esumpcvgval - Mathbox for Thierry Arnoux

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Theorem esumpcvgval 31946

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

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

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

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

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

Step	Hyp	Ref	Expression
1		xrltso 12804	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		nnuz 12550	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 12281	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
5		esumpcvgval.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6		esumpcvgval.2	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵)
7		eqcom 2745	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 ↔ 𝑙 = 𝑘)
8		eqcom 2745	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
9	6, 7, 8	3imtr3i 290	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → 𝐵 = 𝐴)
10	9	cbvmptv 5183	. . . . . . . . . 10 ⊢ (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴)
11	5, 10	fmptd 6970	. . . . . . . . 9 ⊢ (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
12	11	ffvelrnda 6943	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞))
13		elrege0 13115	. . . . . . . . 9 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥)))
14	13	simplbi 497	. . . . . . . 8 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ)
15	12, 14	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ)
16	3, 4, 15	serfre 13680	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ)
17	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
18		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
19	18	peano2nnd 11920	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
20	17, 19	ffvelrnd 6944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞))
21		elrege0 13115	. . . . . . . . . 10 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
22	21	simprbi 496	. . . . . . . . 9 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))
23	20, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))
24	16	ffvelrnda 6943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ)
25	21	simplbi 497	. . . . . . . . . 10 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ)
26	20, 25	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ)
27	24, 26	addge01d 11493	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))))
28	23, 27	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
29	18, 3	eleqtrdi 2849	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
30		seqp1 13664	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘1) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
31	29, 30	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
32	28, 31	breqtrrd 5098	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)))
33		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
34	10	fvmpt2 6868	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
35	33, 5, 34	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
36		rge0ssre 13117	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
37	36, 5	sselid 3915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
38	16	feqmptd 6819	. . . . . . . . . 10 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
39		simpll 763	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
40		elfznn 13214	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
41	40	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
42	39, 41, 35	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
43	37	recnd 10934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
44	39, 41, 43	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
45	42, 29, 44	fsumser 15370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
46	45	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
47	46	mpteq2dva 5170	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
48	38, 47	eqtr2d 2779	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) = seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
49		esumpcvgval.3	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
50	48, 49	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
51	3, 4, 35, 37, 50	isumrecl 15405	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
52		1zzd 12281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
53		fzfid 13621	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
54		fzssuz 13226	. . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ (ℤ_≥‘1)
55	54, 3	sseqtrri 3954	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
57	35	adantlr 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
58	37	adantlr 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
59	5	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
60		elrege0 13115	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
61	60	simprbi 496	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (0[,)+∞) → 0 ≤ 𝐴)
62	59, 61	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
63	50	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
64	3, 52, 53, 56, 57, 58, 62, 63	isumless 15485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴)
65	45, 64	eqbrtrrd 5094	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
66	65	ralrimiva 3107	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
67		brralrspcev 5130	. . . . . . 7 ⊢ ((Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴) → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
68	51, 66, 67	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
69	3, 4, 16, 32, 68	climsup 15309	. . . . 5 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
70	3, 4, 69, 24	climrecl 15220	. . . 4 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
71	70	rexrd 10956	. . 3 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^*)
72		eqid 2738	. . . . . . 7 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
73		sumex 15327	. . . . . . 7 ⊢ Σ𝑘 ∈ 𝑏 𝐴 ∈ V
74	72, 73	elrnmpti 5858	. . . . . 6 ⊢ (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
75		ssnnssfz 31010	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚))
76		fzfid 13621	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin)
77		elfznn 13214	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
78	77, 5	sylan2 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
79	60	simplbi 497	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (0[,)+∞) → 𝐴 ∈ ℝ)
80	78, 79	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
81	80	adantlr 711	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
82	78, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
83	82	adantlr 711	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
84		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚))
85	76, 81, 83, 84	fsumless 15436	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
86	85	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
87	86	reximdv 3201	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
88	87	imp 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
89	75, 88	sylan2 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
90		breq1 5073	. . . . . . . . . 10 ⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
91	90	rexbidv 3225	. . . . . . . . 9 ⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
92	89, 91	syl5ibrcom 246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
93	92	rexlimdva 3212	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
94	93	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
95	74, 94	sylan2b 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
96		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
97		inss2 4160	. . . . . . . . . . . . 13 ⊢ (𝒫 ℕ ∩ Fin) ⊆ Fin
98		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
99	97, 98	sselid 3915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ Fin)
100		simpll 763	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝜑)
101		inss1 4159	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
102		simplr 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
103	101, 102	sselid 3915	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ 𝒫 ℕ)
104	103	elpwid 4541	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ⊆ ℕ)
105		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ 𝑏)
106	104, 105	sseldd 3918	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ ℕ)
107	100, 106, 5	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ (0[,)+∞))
108	107, 79	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ ℝ)
109	99, 108	fsumrecl 15374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
110	109	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
111	96, 110	eqeltrd 2839	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
112	111	r19.29an 3216	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
113	74, 112	sylan2b 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ∈ ℝ)
114	113	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ)
115		fzfid 13621	. . . . . . . 8 ⊢ (𝜑 → (1...𝑚) ∈ Fin)
116	115, 80	fsumrecl 15374	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
117	116	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
118	70	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
119		simprr 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
120	16	frnd 6592	. . . . . . . 8 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
121	120	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
122		1nn 11914	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
123	122	ne0ii 4268	. . . . . . . . 9 ⊢ ℕ ≠ ∅
124		dm0rn0 5823	. . . . . . . . . . 11 ⊢ (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅)
125	16	fdmd 6595	. . . . . . . . . . . 12 ⊢ (𝜑 → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
126	125	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
127	124, 126	bitr3id 284	. . . . . . . . . 10 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
128	127	necon3bid 2987	. . . . . . . . 9 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ ≠ ∅))
129	123, 128	mpbiri 257	. . . . . . . 8 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
130	129	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
131		1z 12280	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
132		seqfn 13661	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
133	131, 132	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1)
134	3	fneq2i 6515	. . . . . . . . . . . . . . 15 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
135	133, 134	mpbir 230	. . . . . . . . . . . . . 14 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ
136		dffn5 6810	. . . . . . . . . . . . . 14 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
137	135, 136	mpbi 229	. . . . . . . . . . . . 13 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
138		fvex 6769	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ V
139	137, 138	elrnmpti 5858	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
140		r19.29 3183	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
141		breq1 5073	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧 ≤ 𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠))
142	141	biimparc 479	. . . . . . . . . . . . . 14 ⊢ (((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
143	142	rexlimivw 3210	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
144	140, 143	syl 17	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
145	139, 144	sylan2b 593	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → 𝑧 ≤ 𝑠)
146	145	ralrimiva 3107	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
147	146	reximi 3174	. . . . . . . . 9 ⊢ (∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
148	68, 147	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
149	148	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
150		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
151		simpll 763	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑)
152	77	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ)
153	151, 152, 35	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
154	150, 3	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ_≥‘1))
155	151, 152, 5	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
156	155, 79	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
157	156	recnd 10934	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ)
158	153, 154, 157	fsumser 15370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
159		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
160	159	rspceeqv 3567	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
161	150, 158, 160	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
162	137, 138	elrnmpti 5858	. . . . . . . . 9 ⊢ (Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
163	161, 162	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
164	163	ad2ant2r 743	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
165		suprub 11866	. . . . . . 7 ⊢ (((ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) ∧ Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → Σ𝑘 ∈ (1...𝑚)𝐴 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
166	121, 130, 149, 164, 165	syl31anc 1371	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
167	114, 117, 118, 119, 166	letrd 11062	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
168	95, 167	rexlimddv 3219	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
169	70	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
170	113, 169	lenltd 11051	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → (𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥))
171	168, 170	mpbid 231	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥)
172		simpr1r 1229	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 = +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
173	172	3anassrs 1358	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
174	71	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^*)
175		pnfnlt 12793	. . . . . . . 8 ⊢ (sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^* → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
176	174, 175	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
177		breq1 5073	. . . . . . . . 9 ⊢ (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
178	177	notbid 317	. . . . . . . 8 ⊢ (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
179	178	adantl 481	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
180	176, 179	mpbird 256	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
181	173, 180	pm2.21dd 194	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
182		simplll 771	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝜑)
183		simpr1l 1228	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ^*)
184	183	3anassrs 1358	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ ℝ^*)
185		simplr 765	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 0 ≤ 𝑥)
186		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞)
187		0xr 10953	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
188		pnfxr 10960	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
189		elico1 13051	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^ ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)))
190	187, 188, 189	mp2an 688	. . . . . . 7 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞))
191	184, 185, 186, 190	syl3anbrc 1341	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ (0[,)+∞))
192		simpr1r 1229	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
193	192	3anassrs 1358	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
194	120	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
195	129	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
196	148	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
197	194, 195, 196	3jca 1126	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠))
198		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ (0[,)+∞))
199	36, 198	sselid 3915	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ ℝ)
200		simprr 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
201		suprlub 11869	. . . . . . . . 9 ⊢ (((ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) ∧ 𝑥 ∈ ℝ) → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦))
202	201	biimpa 476	. . . . . . . 8 ⊢ ((((ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
203	197, 199, 200, 202	syl21anc 834	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
204	40	ssriv 3921	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑛) ⊆ ℕ
205		ovex 7288	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑛) ∈ V
206	205	elpw 4534	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑛) ∈ 𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ)
207	204, 206	mpbir 230	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ 𝒫 ℕ
208		fzfi 13620	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ Fin
209		elin 3899	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑛) ∈ 𝒫 ℕ ∧ (1...𝑛) ∈ Fin))
210	207, 208, 209	mpbir2an 707	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ∈ (𝒫 ℕ ∩ Fin)
211	210	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → (1...𝑛) ∈ (𝒫 ℕ ∩ Fin))
212		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
213	45	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
214	212, 213	eqtr4d 2781	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴)
215		sumeq1 15328	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (1...𝑛) → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
216	215	rspceeqv 3567	. . . . . . . . . . . . . 14 ⊢ (((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
217	211, 214, 216	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
218	217	ex 412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
219	218	rexlimdva 3212	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
220	137, 138	elrnmpti 5858	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
221	72, 73	elrnmpti 5858	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
222	219, 220, 221	3imtr4g 295	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)))
223	222	ssrdv 3923	. . . . . . . . 9 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴))
224		ssrexv 3984	. . . . . . . . 9 ⊢ (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦))
225	223, 224	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦))
226	225	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
227	203, 226	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
228	182, 191, 193, 227	syl12anc 833	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
229		simplrl 773	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ^*)
230		xrlelttric 30977	. . . . . . . 8 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
231	188, 230	mpan 686	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
232		xgepnf 12828	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ↔ 𝑥 = +∞))
233	232	orbi1d 913	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → ((+∞ ≤ 𝑥 ∨ 𝑥 < +∞) ↔ (𝑥 = +∞ ∨ 𝑥 < +∞)))
234	231, 233	mpbid 231	. . . . . 6 ⊢ (𝑥 ∈ ℝ^* → (𝑥 = +∞ ∨ 𝑥 < +∞))
235	229, 234	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → (𝑥 = +∞ ∨ 𝑥 < +∞))
236	181, 228, 235	mpjaodan 955	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
237		0elpw 5273	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 ℕ
238		0fin 8916	. . . . . . . . 9 ⊢ ∅ ∈ Fin
239		elin 3899	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ ∧ ∅ ∈ Fin))
240	237, 238, 239	mpbir2an 707	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 ℕ ∩ Fin)
241		sum0 15361	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0
242	241	eqcomi 2747	. . . . . . . 8 ⊢ 0 = Σ𝑘 ∈ ∅ 𝐴
243		sumeq1 15328	. . . . . . . . 9 ⊢ (𝑏 = ∅ → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴)
244	243	rspceeqv 3567	. . . . . . . 8 ⊢ ((∅ ∈ (𝒫 ℕ ∩ Fin) ∧ 0 = Σ𝑘 ∈ ∅ 𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
245	240, 242, 244	mp2an 688	. . . . . . 7 ⊢ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴
246	72, 73	elrnmpti 5858	. . . . . . 7 ⊢ (0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
247	245, 246	mpbir 230	. . . . . 6 ⊢ 0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
248		breq2 5074	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑥 < 𝑦 ↔ 𝑥 < 0))
249	248	rspcev 3552	. . . . . 6 ⊢ ((0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
250	247, 249	mpan 686	. . . . 5 ⊢ (𝑥 < 0 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
251	250	adantl 481	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
252		xrlelttric 30977	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
253	187, 252	mpan 686	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (0 ≤ 𝑥 ∨ 𝑥 < 0))
254	253	ad2antrl 724	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
255	236, 251, 254	mpjaodan 955	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
256	2, 71, 171, 255	eqsupd 9146	. 2 ⊢ (𝜑 → sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^*, < ) = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
257		nfv 1918	. . 3 ⊢ Ⅎ𝑘𝜑
258		nfcv 2906	. . 3 ⊢ Ⅎ𝑘ℕ
259		nnex 11909	. . . 4 ⊢ ℕ ∈ V
260	259	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
261		icossicc 13097	. . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
262	261, 5	sselid 3915	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
263		elex 3440	. . . . . 6 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → 𝑏 ∈ V)
264	263	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ V)
265	107	fmpttd 6971	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞))
266		esumpfinvallem 31942	. . . . 5 ⊢ ((𝑏 ∈ V ∧ (𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)))
267	264, 265, 266	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)))
268	108	recnd 10934	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ ℂ)
269	99, 268	gsumfsum 20577	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴)
270	267, 269	eqtr3d 2780	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴)
271	257, 258, 260, 262, 270	esumval 31914	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^, < ))
272	3, 4, 35, 43, 69	isumclim 15397	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
273	256, 271, 272	3eqtr4d 2788	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 𝒫 cpw 4530 class class class wbr 5070 ↦ cmpt 5153 Or wor 5493 dom cdm 5580 ran crn 5581 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 supcsup 9129 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 ℕcn 11903 ℤcz 12249 ℤ_≥cuz 12511 [,)cico 13010 [,]cicc 13011 ...cfz 13168 seqcseq 13649 ⇝ cli 15121 Σcsu 15325 ↾_s cress 16867 Σ_g cgsu 17068 ℝ^*_𝑠cxrs 17128 ℂ_fldccnfld 20510 Σ^*cesum 31895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xadd 12778 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-ordt 17129 df-xrs 17130 df-mre 17212 df-mrc 17213 df-acs 17215 df-ps 18199 df-tsr 18200 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-ntr 22079 df-nei 22157 df-cn 22286 df-haus 22374 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-tsms 23186 df-esum 31896

This theorem is referenced by: esumcvg 31954 esumcvgsum 31956

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