esumpcvgval - Mathbox for Thierry Arnoux

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Theorem esumpcvgval 34534

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 13184	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		nnuz 12919	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 12642	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
5		esumpcvgval.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6		esumpcvgval.2	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵)
7		eqcom 2772	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 ↔ 𝑙 = 𝑘)
8		eqcom 2772	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
9	6, 7, 8	3imtr3i 294	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → 𝐵 = 𝐴)
10	9	cbvmptv 5217	. . . . . . . . . 10 ⊢ (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴)
11	5, 10	fmptd 7113	. . . . . . . . 9 ⊢ (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
12	11	ffvelcdmda 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞))
13		elrege0 13499	. . . . . . . . 9 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥)))
14	13	simplbi 502	. . . . . . . 8 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ)
15	12, 14	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ)
16	3, 4, 15	serfre 14087	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ)
17	11	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
18		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
19	18	peano2nnd 12267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
20	17, 19	ffvelcdmd 7084	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞))
21		elrege0 13499	. . . . . . . . . 10 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
22	21	simprbi 503	. . . . . . . . 9 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))
23	20, 22	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))
24	16	ffvelcdmda 7083	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ)
25	21	simplbi 502	. . . . . . . . . 10 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ)
26	20, 25	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ)
27	24, 26	addge01d 11819	. . . . . . . 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 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
29	18, 3	eleqtrdi 2875	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
30		seqp1 14072	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘1) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
31	29, 30	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
32	28, 31	breqtrrd 5141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)))
33		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
34	10	fvmpt2 7005	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
35	33, 5, 34	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
36		rge0ssre 13501	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
37	36, 5	sselid 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
38	16	feqmptd 6953	. . . . . . . . . 10 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
39		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
40		elfznn 13600	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
41	40	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
42	39, 41, 35	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
43	37	recnd 11254	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
44	39, 41, 43	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
45	42, 29, 44	fsumser 15806	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
46	45	eqcomd 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
47	46	mpteq2dva 5206	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
48	38, 47	eqtr2d 2801	. . . . . . . . 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 2866	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
51	3, 4, 35, 37, 50	isumrecl 15841	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
52		1zzd 12642	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
53		fzfid 14029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
54		fzssuz 13612	. . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ (ℤ_≥‘1)
55	54, 3	sseqtrri 3987	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
57	35	adantlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
58	37	adantlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
59	5	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
60		elrege0 13499	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
61	60	simprbi 503	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (0[,)+∞) → 0 ≤ 𝐴)
62	59, 61	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
63	50	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
64	3, 52, 53, 56, 57, 58, 62, 63	isumless 15924	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴)
65	45, 64	eqbrtrrd 5137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
66	65	ralrimiva 3159	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
67		brralrspcev 5173	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
69	3, 4, 16, 32, 68	climsup 15747	. . . . 5 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
70	3, 4, 69, 24	climrecl 15660	. . . 4 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
71	70	rexrd 11276	. . 3 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^*)
72		eqid 2765	. . . . . . 7 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
73		sumex 15765	. . . . . . 7 ⊢ Σ𝑘 ∈ 𝑏 𝐴 ∈ V
74	72, 73	elrnmpti 5954	. . . . . 6 ⊢ (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
75		ssnnssfz 33204	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚))
76		fzfid 14029	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin)
77		elfznn 13600	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
78	77, 5	sylan2 605	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
79	60	simplbi 502	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (0[,)+∞) → 𝐴 ∈ ℝ)
80	78, 79	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
81	80	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
82	78, 61	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
83	82	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
84		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚))
85	76, 81, 83, 84	fsumless 15873	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
86	85	ex 418	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
87	86	reximdv 3182	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
88	87	imp 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
89	75, 88	sylan2 605	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
90		breq1 5114	. . . . . . . . . 10 ⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
91	90	rexbidv 3191	. . . . . . . . 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 250	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
93	92	rexlimdva 3168	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
94	93	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
95	74, 94	sylan2b 606	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
96		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
97		inss2 4190	. . . . . . . . . . . . 13 ⊢ (𝒫 ℕ ∩ Fin) ⊆ Fin
98		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
99	97, 98	sselid 3936	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ Fin)
100		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝜑)
101		inss1 4189	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
102		simplr 781	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
103	101, 102	sselid 3936	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ 𝒫 ℕ)
104	103	elpwid 4573	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ⊆ ℕ)
105		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ 𝑏)
106	104, 105	sseldd 3939	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ ℕ)
107	100, 106, 5	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ (0[,)+∞))
108	107, 79	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ ℝ)
109	99, 108	fsumrecl 15810	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
110	109	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
111	96, 110	eqeltrd 2865	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
112	111	r19.29an 3171	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
113	74, 112	sylan2b 606	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ∈ ℝ)
114	113	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ)
115		fzfid 14029	. . . . . . . 8 ⊢ (𝜑 → (1...𝑚) ∈ Fin)
116	115, 80	fsumrecl 15810	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
117	116	ad2antrr 739	. . . . . 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 739	. . . . . 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 785	. . . . . 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 6718	. . . . . . . 8 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
121	120	ad2antrr 739	. . . . . . 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 12261	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
123	122	ne0ii 4297	. . . . . . . . 9 ⊢ ℕ ≠ ∅
124		dm0rn0 5916	. . . . . . . . . . 11 ⊢ (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅)
125	16	fdmd 6720	. . . . . . . . . . . 12 ⊢ (𝜑 → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
126	125	eqeq1d 2767	. . . . . . . . . . 11 ⊢ (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
127	124, 126	bitr3id 288	. . . . . . . . . 10 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
128	127	necon3bid 3004	. . . . . . . . 9 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ ≠ ∅))
129	123, 128	mpbiri 261	. . . . . . . 8 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
130	129	ad2antrr 739	. . . . . . 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 12641	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
132		seqfn 14069	. . . . . . . . . . . . . . . 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 6637	. . . . . . . . . . . . . . 15 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
135	133, 134	mpbir 234	. . . . . . . . . . . . . 14 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ
136		dffn5 6943	. . . . . . . . . . . . . 14 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
137	135, 136	mpbi 233	. . . . . . . . . . . . 13 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
138		fvex 6898	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ V
139	137, 138	elrnmpti 5954	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
140		r19.29 3130	. . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧 ≤ 𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠))
142	141	biimparc 485	. . . . . . . . . . . . . 14 ⊢ (((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
143	142	rexlimivw 3164	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
144	140, 143	syl 18	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
145	139, 144	sylan2b 606	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → 𝑧 ≤ 𝑠)
146	145	ralrimiva 3159	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
147	146	reximi 3105	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
149	148	ad2antrr 739	. . . . . . 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 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
151		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑)
152	77	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ)
153	151, 152, 35	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
154	150, 3	eleqtrdi 2875	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ_≥‘1))
155	151, 152, 5	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
156	155, 79	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
157	156	recnd 11254	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ)
158	153, 154, 157	fsumser 15806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
159		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
160	159	rspceeqv 3606	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
162	137, 138	elrnmpti 5954	. . . . . . . . 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 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
164	163	ad2ant2r 760	. . . . . . 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 12193	. . . . . . 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 1400	. . . . . 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 11384	. . . . 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 3174	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
169	70	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
170	113, 169	lenltd 11373	. . . 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 235	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥)
172		simpr1r 1250	. . . . . . 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 1381	. . . . . 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 743	. . . . . . . 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 13171	. . . . . . . 8 ⊢ (sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^* → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
176	174, 175	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
177		breq1 5114	. . . . . . . . 9 ⊢ (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
178	177	notbid 321	. . . . . . . 8 ⊢ (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
179	178	adantl 487	. . . . . . 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 260	. . . . . 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 198	. . . . 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 787	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝜑)
183		simpr1l 1249	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ^*)
184	183	3anassrs 1381	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ ℝ^*)
185		simplr 781	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 0 ≤ 𝑥)
186		simpr 490	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞)
187		0xr 11273	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
188		pnfxr 11280	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
189		elico1 13433	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^ ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)))
190	187, 188, 189	mp2an 705	. . . . . . 7 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞))
191	184, 185, 186, 190	syl3anbrc 1362	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ (0[,)+∞))
192		simpr1r 1250	. . . . . . 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 1381	. . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
195	129	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
196	148	adantr 486	. . . . . . . . 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 1146	. . . . . . . 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 783	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ (0[,)+∞))
199	36, 198	sselid 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ ℝ)
200		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
201		suprlub 12196	. . . . . . . . 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 482	. . . . . . . 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 851	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
204	40	ssriv 3942	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑛) ⊆ ℕ
205		ovex 7452	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑛) ∈ V
206	205	elpw 4568	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑛) ∈ 𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ)
207	204, 206	mpbir 234	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ 𝒫 ℕ
208		fzfi 14028	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ Fin
209		elin 3922	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑛) ∈ 𝒫 ℕ ∧ (1...𝑛) ∈ Fin))
210	207, 208, 209	mpbir2an 724	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
213	45	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
214	212, 213	eqtr4d 2803	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴)
215		sumeq1 15766	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (1...𝑛) → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
216	215	rspceeqv 3606	. . . . . . . . . . . . . 14 ⊢ (((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
217	211, 214, 216	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
218	217	ex 418	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
219	218	rexlimdva 3168	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
220	137, 138	elrnmpti 5954	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
221	72, 73	elrnmpti 5954	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
222	219, 220, 221	3imtr4g 299	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)))
223	222	ssrdv 3944	. . . . . . . . 9 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴))
224		ssrexv 4008	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦))
226	225	imp 412	. . . . . . 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 603	. . . . . 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 850	. . . . 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 789	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ^*)
230		xrlelttric 33169	. . . . . . . 8 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
231	188, 230	mpan 703	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
232		xgepnf 13209	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ↔ 𝑥 = +∞))
233	232	orbi1d 930	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → ((+∞ ≤ 𝑥 ∨ 𝑥 < +∞) ↔ (𝑥 = +∞ ∨ 𝑥 < +∞)))
234	231, 233	mpbid 235	. . . . . 6 ⊢ (𝑥 ∈ ℝ^* → (𝑥 = +∞ ∨ 𝑥 < +∞))
235	229, 234	syl 18	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → (𝑥 = +∞ ∨ 𝑥 < +∞))
236	181, 228, 235	mpjaodan 973	. . . 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 5328	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 ℕ
238		0fi 9046	. . . . . . . . 9 ⊢ ∅ ∈ Fin
239		elin 3922	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ ∧ ∅ ∈ Fin))
240	237, 238, 239	mpbir2an 724	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 ℕ ∩ Fin)
241		sum0 15797	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0
242	241	eqcomi 2774	. . . . . . . 8 ⊢ 0 = Σ𝑘 ∈ ∅ 𝐴
243		sumeq1 15766	. . . . . . . . 9 ⊢ (𝑏 = ∅ → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴)
244	243	rspceeqv 3606	. . . . . . . 8 ⊢ ((∅ ∈ (𝒫 ℕ ∩ Fin) ∧ 0 = Σ𝑘 ∈ ∅ 𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
245	240, 242, 244	mp2an 705	. . . . . . 7 ⊢ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴
246	72, 73	elrnmpti 5954	. . . . . . 7 ⊢ (0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
247	245, 246	mpbir 234	. . . . . 6 ⊢ 0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
248		breq2 5115	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑥 < 𝑦 ↔ 𝑥 < 0))
249	248	rspcev 3583	. . . . . 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 703	. . . . 5 ⊢ (𝑥 < 0 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
251	250	adantl 487	. . . 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 33169	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
253	187, 252	mpan 703	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (0 ≤ 𝑥 ∨ 𝑥 < 0))
254	253	ad2antrl 741	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
255	236, 251, 254	mpjaodan 973	. . 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 9424	. 2 ⊢ (𝜑 → sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^*, < ) = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
257		nfv 1947	. . 3 ⊢ Ⅎ𝑘𝜑
258		nfcv 2927	. . 3 ⊢ Ⅎ𝑘ℕ
259		nnex 12256	. . . 4 ⊢ ℕ ∈ V
260	259	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
261		icossicc 13481	. . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
262	261, 5	sselid 3936	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
263		elex 3478	. . . . . 6 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → 𝑏 ∈ V)
264	263	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ V)
265	107	fmpttd 7114	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞))
266		esumpfinvallem 34530	. . . . 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 596	. . . 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 11254	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ ℂ)
269	99, 268	gsumfsum 21636	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴)
270	267, 269	eqtr3d 2802	. . 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 34502	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^, < ))
272	3, 4, 35, 43, 69	isumclim 15833	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
273	256, 271, 272	3eqtr4d 2810	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 Vcvv 3457 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 𝒫 cpw 4564 class class class wbr 5111 ↦ cmpt 5194 Or wor 5570 dom cdm 5663 ran crn 5664 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 Fincfn 8949 supcsup 9407 ℂcc 11115 ℝcr 11116 0cc0 11117 1c1 11118 + caddc 11120 +∞cpnf 11257 ℝ^*cxr 11259 < clt 11260 ≤ cle 11261 ℕcn 12250 ℤcz 12608 ℤ_≥cuz 12880 [,)cico 13392 [,]cicc 13393 ...cfz 13553 seqcseq 14057 ⇝ cli 15561 Σcsu 15763 ↾_s cress 17314 Σ_g cgsu 17517 ℝ^*_𝑠cxrs 17578 ℂ_fldccnfld 21574 Σ^*cesum 34483

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195 ax-addf 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xadd 13156 df-ioo 13394 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-rlim 15566 df-sum 15764 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-starv 17349 df-tset 17353 df-ple 17354 df-ds 17356 df-unif 17357 df-rest 17499 df-topn 17500 df-0g 17518 df-gsum 17519 df-topgen 17520 df-ordt 17579 df-xrs 17580 df-mre 17662 df-mrc 17663 df-acs 17665 df-ps 18646 df-tsr 18647 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-grp 19049 df-minusg 19050 df-cntz 19433 df-cmn 19898 df-abl 19899 df-mgp 20263 df-ur 20310 df-ring 20363 df-cring 20364 df-fbas 21571 df-fg 21572 df-cnfld 21575 df-top 23103 df-topon 23120 df-topsp 23142 df-bases 23155 df-ntr 23229 df-nei 23307 df-cn 23436 df-haus 23524 df-fil 24056 df-fm 24148 df-flim 24149 df-flf 24150 df-tsms 24337 df-esum 34484

This theorem is used by: esumcvg 34542 esumcvgsum 34544

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