esumpcvgval - Mathbox for Thierry Arnoux

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Theorem esumpcvgval 29467

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 11850	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		nnuz 11599	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 11285	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
5		esumpcvgval.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6		esumpcvgval.2	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵)
7		eqcom 2617	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 ↔ 𝑙 = 𝑘)
8		eqcom 2617	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
9	6, 7, 8	3imtr3i 279	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → 𝐵 = 𝐴)
10	9	cbvmptv 4678	. . . . . . . . . 10 ⊢ (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴)
11	5, 10	fmptd 6292	. . . . . . . . 9 ⊢ (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
12	11	ffvelrnda 6267	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞))
13		elrege0 12149	. . . . . . . . 9 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥)))
14	13	simplbi 475	. . . . . . . 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 12692	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ)
17	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
18		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
19	18	peano2nnd 10914	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
20	17, 19	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞))
21		elrege0 12149	. . . . . . . . . 10 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
22	21	simprbi 479	. . . . . . . . 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 6267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ)
25	21	simplbi 475	. . . . . . . . . 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 10494	. . . . . . . 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 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
29	18, 3	syl6eleq 2698	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
30		seqp1 12678	. . . . . . . 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 4611	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)))
33		simpr 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
34	10	fvmpt2 6200	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
35	33, 5, 34	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
36		rge0ssre 12151	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
37	36, 5	sseldi 3566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
38	16	feqmptd 6159	. . . . . . . . . 10 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
39		simpll 786	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
40		elfznn 12241	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
41	40	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
42	39, 41, 35	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
43	37	recnd 9947	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
44	39, 41, 43	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
45	42, 29, 44	fsumser 14308	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
46	45	eqcomd 2616	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
47	46	mpteq2dva 4672	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
48	38, 47	eqtr2d 2645	. . . . . . . . 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 2689	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
51	3, 4, 35, 37, 50	isumrecl 14338	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
52		1zzd 11285	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
53		fzfid 12634	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
54		fzssuz 12253	. . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ (ℤ_≥‘1)
55	54, 3	sseqtr4i 3601	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
57	35	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
58	37	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
59	5	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
60		elrege0 12149	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
61	60	simprbi 479	. . . . . . . . . . 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 14416	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴)
65	45, 64	eqbrtrrd 4607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
66	65	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
67		breq2 4587	. . . . . . . . 9 ⊢ (𝑠 = Σ𝑘 ∈ ℕ 𝐴 → ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴))
68	67	ralbidv 2969	. . . . . . . 8 ⊢ (𝑠 = Σ𝑘 ∈ ℕ 𝐴 → (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ↔ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴))
69	68	rspcev 3282	. . . . . . 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)
70	51, 66, 69	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
71	3, 4, 16, 32, 70	climsup 14248	. . . . 5 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
72	3, 4, 71, 24	climrecl 14162	. . . 4 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
73	72	rexrd 9968	. . 3 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^*)
74		eqid 2610	. . . . . . 7 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
75		sumex 14266	. . . . . . 7 ⊢ Σ𝑘 ∈ 𝑏 𝐴 ∈ V
76	74, 75	elrnmpti 5297	. . . . . 6 ⊢ (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
77		ssnnssfz 28937	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚))
78		fzfid 12634	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin)
79		elfznn 12241	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
80	79, 5	sylan2 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
81	60	simplbi 475	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (0[,)+∞) → 𝐴 ∈ ℝ)
82	80, 81	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
83	82	adantlr 747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
84	80, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
85	84	adantlr 747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
86		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚))
87	78, 83, 85, 86	fsumless 14369	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
88	87	ex 449	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
89	88	reximdv 2999	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
90	89	imp 444	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
91	77, 90	sylan2 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
92		breq1 4586	. . . . . . . . . 10 ⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
93	92	rexbidv 3034	. . . . . . . . 9 ⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
94	91, 93	syl5ibrcom 236	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
95	94	rexlimdva 3013	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
96	95	imp 444	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
97	76, 96	sylan2b 491	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
98		simpr 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
99		inss2 3796	. . . . . . . . . . . . 13 ⊢ (𝒫 ℕ ∩ Fin) ⊆ Fin
100		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
101	99, 100	sseldi 3566	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ Fin)
102		simpll 786	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝜑)
103		inss1 3795	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
104		simplr 788	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
105	103, 104	sseldi 3566	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ 𝒫 ℕ)
106	105	elpwid 4118	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ⊆ ℕ)
107		simpr 476	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ 𝑏)
108	106, 107	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ ℕ)
109	102, 108, 5	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ (0[,)+∞))
110	109, 81	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ ℝ)
111	101, 110	fsumrecl 14312	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
112	111	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
113	98, 112	eqeltrd 2688	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
114	113	r19.29an 3059	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
115	76, 114	sylan2b 491	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ∈ ℝ)
116	115	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ)
117		fzfid 12634	. . . . . . . 8 ⊢ (𝜑 → (1...𝑚) ∈ Fin)
118	117, 82	fsumrecl 14312	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
119	118	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
120	72	ad2antrr 758	. . . . . 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)
121		simprr 792	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
122		frn 5966	. . . . . . . . 9 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
123	16, 122	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
124	123	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
125		1nn 10908	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
126	125	ne0ii 3882	. . . . . . . . 9 ⊢ ℕ ≠ ∅
127		dm0rn0 5263	. . . . . . . . . . 11 ⊢ (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅)
128		fdm 5964	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
129	16, 128	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
130	129	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
131	127, 130	syl5bbr 273	. . . . . . . . . 10 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
132	131	necon3bid 2826	. . . . . . . . 9 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ ≠ ∅))
133	126, 132	mpbiri 247	. . . . . . . 8 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
134	133	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
135		1z 11284	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
136		seqfn 12675	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
137	135, 136	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1)
138	3	fneq2i 5900	. . . . . . . . . . . . . . 15 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
139	137, 138	mpbir 220	. . . . . . . . . . . . . 14 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ
140		dffn5 6151	. . . . . . . . . . . . . 14 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
141	139, 140	mpbi 219	. . . . . . . . . . . . 13 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
142		fvex 6113	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ V
143	141, 142	elrnmpti 5297	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
144		r19.29 3054	. . . . . . . . . . . . 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)))
145		breq1 4586	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧 ≤ 𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠))
146	145	biimparc 503	. . . . . . . . . . . . . 14 ⊢ (((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
147	146	rexlimivw 3011	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
148	144, 147	syl 17	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
149	143, 148	sylan2b 491	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → 𝑧 ≤ 𝑠)
150	149	ralrimiva 2949	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
151	150	reximi 2994	. . . . . . . . 9 ⊢ (∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
152	70, 151	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
153	152	ad2antrr 758	. . . . . . 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)
154		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
155		simpll 786	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑)
156	79	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ)
157	155, 156, 35	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
158	154, 3	syl6eleq 2698	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ_≥‘1))
159	155, 156, 5	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
160	159, 81	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
161	160	recnd 9947	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ)
162	157, 158, 161	fsumser 14308	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
163		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
164	163	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ↔ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)))
165	164	rspcev 3282	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
166	154, 162, 165	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
167	141, 142	elrnmpti 5297	. . . . . . . . 9 ⊢ (Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
168	166, 167	sylibr 223	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
169	168	ad2ant2r 779	. . . . . . 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)))
170		suprub 10863	. . . . . . 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, < ))
171	124, 134, 153, 169, 170	syl31anc 1321	. . . . . 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, < ))
172	116, 119, 120, 121, 171	letrd 10073	. . . . 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, < ))
173	97, 172	rexlimddv 3017	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
174	72	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
175	115, 174	lenltd 10062	. . . 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))
176	173, 175	mpbid 221	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥)
177		simpr1r 1112	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 = +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
178	177	3anassrs 1282	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
179	73	ad3antrrr 762	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^*)
180		pnfnlt 11838	. . . . . . . 8 ⊢ (sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^* → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
181	179, 180	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
182		breq1 4586	. . . . . . . . 9 ⊢ (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
183	182	notbid 307	. . . . . . . 8 ⊢ (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
184	183	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, < )))
185	181, 184	mpbird 246	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
186	178, 185	pm2.21dd 185	. . . . 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)
187		simplll 794	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝜑)
188		simpr1l 1111	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ^*)
189	188	3anassrs 1282	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ ℝ^*)
190		simplr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 0 ≤ 𝑥)
191		simpr 476	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞)
192		0xr 9965	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
193		pnfxr 9971	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
194		elico1 12089	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^ ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)))
195	192, 193, 194	mp2an 704	. . . . . . 7 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞))
196	189, 190, 191, 195	syl3anbrc 1239	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ (0[,)+∞))
197		simpr1r 1112	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
198	197	3anassrs 1282	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
199	123	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
200	133	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
201	152	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
202	199, 200, 201	3jca 1235	. . . . . . . 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))
203		simprl 790	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ (0[,)+∞))
204	36, 203	sseldi 3566	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ ℝ)
205		simprr 792	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
206		suprlub 10864	. . . . . . . . 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))
207	206	biimpa 500	. . . . . . . 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)
208	202, 204, 205, 207	syl21anc 1317	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
209	40	ssriv 3572	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑛) ⊆ ℕ
210		ovex 6577	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑛) ∈ V
211	210	elpw 4114	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑛) ∈ 𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ)
212	209, 211	mpbir 220	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ 𝒫 ℕ
213		fzfi 12633	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ Fin
214		elin 3758	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑛) ∈ 𝒫 ℕ ∧ (1...𝑛) ∈ Fin))
215	212, 213, 214	mpbir2an 957	. . . . . . . . . . . . . . 15 ⊢ (1...𝑛) ∈ (𝒫 ℕ ∩ Fin)
216	215	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → (1...𝑛) ∈ (𝒫 ℕ ∩ Fin))
217		simpr 476	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
218	45	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
219	217, 218	eqtr4d 2647	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴)
220		sumeq1 14267	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (1...𝑛) → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
221	220	eqeq2d 2620	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (1...𝑛) → (𝑦 = Σ𝑘 ∈ 𝑏 𝐴 ↔ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴))
222	221	rspcev 3282	. . . . . . . . . . . . . 14 ⊢ (((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
223	216, 219, 222	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
224	223	ex 449	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
225	224	rexlimdva 3013	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
226	141, 142	elrnmpti 5297	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
227	74, 75	elrnmpti 5297	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
228	225, 226, 227	3imtr4g 284	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)))
229	228	ssrdv 3574	. . . . . . . . 9 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴))
230		ssrexv 3630	. . . . . . . . 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))
231	229, 230	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦))
232	231	imp 444	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
233	208, 232	syldan 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
234	187, 196, 198, 233	syl12anc 1316	. . . . 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)
235		simplrl 796	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ^*)
236		xrlelttric 28906	. . . . . . . 8 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
237	193, 236	mpan 702	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
238		xgepnf 28904	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ↔ 𝑥 = +∞))
239	238	orbi1d 735	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → ((+∞ ≤ 𝑥 ∨ 𝑥 < +∞) ↔ (𝑥 = +∞ ∨ 𝑥 < +∞)))
240	237, 239	mpbid 221	. . . . . 6 ⊢ (𝑥 ∈ ℝ^* → (𝑥 = +∞ ∨ 𝑥 < +∞))
241	235, 240	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → (𝑥 = +∞ ∨ 𝑥 < +∞))
242	186, 234, 241	mpjaodan 823	. . . 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)
243		0elpw 4760	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 ℕ
244		0fin 8073	. . . . . . . . 9 ⊢ ∅ ∈ Fin
245		elin 3758	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ ∧ ∅ ∈ Fin))
246	243, 244, 245	mpbir2an 957	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 ℕ ∩ Fin)
247		sum0 14299	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0
248	247	eqcomi 2619	. . . . . . . 8 ⊢ 0 = Σ𝑘 ∈ ∅ 𝐴
249		sumeq1 14267	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴)
250	249	eqeq2d 2620	. . . . . . . . 9 ⊢ (𝑏 = ∅ → (0 = Σ𝑘 ∈ 𝑏 𝐴 ↔ 0 = Σ𝑘 ∈ ∅ 𝐴))
251	250	rspcev 3282	. . . . . . . 8 ⊢ ((∅ ∈ (𝒫 ℕ ∩ Fin) ∧ 0 = Σ𝑘 ∈ ∅ 𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
252	246, 248, 251	mp2an 704	. . . . . . 7 ⊢ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴
253	74, 75	elrnmpti 5297	. . . . . . 7 ⊢ (0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
254	252, 253	mpbir 220	. . . . . 6 ⊢ 0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
255		breq2 4587	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑥 < 𝑦 ↔ 𝑥 < 0))
256	255	rspcev 3282	. . . . . 6 ⊢ ((0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
257	254, 256	mpan 702	. . . . 5 ⊢ (𝑥 < 0 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
258	257	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)
259		xrlelttric 28906	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
260	192, 259	mpan 702	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (0 ≤ 𝑥 ∨ 𝑥 < 0))
261	260	ad2antrl 760	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
262	242, 258, 261	mpjaodan 823	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
263	2, 73, 176, 262	eqsupd 8246	. 2 ⊢ (𝜑 → sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^*, < ) = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
264		nfv 1830	. . 3 ⊢ Ⅎ𝑘𝜑
265		nfcv 2751	. . 3 ⊢ Ⅎ𝑘ℕ
266		nnex 10903	. . . 4 ⊢ ℕ ∈ V
267	266	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
268		icossicc 12131	. . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
269	268, 5	sseldi 3566	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
270		elex 3185	. . . . . 6 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → 𝑏 ∈ V)
271	270	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ V)
272		eqid 2610	. . . . . 6 ⊢ (𝑘 ∈ 𝑏 ↦ 𝐴) = (𝑘 ∈ 𝑏 ↦ 𝐴)
273	109, 272	fmptd 6292	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞))
274		esumpfinvallem 29463	. . . . 5 ⊢ ((𝑏 ∈ V ∧ (𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)))
275	271, 273, 274	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)))
276	110	recnd 9947	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ ℂ)
277	101, 276	gsumfsum 19632	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴)
278	275, 277	eqtr3d 2646	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴)
279	264, 265, 267, 269, 278	esumval 29435	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^, < ))
280	3, 4, 35, 43, 71	isumclim 14330	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
281	263, 279, 280	3eqtr4d 2654	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 Or wor 4958 dom cdm 5038 ran crn 5039 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 supcsup 8229 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 ℕcn 10897 ℤcz 11254 ℤ_≥cuz 11563 [,)cico 12048 [,]cicc 12049 ...cfz 12197 seqcseq 12663 ⇝ cli 14063 Σcsu 14264 ↾_s cress 15696 Σ_g cgsu 15924 ℝ^*_𝑠cxrs 15983 ℂ_fldccnfld 19567 Σ^*cesum 29416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-ordt 15984 df-xrs 15985 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-ntr 20634 df-nei 20712 df-cn 20841 df-haus 20929 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tsms 21740 df-esum 29417

This theorem is referenced by: esumcvg 29475 esumcvgsum 29477

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