411	. . . . . . . . . . 11 ⊢ (0[,]+∞) ⊆ ℝ^*
269		resttopon 22885	. . . . . . . . . . 11 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ (0[,]+∞) ⊆ ℝ^) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)))
270	267, 268, 269	mp2an 688	. . . . . . . . . 10 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))
271	266, 270	eqeltri 2827	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
272	271	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ (TopOn‘(0[,]+∞)))
273		0xr 11265	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
274		pnfxr 11272	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
275		0lepnf 13116	. . . . . . . . . 10 ⊢ 0 ≤ +∞
276		ubicc2 13446	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
277	273, 274, 275, 276	mp3an 1459	. . . . . . . . 9 ⊢ +∞ ∈ (0[,]+∞)
278	277	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → +∞ ∈ (0[,]+∞))
279	40	nnzd 12589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
280		eqid 2730	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑘) = (ℤ_≥‘𝑘)
281	280	lmconst 22985	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘(0[,]+∞)) ∧ +∞ ∈ (0[,]+∞) ∧ 𝑘 ∈ ℤ) → ((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞)
282	272, 278, 279, 281	syl3anc 1369	. . . . . . 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 247	. . . . . . 7 ⊢ ((𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞}) → (((ℤ_≥‘𝑘) × {+∞})(⇝_𝑡‘𝐽)+∞ → (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞))
285	282, 284	mpan9 505	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})) → (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞)
286		ovexd 7446	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0[,]+∞) ∈ V)
287		cnex 11193	. . . . . . . . . 10 ⊢ ℂ ∈ V
288	287	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ℂ ∈ V)
289	56	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(0[,]+∞))
290		nnsscn 12221	. . . . . . . . . 10 ⊢ ℕ ⊆ ℂ
291	290	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ℕ ⊆ ℂ)
292		elpm2r 8841	. . . . . . . . 9 ⊢ ((((0[,]+∞) ∈ V ∧ ℂ ∈ V) ∧ (𝐹:ℕ⟶(0[,]+∞) ∧ ℕ ⊆ ℂ)) → 𝐹 ∈ ((0[,]+∞) ↑_pm ℂ))
293	286, 288, 289, 291, 292	syl22anc 835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹 ∈ ((0[,]+∞) ↑_pm ℂ))
294	272, 293, 279	lmres 23024	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹(⇝_𝑡‘𝐽)+∞ ↔ (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞))
295	294	biimpar 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐹 ↾ (ℤ_≥‘𝑘))(⇝_𝑡‘𝐽)+∞) → 𝐹(⇝_𝑡‘𝐽)+∞)
296	285, 295	syldan 589	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})) → 𝐹(⇝_𝑡‘𝐽)+∞)
297	296	r19.29an 3156	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ (𝐹 ↾ (ℤ_≥‘𝑘)) = ((ℤ_≥‘𝑘) × {+∞})) → 𝐹(⇝_𝑡‘𝐽)+∞)
298	264, 297	syldan 589	. . 3 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → 𝐹(⇝_𝑡‘𝐽)+∞)
299		nfv 1915	. . . . 5 ⊢ Ⅎ𝑘𝜑
300		nfre1 3280	. . . . 5 ⊢ Ⅎ𝑘∃𝑘 ∈ ℕ 𝐴 = +∞
301	299, 300	nfan 1900	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞)
302	127	a1i 11	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → ℕ ∈ V)
303	41	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
304		simpr 483	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ℕ 𝐴 = +∞) → ∃𝑘 ∈ ℕ 𝐴 = +∞)
305	301, 302, 303, 304	esumpinfval 33369	. . 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 2814	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ ℕ ↔ 𝑚 ∈ ℕ))
308	307	anbi2d 627	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝜑 ∧ 𝑘 ∈ ℕ) ↔ (𝜑 ∧ 𝑚 ∈ ℕ)))
309	7	eleq1d 2816	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝐴 ∈ (0[,]+∞) ↔ 𝐵 ∈ (0[,]+∞)))
310	308, 309	imbi12d 343	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ (0[,]+∞))))
311	310, 41	chvarvv 2000	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ (0[,]+∞))
312		eliccelico 32255	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞)))
313	273, 274, 275, 312	mp3an 1459	. . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
314	311, 313	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
315	314	ralrimiva 3144	. . . 4 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝐵 ∈ (0[,)+∞) ∨ 𝐵 = +∞))
316		r19.30 3118	. . . 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 2732	. . . . 5 ⊢ (𝑘 = 𝑚 → (𝐴 = +∞ ↔ 𝐵 = +∞))
319	318	cbvrexvw 3233	. . . 4 ⊢ (∃𝑘 ∈ ℕ 𝐴 = +∞ ↔ ∃𝑚 ∈ ℕ 𝐵 = +∞)
320	319	orbi2i 909	. . 3 ⊢ ((∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑘 ∈ ℕ 𝐴 = +∞) ↔ (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑚 ∈ ℕ 𝐵 = +∞))
321	317, 320	sylibr 233	. 2 ⊢ (𝜑 → (∀𝑚 ∈ ℕ 𝐵 ∈ (0[,)+∞) ∨ ∃𝑘 ∈ ℕ 𝐴 = +∞))
322	217, 306, 321	mpjaodan 955	1 ⊢ (𝜑 → 𝐹(⇝_𝑡‘𝐽)Σ^*𝑘 ∈ ℕ𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 ∧ w3a 1085 = wceq 1539 [wsb 2065 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 dom cdm 5675 ran crn 5676 ↾ cres 5677 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ↑_pm cpm 8823 Fincfn 8941 supcsup 9437 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 +∞cpnf 11249 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 ℕcn 12216 ℤcz 12562 ℤ_≥cuz 12826 [,)cico 13330 [,]cicc 13331 ...cfz 13488 seqcseq 13970 ⇝ cli 15432 Σcsu 15636 ↾_s cress 17177 ↾_t crest 17370 TopOpenctopn 17371 Σ_g cgsu 17390 ordTopcordt 17449 ℝ^*_𝑠cxrs 17450 ℂ_fldccnfld 21144 TopOnctopon 22632 ⇝_𝑡clm 22950 Σ^*cesum 33323

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-ordt 17451 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-ps 18523 df-tsr 18524 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-subrng 20434 df-subrg 20459 df-abv 20568 df-lmod 20616 df-scaf 20617 df-sra 20930 df-rgmod 20931 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-lm 22953 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-tmd 23796 df-tgp 23797 df-tsms 23851 df-trg 23884 df-xms 24046 df-ms 24047 df-tms 24048 df-nm 24311 df-ngp 24312 df-nrg 24314 df-nlm 24315 df-ii 24617 df-cncf 24618 df-limc 25615 df-dv 25616 df-log 26301 df-esum 33324

This theorem is referenced by: esumcvg2 33383

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