esumpcvgval - Mathbox for Thierry Arnoux

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Theorem esumpcvgval 30140

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 11974	. . . 4 ⊢ < Or ℝ^*
2	1	a1i 11	. . 3 ⊢ (𝜑 → < Or ℝ^*)
3		nnuz 11723	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 11408	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
5		esumpcvgval.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6		esumpcvgval.2	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵)
7		eqcom 2629	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 ↔ 𝑙 = 𝑘)
8		eqcom 2629	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
9	6, 7, 8	3imtr3i 280	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → 𝐵 = 𝐴)
10	9	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴)
11	5, 10	fmptd 6385	. . . . . . . . 9 ⊢ (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
12	11	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞))
13		elrege0 12278	. . . . . . . . 9 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥)))
14	13	simplbi 476	. . . . . . . 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 12830	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ)
17	11	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
18		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
19	18	peano2nnd 11037	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
20	17, 19	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞))
21		elrege0 12278	. . . . . . . . . 10 ⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
22	21	simprbi 480	. . . . . . . . 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 6359	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ)
25	21	simplbi 476	. . . . . . . . . 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 10615	. . . . . . . 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 222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
29	18, 3	syl6eleq 2711	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
30		seqp1 12816	. . . . . . . 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 4681	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)))
33		simpr 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
34	10	fvmpt2 6291	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
35	33, 5, 34	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
36		rge0ssre 12280	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
37	36, 5	sseldi 3601	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
38	16	feqmptd 6249	. . . . . . . . . 10 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
39		simpll 790	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
40		elfznn 12370	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
41	40	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
42	39, 41, 35	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
43	37	recnd 10068	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
44	39, 41, 43	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
45	42, 29, 44	fsumser 14461	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
46	45	eqcomd 2628	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
47	46	mpteq2dva 4744	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
48	38, 47	eqtr2d 2657	. . . . . . . . 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 2702	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
51	3, 4, 35, 37, 50	isumrecl 14496	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
52		1zzd 11408	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
53		fzfid 12772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
54		fzssuz 12382	. . . . . . . . . . . 12 ⊢ (1...𝑛) ⊆ (ℤ_≥‘1)
55	54, 3	sseqtr4i 3638	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
57	35	adantlr 751	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
58	37	adantlr 751	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
59	5	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
60		elrege0 12278	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
61	60	simprbi 480	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (0[,)+∞) → 0 ≤ 𝐴)
62	59, 61	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
63	50	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
64	3, 52, 53, 56, 57, 58, 62, 63	isumless 14577	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴)
65	45, 64	eqbrtrrd 4677	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
66	65	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
67		breq2 4657	. . . . . . . . 9 ⊢ (𝑠 = Σ𝑘 ∈ ℕ 𝐴 → ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴))
68	67	ralbidv 2986	. . . . . . . 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 3309	. . . . . . 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 693	. . . . . 6 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
71	3, 4, 16, 32, 70	climsup 14400	. . . . 5 ⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
72	3, 4, 71, 24	climrecl 14314	. . . 4 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
73	72	rexrd 10089	. . 3 ⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ^*)
74		eqid 2622	. . . . . . 7 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
75		sumex 14418	. . . . . . 7 ⊢ Σ𝑘 ∈ 𝑏 𝐴 ∈ V
76	74, 75	elrnmpti 5376	. . . . . 6 ⊢ (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
77		ssnnssfz 29549	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚))
78		fzfid 12772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin)
79		elfznn 12370	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
80	79, 5	sylan2 491	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
81	60	simplbi 476	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (0[,)+∞) → 𝐴 ∈ ℝ)
82	80, 81	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
83	82	adantlr 751	. . . . . . . . . . . . . 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 751	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
86		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚))
87	78, 83, 85, 86	fsumless 14528	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
88	87	ex 450	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
89	88	reximdv 3016	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
90	89	imp 445	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
91	77, 90	sylan2 491	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
92		breq1 4656	. . . . . . . . . 10 ⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
93	92	rexbidv 3052	. . . . . . . . 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 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
95	94	rexlimdva 3031	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
96	95	imp 445	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
97	76, 96	sylan2b 492	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
98		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 = Σ𝑘 ∈ 𝑏 𝐴)
99		inss2 3834	. . . . . . . . . . . . 13 ⊢ (𝒫 ℕ ∩ Fin) ⊆ Fin
100		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
101	99, 100	sseldi 3601	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ Fin)
102		simpll 790	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝜑)
103		inss1 3833	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
104		simplr 792	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
105	103, 104	sseldi 3601	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ∈ 𝒫 ℕ)
106	105	elpwid 4170	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑏 ⊆ ℕ)
107		simpr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ 𝑏)
108	106, 107	sseldd 3604	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝑘 ∈ ℕ)
109	102, 108, 5	syl2anc 693	. . . . . . . . . . . . 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 14465	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
112	111	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ)
113	98, 112	eqeltrd 2701	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
114	113	r19.29an 3077	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ)
115	76, 114	sylan2b 492	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ∈ ℝ)
116	115	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ)
117		fzfid 12772	. . . . . . . 8 ⊢ (𝜑 → (1...𝑚) ∈ Fin)
118	117, 82	fsumrecl 14465	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
119	118	ad2antrr 762	. . . . . 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 762	. . . . . 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 796	. . . . . 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 6053	. . . . . . . . 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 762	. . . . . . 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 11031	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
126	125	ne0ii 3923	. . . . . . . . 9 ⊢ ℕ ≠ ∅
127		dm0rn0 5342	. . . . . . . . . . 11 ⊢ (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅)
128		fdm 6051	. . . . . . . . . . . . 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 2624	. . . . . . . . . . 11 ⊢ (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
131	127, 130	syl5bbr 274	. . . . . . . . . 10 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
132	131	necon3bid 2838	. . . . . . . . 9 ⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ ≠ ∅))
133	126, 132	mpbiri 248	. . . . . . . 8 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
134	133	ad2antrr 762	. . . . . . 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 11407	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
136		seqfn 12813	. . . . . . . . . . . . . . . 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 5986	. . . . . . . . . . . . . . 15 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ_≥‘1))
139	137, 138	mpbir 221	. . . . . . . . . . . . . 14 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ
140		dffn5 6241	. . . . . . . . . . . . . 14 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
141	139, 140	mpbi 220	. . . . . . . . . . . . 13 ⊢ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
142		fvex 6201	. . . . . . . . . . . . 13 ⊢ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ V
143	141, 142	elrnmpti 5376	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
144		r19.29 3072	. . . . . . . . . . . . 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 4656	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧 ≤ 𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠))
146	145	biimparc 504	. . . . . . . . . . . . . 14 ⊢ (((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠)
147	146	rexlimivw 3029	. . . . . . . . . . . . 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 492	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → 𝑧 ≤ 𝑠)
150	149	ralrimiva 2966	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)
151	150	reximi 3011	. . . . . . . . 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 762	. . . . . . 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 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
155		simpll 790	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑)
156	79	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ)
157	155, 156, 35	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
158	154, 3	syl6eleq 2711	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ_≥‘1))
159	155, 156, 5	syl2anc 693	. . . . . . . . . . . . 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 10068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ)
162	157, 158, 161	fsumser 14461	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
163		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
164	163	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ↔ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)))
165	164	rspcev 3309	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
167	141, 142	elrnmpti 5376	. . . . . . . . 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 224	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
169	168	ad2ant2r 783	. . . . . . 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 10984	. . . . . . 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 1329	. . . . . 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 10194	. . . . 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 3035	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
174	72	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
175	115, 174	lenltd 10183	. . . 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 222	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥)
177		simpr1r 1119	. . . . . . 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 1290	. . . . . 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 766	. . . . . . . 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 11962	. . . . . . . 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 4656	. . . . . . . . 9 ⊢ (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
183	182	notbid 308	. . . . . . . 8 ⊢ (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
184	183	adantl 482	. . . . . . 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 247	. . . . . 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 186	. . . . 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 798	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝜑)
188		simpr1l 1118	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ^*)
189	188	3anassrs 1290	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ ℝ^*)
190		simplr 792	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 0 ≤ 𝑥)
191		simpr 477	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞)
192		0xr 10086	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
193		pnfxr 10092	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
194		elico1 12218	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^ ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)))
195	192, 193, 194	mp2an 708	. . . . . . 7 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ^* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞))
196	189, 190, 191, 195	syl3anbrc 1246	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ (0[,)+∞))
197		simpr1r 1119	. . . . . . 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 1290	. . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
200	133	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
201	152	adantr 481	. . . . . . . . 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 1242	. . . . . . . 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 794	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ (0[,)+∞))
204	36, 203	sseldi 3601	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ ℝ)
205		simprr 796	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
206		suprlub 10987	. . . . . . . . 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 501	. . . . . . . 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 1325	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
209	40	ssriv 3607	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑛) ⊆ ℕ
210		ovex 6678	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑛) ∈ V
211	210	elpw 4164	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑛) ∈ 𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ)
212	209, 211	mpbir 221	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ 𝒫 ℕ
213		fzfi 12771	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑛) ∈ Fin
214		elin 3796	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑛) ∈ 𝒫 ℕ ∧ (1...𝑛) ∈ Fin))
215	212, 213, 214	mpbir2an 955	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
218	45	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
219	217, 218	eqtr4d 2659	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴)
220		sumeq1 14419	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (1...𝑛) → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
221	220	eqeq2d 2632	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (1...𝑛) → (𝑦 = Σ𝑘 ∈ 𝑏 𝐴 ↔ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴))
222	221	rspcev 3309	. . . . . . . . . . . . . 14 ⊢ (((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
223	216, 219, 222	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
224	223	ex 450	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
225	224	rexlimdva 3031	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴))
226	141, 142	elrnmpti 5376	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
227	74, 75	elrnmpti 5376	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)
228	225, 226, 227	3imtr4g 285	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)))
229	228	ssrdv 3609	. . . . . . . . 9 ⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴))
230		ssrexv 3667	. . . . . . . . 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 445	. . . . . . 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 487	. . . . . 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 1324	. . . . 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 800	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ^*)
236		xrlelttric 29517	. . . . . . . 8 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
237	193, 236	mpan 706	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ∨ 𝑥 < +∞))
238		xgepnf 11996	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* → (+∞ ≤ 𝑥 ↔ 𝑥 = +∞))
239	238	orbi1d 739	. . . . . . 7 ⊢ (𝑥 ∈ ℝ^* → ((+∞ ≤ 𝑥 ∨ 𝑥 < +∞) ↔ (𝑥 = +∞ ∨ 𝑥 < +∞)))
240	237, 239	mpbid 222	. . . . . 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 827	. . . 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 4834	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 ℕ
244		0fin 8188	. . . . . . . . 9 ⊢ ∅ ∈ Fin
245		elin 3796	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ ∧ ∅ ∈ Fin))
246	243, 244, 245	mpbir2an 955	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 ℕ ∩ Fin)
247		sum0 14452	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0
248	247	eqcomi 2631	. . . . . . . 8 ⊢ 0 = Σ𝑘 ∈ ∅ 𝐴
249		sumeq1 14419	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴)
250	249	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑏 = ∅ → (0 = Σ𝑘 ∈ 𝑏 𝐴 ↔ 0 = Σ𝑘 ∈ ∅ 𝐴))
251	250	rspcev 3309	. . . . . . . 8 ⊢ ((∅ ∈ (𝒫 ℕ ∩ Fin) ∧ 0 = Σ𝑘 ∈ ∅ 𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
252	246, 248, 251	mp2an 708	. . . . . . 7 ⊢ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴
253	74, 75	elrnmpti 5376	. . . . . . 7 ⊢ (0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴)
254	252, 253	mpbir 221	. . . . . 6 ⊢ 0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)
255		breq2 4657	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑥 < 𝑦 ↔ 𝑥 < 0))
256	255	rspcev 3309	. . . . . 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 706	. . . . 5 ⊢ (𝑥 < 0 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)
258	257	adantl 482	. . . 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 29517	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
260	192, 259	mpan 706	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (0 ≤ 𝑥 ∨ 𝑥 < 0))
261	260	ad2antrl 764	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (0 ≤ 𝑥 ∨ 𝑥 < 0))
262	242, 258, 261	mpjaodan 827	. . 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 8363	. 2 ⊢ (𝜑 → sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^*, < ) = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
264		nfv 1843	. . 3 ⊢ Ⅎ𝑘𝜑
265		nfcv 2764	. . 3 ⊢ Ⅎ𝑘ℕ
266		nnex 11026	. . . 4 ⊢ ℕ ∈ V
267	266	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
268		icossicc 12260	. . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
269	268, 5	sseldi 3601	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
270		elex 3212	. . . . . 6 ⊢ (𝑏 ∈ (𝒫 ℕ ∩ Fin) → 𝑏 ∈ V)
271	270	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ V)
272		eqid 2622	. . . . . 6 ⊢ (𝑘 ∈ 𝑏 ↦ 𝐴) = (𝑘 ∈ 𝑏 ↦ 𝐴)
273	109, 272	fmptd 6385	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞))
274		esumpfinvallem 30136	. . . . 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 693	. . . 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 10068	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘 ∈ 𝑏) → 𝐴 ∈ ℂ)
277	101, 276	gsumfsum 19813	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂ_fld Σ_g (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴)
278	275, 277	eqtr3d 2658	. . 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 30108	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴), ℝ^, < ))
280	3, 4, 35, 43, 71	isumclim 14488	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
281	263, 279, 280	3eqtr4d 2666	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 class class class wbr 4653 ↦ cmpt 4729 Or wor 5034 dom cdm 5114 ran crn 5115 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 supcsup 8346 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℕcn 11020 ℤcz 11377 ℤ_≥cuz 11687 [,)cico 12177 [,]cicc 12178 ...cfz 12326 seqcseq 12801 ⇝ cli 14215 Σcsu 14416 ↾_s cress 15858 Σ_g cgsu 16101 ℝ^*_𝑠cxrs 16160 ℂ_fldccnfld 19746 Σ^*cesum 30089

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-ordt 16161 df-xrs 16162 df-mre 16246 df-mrc 16247 df-acs 16249 df-ps 17200 df-tsr 17201 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-ntr 20824 df-nei 20902 df-cn 21031 df-haus 21119 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 df-esum 30090

This theorem is referenced by: esumcvg 30148 esumcvgsum 30150

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