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Theorem climcnds 15817

Description: The Cauchy condensation test. If 𝑎(𝑘) is a decreasing sequence of nonnegative terms, then Σ𝑘 ∈ ℕ𝑎(𝑘) converges iff Σ𝑛 ∈ ℕ₀2↑𝑛 · 𝑎(2↑𝑛) converges. (Contributed by Mario Carneiro, 18-Jul-2014.) (Proof shortened by AV, 10-Jul-2022.)

**Hypotheses**
Ref	Expression
climcnds.1	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
climcnds.2	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐹‘𝑘))
climcnds.3	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
climcnds.4	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))

**Assertion**
Ref	Expression
climcnds	⊢ (𝜑 → (seq1( + , 𝐹) ∈ dom ⇝ ↔ seq0( + , 𝐺) ∈ dom ⇝ ))

Distinct variable groups: 𝑘,𝑛,𝐹 𝑘,𝐺,𝑛 𝜑,𝑘,𝑛

Proof of Theorem climcnds Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12836	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12564	. . . . 5 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → 1 ∈ ℤ)
3		1zzd 12564	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
4		nnnn0 12449	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
5		climcnds.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
6		2nn 12259	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
7		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
8		nnexpcl 14039	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
9	6, 7, 8	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
10	9	nnred 12201	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℝ)
11		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑘 = (2↑𝑛) → (𝐹‘𝑘) = (𝐹‘(2↑𝑛)))
12	11	eleq1d 2813	. . . . . . . . . . 11 ⊢ (𝑘 = (2↑𝑛) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ))
13		climcnds.1	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
14	13	ralrimiva 3125	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
15	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
16	12, 15, 9	rspcdva 3589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(2↑𝑛)) ∈ ℝ)
17	10, 16	remulcld 11204	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
18	5, 17	eqeltrd 2828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ ℝ)
19	4, 18	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
20	1, 3, 19	serfre 13996	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
21	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → seq1( + , 𝐺):ℕ⟶ℝ)
22		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
23	22, 1	eleqtrdi 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ_≥‘1))
24		nnz 12550	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
25	24	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
26		uzid 12808	. . . . . . . 8 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
27		peano2uz 12860	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
28	25, 26, 27	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
29		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝜑)
30		elfznn 13514	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(𝑗 + 1)) → 𝑛 ∈ ℕ)
31	29, 30, 19	syl2an 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑗 + 1))) → (𝐺‘𝑛) ∈ ℝ)
32		simpll 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 𝜑)
33		elfz1eq 13496	. . . . . . . . . 10 ⊢ (𝑛 ∈ ((𝑗 + 1)...(𝑗 + 1)) → 𝑛 = (𝑗 + 1))
34	33	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 𝑛 = (𝑗 + 1))
35		nnnn0 12449	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ₀)
36		peano2nn0 12482	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ₀)
37	35, 36	syl 17	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ₀)
38	37	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ((𝑗 + 1)...(𝑗 + 1))) → (𝑗 + 1) ∈ ℕ₀)
39	34, 38	eqeltrd 2828	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 𝑛 ∈ ℕ₀)
40	9	nnnn0d 12503	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ₀)
41	40	nn0ge0d 12506	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 0 ≤ (2↑𝑛))
42	11	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑘 = (2↑𝑛) → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘(2↑𝑛))))
43		climcnds.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐹‘𝑘))
44	43	ralrimiva 3125	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ ℕ 0 ≤ (𝐹‘𝑘))
45	44	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ∀𝑘 ∈ ℕ 0 ≤ (𝐹‘𝑘))
46	42, 45, 9	rspcdva 3589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 0 ≤ (𝐹‘(2↑𝑛)))
47	10, 16, 41, 46	mulge0d 11755	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 0 ≤ ((2↑𝑛) · (𝐹‘(2↑𝑛))))
48	47, 5	breqtrrd 5135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 0 ≤ (𝐺‘𝑛))
49	32, 39, 48	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 0 ≤ (𝐺‘𝑛))
50	23, 28, 31, 49	sermono 13999	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ≤ (seq1( + , 𝐺)‘(𝑗 + 1)))
51	50	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ≤ (seq1( + , 𝐺)‘(𝑗 + 1)))
52		2re 12260	. . . . . . 7 ⊢ 2 ∈ ℝ
53		eqidd 2730	. . . . . . . 8 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝐹‘𝑘))
54	13	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
55		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → seq1( + , 𝐹) ∈ dom ⇝ )
56	1, 2, 53, 54, 55	isumrecl 15731	. . . . . . 7 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → Σ𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
57		remulcl 11153	. . . . . . 7 ⊢ ((2 ∈ ℝ ∧ Σ𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ) → (2 · Σ𝑘 ∈ ℕ (𝐹‘𝑘)) ∈ ℝ)
58	52, 56, 57	sylancr 587	. . . . . 6 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → (2 · Σ𝑘 ∈ ℕ (𝐹‘𝑘)) ∈ ℝ)
59	21	ffvelcdmda 7056	. . . . . . . 8 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ∈ ℝ)
60	1, 3, 13	serfre 13996	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℝ)
61	60	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → seq1( + , 𝐹):ℕ⟶ℝ)
62	35	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ₀)
63		nnexpcl 14039	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (2↑𝑗) ∈ ℕ)
64	6, 62, 63	sylancr 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ)
65	61, 64	ffvelcdmd 7057	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
66		remulcl 11153	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
67	52, 65, 66	sylancr 587	. . . . . . . 8 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
68	58	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (2 · Σ𝑘 ∈ ℕ (𝐹‘𝑘)) ∈ ℝ)
69		climcnds.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
70	13, 43, 69, 5	climcndslem2 15816	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))))
71	70	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))))
72		eqidd 2730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹‘𝑘) = (𝐹‘𝑘))
73	64, 1	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ (ℤ_≥‘1))
74		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 𝜑)
75		elfznn 13514	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(2↑𝑗)) → 𝑘 ∈ ℕ)
76	13	recnd 11202	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
77	74, 75, 76	syl2an 596	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹‘𝑘) ∈ ℂ)
78	72, 73, 77	fsumser 15696	. . . . . . . . . 10 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑𝑗))(𝐹‘𝑘) = (seq1( + , 𝐹)‘(2↑𝑗)))
79		1zzd 12564	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 1 ∈ ℤ)
80		fzfid 13938	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (1...(2↑𝑗)) ∈ Fin)
81	75	ssriv 3950	. . . . . . . . . . . 12 ⊢ (1...(2↑𝑗)) ⊆ ℕ
82	81	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (1...(2↑𝑗)) ⊆ ℕ)
83		eqidd 2730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝐹‘𝑘))
84	13	ad4ant14 752	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
85	43	ad4ant14 752	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐹‘𝑘))
86		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → seq1( + , 𝐹) ∈ dom ⇝ )
87	1, 79, 80, 82, 83, 84, 85, 86	isumless 15811	. . . . . . . . . 10 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑𝑗))(𝐹‘𝑘) ≤ Σ𝑘 ∈ ℕ (𝐹‘𝑘))
88	78, 87	eqbrtrrd 5131	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ≤ Σ𝑘 ∈ ℕ (𝐹‘𝑘))
89	56	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
90		2rp 12956	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
91	90	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 2 ∈ ℝ⁺)
92	65, 89, 91	lemul2d 13039	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → ((seq1( + , 𝐹)‘(2↑𝑗)) ≤ Σ𝑘 ∈ ℕ (𝐹‘𝑘) ↔ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ≤ (2 · Σ𝑘 ∈ ℕ (𝐹‘𝑘))))
93	88, 92	mpbid 232	. . . . . . . 8 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ≤ (2 · Σ𝑘 ∈ ℕ (𝐹‘𝑘)))
94	59, 67, 68, 71, 93	letrd 11331	. . . . . . 7 ⊢ (((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ≤ (2 · Σ𝑘 ∈ ℕ (𝐹‘𝑘)))
95	94	ralrimiva 3125	. . . . . 6 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → ∀𝑗 ∈ ℕ (seq1( + , 𝐺)‘𝑗) ≤ (2 · Σ𝑘 ∈ ℕ (𝐹‘𝑘)))
96		brralrspcev 5167	. . . . . 6 ⊢ (((2 · Σ𝑘 ∈ ℕ (𝐹‘𝑘)) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (seq1( + , 𝐺)‘𝑗) ≤ (2 · Σ𝑘 ∈ ℕ (𝐹‘𝑘))) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (seq1( + , 𝐺)‘𝑗) ≤ 𝑥)
97	58, 95, 96	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (seq1( + , 𝐺)‘𝑗) ≤ 𝑥)
98	1, 2, 21, 51, 97	climsup 15636	. . . 4 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → seq1( + , 𝐺) ⇝ sup(ran seq1( + , 𝐺), ℝ, < ))
99		climrel 15458	. . . . 5 ⊢ Rel ⇝
100	99	releldmi 5912	. . . 4 ⊢ (seq1( + , 𝐺) ⇝ sup(ran seq1( + , 𝐺), ℝ, < ) → seq1( + , 𝐺) ∈ dom ⇝ )
101	98, 100	syl 17	. . 3 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → seq1( + , 𝐺) ∈ dom ⇝ )
102		nn0uz 12835	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
103		1nn0 12458	. . . . . 6 ⊢ 1 ∈ ℕ₀
104	103	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
105	18	recnd 11202	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ ℂ)
106	102, 104, 105	iserex 15623	. . . 4 ⊢ (𝜑 → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq1( + , 𝐺) ∈ dom ⇝ ))
107	106	biimpar 477	. . 3 ⊢ ((𝜑 ∧ seq1( + , 𝐺) ∈ dom ⇝ ) → seq0( + , 𝐺) ∈ dom ⇝ )
108	101, 107	syldan 591	. 2 ⊢ ((𝜑 ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → seq0( + , 𝐺) ∈ dom ⇝ )
109		1zzd 12564	. . . 4 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → 1 ∈ ℤ)
110	60	adantr 480	. . . 4 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → seq1( + , 𝐹):ℕ⟶ℝ)
111		elfznn 13514	. . . . . . 7 ⊢ (𝑘 ∈ (1...(𝑗 + 1)) → 𝑘 ∈ ℕ)
112	29, 111, 13	syl2an 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(𝑗 + 1))) → (𝐹‘𝑘) ∈ ℝ)
113		simpll 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 𝜑)
114		peano2nn 12198	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
115	114	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
116		elfz1eq 13496	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1)) → 𝑘 = (𝑗 + 1))
117		eleq1 2816	. . . . . . . . 9 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 ∈ ℕ ↔ (𝑗 + 1) ∈ ℕ))
118	117	biimparc 479	. . . . . . . 8 ⊢ (((𝑗 + 1) ∈ ℕ ∧ 𝑘 = (𝑗 + 1)) → 𝑘 ∈ ℕ)
119	115, 116, 118	syl2an 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 𝑘 ∈ ℕ)
120	113, 119, 43	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ ((𝑗 + 1)...(𝑗 + 1))) → 0 ≤ (𝐹‘𝑘))
121	23, 28, 112, 120	sermono 13999	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘𝑗) ≤ (seq1( + , 𝐹)‘(𝑗 + 1)))
122	121	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘𝑗) ≤ (seq1( + , 𝐹)‘(𝑗 + 1)))
123		0zd 12541	. . . . . 6 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → 0 ∈ ℤ)
124		eqidd 2730	. . . . . 6 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) = (𝐺‘𝑛))
125	18	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ ℝ)
126		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → seq0( + , 𝐺) ∈ dom ⇝ )
127	102, 123, 124, 125, 126	isumrecl 15731	. . . . 5 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → Σ𝑛 ∈ ℕ₀ (𝐺‘𝑛) ∈ ℝ)
128	110	ffvelcdmda 7056	. . . . . . 7 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘𝑗) ∈ ℝ)
129		0zd 12541	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
130	102, 129, 18	serfre 13996	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐺):ℕ₀⟶ℝ)
131	130	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → seq0( + , 𝐺):ℕ₀⟶ℝ)
132		ffvelcdm 7053	. . . . . . . 8 ⊢ ((seq0( + , 𝐺):ℕ₀⟶ℝ ∧ 𝑗 ∈ ℕ₀) → (seq0( + , 𝐺)‘𝑗) ∈ ℝ)
133	131, 35, 132	syl2an 596	. . . . . . 7 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq0( + , 𝐺)‘𝑗) ∈ ℝ)
134	127	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → Σ𝑛 ∈ ℕ₀ (𝐺‘𝑛) ∈ ℝ)
135	110	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → seq1( + , 𝐹):ℕ⟶ℝ)
136		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
137	24	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
138	37	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ₀)
139	138	nn0red 12504	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℝ)
140		nnexpcl 14039	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℕ)
141	6, 138, 140	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ)
142	141	nnred 12201	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℝ)
143		2z 12565	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
144		uzid 12808	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
145	143, 144	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℤ_≥‘2)
146		bernneq3 14196	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ (𝑗 + 1) ∈ ℕ₀) → (𝑗 + 1) < (2↑(𝑗 + 1)))
147	145, 138, 146	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) < (2↑(𝑗 + 1)))
148	139, 142, 147	ltled 11322	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ≤ (2↑(𝑗 + 1)))
149	137	peano2zd 12641	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℤ)
150	141	nnzd 12556	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℤ)
151		eluz 12807	. . . . . . . . . . . . 13 ⊢ (((𝑗 + 1) ∈ ℤ ∧ (2↑(𝑗 + 1)) ∈ ℤ) → ((2↑(𝑗 + 1)) ∈ (ℤ_≥‘(𝑗 + 1)) ↔ (𝑗 + 1) ≤ (2↑(𝑗 + 1))))
152	149, 150, 151	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → ((2↑(𝑗 + 1)) ∈ (ℤ_≥‘(𝑗 + 1)) ↔ (𝑗 + 1) ≤ (2↑(𝑗 + 1))))
153	148, 152	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ (ℤ_≥‘(𝑗 + 1)))
154		eluzp1m1 12819	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ (2↑(𝑗 + 1)) ∈ (ℤ_≥‘(𝑗 + 1))) → ((2↑(𝑗 + 1)) − 1) ∈ (ℤ_≥‘𝑗))
155	137, 153, 154	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → ((2↑(𝑗 + 1)) − 1) ∈ (ℤ_≥‘𝑗))
156		eluznn 12877	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℕ ∧ ((2↑(𝑗 + 1)) − 1) ∈ (ℤ_≥‘𝑗)) → ((2↑(𝑗 + 1)) − 1) ∈ ℕ)
157	136, 155, 156	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → ((2↑(𝑗 + 1)) − 1) ∈ ℕ)
158	135, 157	ffvelcdmd 7057	. . . . . . . 8 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ∈ ℝ)
159	23	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ_≥‘1))
160		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 𝜑)
161		elfznn 13514	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ ℕ)
162	160, 161, 13	syl2an 596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) ∈ ℝ)
163	114	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
164		elfzuz 13481	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝑗 + 1)...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ (ℤ_≥‘(𝑗 + 1)))
165		eluznn 12877	. . . . . . . . . . 11 ⊢ (((𝑗 + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝑗 + 1))) → 𝑘 ∈ ℕ)
166	163, 164, 165	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ ((𝑗 + 1)...((2↑(𝑗 + 1)) − 1))) → 𝑘 ∈ ℕ)
167	160, 166, 43	syl2an2r 685	. . . . . . . . 9 ⊢ ((((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ ((𝑗 + 1)...((2↑(𝑗 + 1)) − 1))) → 0 ≤ (𝐹‘𝑘))
168	159, 155, 162, 167	sermono 13999	. . . . . . . 8 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘𝑗) ≤ (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)))
169	35	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ₀)
170	13, 43, 69, 5	climcndslem1 15815	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗))
171	160, 169, 170	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗))
172	128, 158, 133, 168, 171	letrd 11331	. . . . . . 7 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘𝑗) ≤ (seq0( + , 𝐺)‘𝑗))
173		eqidd 2730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (0...𝑗)) → (𝐺‘𝑛) = (𝐺‘𝑛))
174	169, 102	eleqtrdi 2838	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ_≥‘0))
175		elfznn0 13581	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...𝑗) → 𝑛 ∈ ℕ₀)
176	160, 175, 105	syl2an 596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (0...𝑗)) → (𝐺‘𝑛) ∈ ℂ)
177	173, 174, 176	fsumser 15696	. . . . . . . 8 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → Σ𝑛 ∈ (0...𝑗)(𝐺‘𝑛) = (seq0( + , 𝐺)‘𝑗))
178		0zd 12541	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → 0 ∈ ℤ)
179		fzfid 13938	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (0...𝑗) ∈ Fin)
180	175	ssriv 3950	. . . . . . . . . 10 ⊢ (0...𝑗) ⊆ ℕ₀
181	180	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (0...𝑗) ⊆ ℕ₀)
182		eqidd 2730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) = (𝐺‘𝑛))
183	18	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ ℝ)
184	48	ad4ant14 752	. . . . . . . . 9 ⊢ ((((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ₀) → 0 ≤ (𝐺‘𝑛))
185		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → seq0( + , 𝐺) ∈ dom ⇝ )
186	102, 178, 179, 181, 182, 183, 184, 185	isumless 15811	. . . . . . . 8 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → Σ𝑛 ∈ (0...𝑗)(𝐺‘𝑛) ≤ Σ𝑛 ∈ ℕ₀ (𝐺‘𝑛))
187	177, 186	eqbrtrrd 5131	. . . . . . 7 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq0( + , 𝐺)‘𝑗) ≤ Σ𝑛 ∈ ℕ₀ (𝐺‘𝑛))
188	128, 133, 134, 172, 187	letrd 11331	. . . . . 6 ⊢ (((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘𝑗) ≤ Σ𝑛 ∈ ℕ₀ (𝐺‘𝑛))
189	188	ralrimiva 3125	. . . . 5 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → ∀𝑗 ∈ ℕ (seq1( + , 𝐹)‘𝑗) ≤ Σ𝑛 ∈ ℕ₀ (𝐺‘𝑛))
190		brralrspcev 5167	. . . . 5 ⊢ ((Σ𝑛 ∈ ℕ₀ (𝐺‘𝑛) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (seq1( + , 𝐹)‘𝑗) ≤ Σ𝑛 ∈ ℕ₀ (𝐺‘𝑛)) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (seq1( + , 𝐹)‘𝑗) ≤ 𝑥)
191	127, 189, 190	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (seq1( + , 𝐹)‘𝑗) ≤ 𝑥)
192	1, 109, 110, 122, 191	climsup 15636	. . 3 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → seq1( + , 𝐹) ⇝ sup(ran seq1( + , 𝐹), ℝ, < ))
193	99	releldmi 5912	. . 3 ⊢ (seq1( + , 𝐹) ⇝ sup(ran seq1( + , 𝐹), ℝ, < ) → seq1( + , 𝐹) ∈ dom ⇝ )
194	192, 193	syl 17	. 2 ⊢ ((𝜑 ∧ seq0( + , 𝐺) ∈ dom ⇝ ) → seq1( + , 𝐹) ∈ dom ⇝ )
195	108, 194	impbida 800	1 ⊢ (𝜑 → (seq1( + , 𝐹) ∈ dom ⇝ ↔ seq0( + , 𝐺) ∈ dom ⇝ ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ⊆ wss 3914 class class class wbr 5107 dom cdm 5638 ran crn 5639 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 supcsup 9391 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 ...cfz 13468 seqcseq 13966 ↑cexp 14026 ⇝ cli 15450 Σcsu 15652

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-er 8671 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-ico 13312 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455 df-sum 15653

This theorem is referenced by: harmonic 15825 zetacvg 26925

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