geomcau - Mathbox for Jeff Madsen

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Theorem geomcau 37094

Description: If the distance between consecutive points in a sequence is bounded by a geometric sequence, then the sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)

**Hypotheses**
Ref	Expression
lmclim2.2	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
lmclim2.3	⊢ (𝜑 → 𝐹:ℕ⟶𝑋)
geomcau.4	⊢ (𝜑 → 𝐴 ∈ ℝ)
geomcau.5	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
geomcau.6	⊢ (𝜑 → 𝐵 < 1)
geomcau.7	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (𝐵↑𝑘)))

**Assertion**
Ref	Expression
geomcau	⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))

Distinct variable groups: 𝐷,𝑘 𝑘,𝐹 𝑘,𝑋 𝐴,𝑘 𝐵,𝑘 𝜑,𝑘

Proof of Theorem geomcau Dummy variables 𝑗 𝑛 𝑥 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12872	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12600	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
3		geomcau.5	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
4	3	rpcnd 13025	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
5	3	rpred 13023	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6	3	rpge0d 13027	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝐵)
7	5, 6	absidd 15376	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐵) = 𝐵)
8		geomcau.6	. . . . . . . 8 ⊢ (𝜑 → 𝐵 < 1)
9	7, 8	eqbrtrd 5170	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐵) < 1)
10	4, 9	expcnv 15817	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ (𝐵↑𝑚)) ⇝ 0)
11		geomcau.4	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12		1re 11221	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
13		resubcl 11531	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1 − 𝐵) ∈ ℝ)
14	12, 5, 13	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (1 − 𝐵) ∈ ℝ)
15		posdif 11714	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐵 < 1 ↔ 0 < (1 − 𝐵)))
16	5, 12, 15	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 < 1 ↔ 0 < (1 − 𝐵)))
17	8, 16	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → 0 < (1 − 𝐵))
18	14, 17	elrpd 13020	. . . . . . . 8 ⊢ (𝜑 → (1 − 𝐵) ∈ ℝ⁺)
19	11, 18	rerpdivcld 13054	. . . . . . 7 ⊢ (𝜑 → (𝐴 / (1 − 𝐵)) ∈ ℝ)
20	19	recnd 11249	. . . . . 6 ⊢ (𝜑 → (𝐴 / (1 − 𝐵)) ∈ ℂ)
21		nnex 12225	. . . . . . . 8 ⊢ ℕ ∈ V
22	21	mptex 7227	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵)))) ∈ V
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵)))) ∈ V)
24		nnnn0 12486	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
25	24	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
26		oveq2 7420	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐵↑𝑚) = (𝐵↑𝑛))
27		eqid 2731	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ₀ ↦ (𝐵↑𝑚)) = (𝑚 ∈ ℕ₀ ↦ (𝐵↑𝑚))
28		ovex 7445	. . . . . . . . 9 ⊢ (𝐵↑𝑛) ∈ V
29	26, 27, 28	fvmpt 6998	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((𝑚 ∈ ℕ₀ ↦ (𝐵↑𝑚))‘𝑛) = (𝐵↑𝑛))
30	25, 29	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ₀ ↦ (𝐵↑𝑚))‘𝑛) = (𝐵↑𝑛))
31		nnz 12586	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
32		rpexpcl 14053	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑛 ∈ ℤ) → (𝐵↑𝑛) ∈ ℝ⁺)
33	3, 31, 32	syl2an 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵↑𝑛) ∈ ℝ⁺)
34	33	rpcnd 13025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵↑𝑛) ∈ ℂ)
35	30, 34	eqeltrd 2832	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ₀ ↦ (𝐵↑𝑚))‘𝑛) ∈ ℂ)
36	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (1 − 𝐵)) ∈ ℂ)
37	34, 36	mulcomd 11242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵↑𝑛) · (𝐴 / (1 − 𝐵))) = ((𝐴 / (1 − 𝐵)) · (𝐵↑𝑛)))
38	26	oveq1d 7427	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵))) = ((𝐵↑𝑛) · (𝐴 / (1 − 𝐵))))
39		eqid 2731	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵)))) = (𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵))))
40		ovex 7445	. . . . . . . . 9 ⊢ ((𝐵↑𝑛) · (𝐴 / (1 − 𝐵))) ∈ V
41	38, 39, 40	fvmpt 6998	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵))))‘𝑛) = ((𝐵↑𝑛) · (𝐴 / (1 − 𝐵))))
42	41	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵))))‘𝑛) = ((𝐵↑𝑛) · (𝐴 / (1 − 𝐵))))
43	30	oveq2d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴 / (1 − 𝐵)) · ((𝑚 ∈ ℕ₀ ↦ (𝐵↑𝑚))‘𝑛)) = ((𝐴 / (1 − 𝐵)) · (𝐵↑𝑛)))
44	37, 42, 43	3eqtr4d 2781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵))))‘𝑛) = ((𝐴 / (1 − 𝐵)) · ((𝑚 ∈ ℕ₀ ↦ (𝐵↑𝑚))‘𝑛)))
45	1, 2, 10, 20, 23, 35, 44	climmulc2 15588	. . . . 5 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵)))) ⇝ ((𝐴 / (1 − 𝐵)) · 0))
46	20	mul01d 11420	. . . . 5 ⊢ (𝜑 → ((𝐴 / (1 − 𝐵)) · 0) = 0)
47	45, 46	breqtrd 5174	. . . 4 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵)))) ⇝ 0)
48	33	rpred 13023	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵↑𝑛) ∈ ℝ)
49	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (1 − 𝐵)) ∈ ℝ)
50	48, 49	remulcld 11251	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵↑𝑛) · (𝐴 / (1 − 𝐵))) ∈ ℝ)
51	50	recnd 11249	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵↑𝑛) · (𝐴 / (1 − 𝐵))) ∈ ℂ)
52	1, 2, 23, 42, 51	clim0c 15458	. . . 4 ⊢ (𝜑 → ((𝑚 ∈ ℕ ↦ ((𝐵↑𝑚) · (𝐴 / (1 − 𝐵)))) ⇝ 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘((𝐵↑𝑛) · (𝐴 / (1 − 𝐵)))) < 𝑥))
53	47, 52	mpbid 231	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘((𝐵↑𝑛) · (𝐴 / (1 − 𝐵)))) < 𝑥)
54		nnz 12586	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
55	54	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
56		uzid 12844	. . . . . . 7 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
57		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝐵↑𝑛) = (𝐵↑𝑗))
58	57	fvoveq1d 7434	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (abs‘((𝐵↑𝑛) · (𝐴 / (1 − 𝐵)))) = (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))))
59	58	breq1d 5158	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ((abs‘((𝐵↑𝑛) · (𝐴 / (1 − 𝐵)))) < 𝑥 ↔ (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) < 𝑥))
60	59	rspcv 3608	. . . . . . 7 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘((𝐵↑𝑛) · (𝐴 / (1 − 𝐵)))) < 𝑥 → (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) < 𝑥))
61	55, 56, 60	3syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℕ) → (∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘((𝐵↑𝑛) · (𝐴 / (1 − 𝐵)))) < 𝑥 → (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) < 𝑥))
62		lmclim2.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
63	62	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝐷 ∈ (Met‘𝑋))
64		lmclim2.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋)
65		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑗 ∈ ℕ)
66		ffvelcdm 7083	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶𝑋 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) ∈ 𝑋)
67	64, 65, 66	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ∈ 𝑋)
68		eluznn 12909	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ ℕ)
69		ffvelcdm 7083	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶𝑋 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑋)
70	64, 68, 69	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑛) ∈ 𝑋)
71		metcl 24159	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑗) ∈ 𝑋 ∧ (𝐹‘𝑛) ∈ 𝑋) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ∈ ℝ)
72	63, 67, 70, 71	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ∈ ℝ)
73		eqid 2731	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑗) = (ℤ_≥‘𝑗)
74		nnnn0 12486	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ₀)
75	74	ad2antrl 725	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℕ₀)
76	75	nn0zd 12591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℤ)
77		oveq2 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑘 → (𝐵↑𝑚) = (𝐵↑𝑘))
78	77	oveq2d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑘 → (𝐴 · (𝐵↑𝑚)) = (𝐴 · (𝐵↑𝑘)))
79		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚))) = (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))
80		ovex 7445	. . . . . . . . . . . . . . 15 ⊢ (𝐴 · (𝐵↑𝑘)) ∈ V
81	78, 79, 80	fvmpt 6998	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → ((𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))‘𝑘) = (𝐴 · (𝐵↑𝑘)))
82	81	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))‘𝑘) = (𝐴 · (𝐵↑𝑘)))
83	11	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐴 ∈ ℝ)
84	5	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐵 ∈ ℝ)
85		eluznn0 12908	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ ℕ₀)
86	75, 85	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ ℕ₀)
87	84, 86	reexpcld 14135	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐵↑𝑘) ∈ ℝ)
88	83, 87	remulcld 11251	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐴 · (𝐵↑𝑘)) ∈ ℝ)
89	88	recnd 11249	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐴 · (𝐵↑𝑘)) ∈ ℂ)
90	11	recnd 11249	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
91	90	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝐴 ∈ ℂ)
92	4	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝐵 ∈ ℂ)
93	9	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘𝐵) < 1)
94		eqid 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐵↑𝑚)) = (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐵↑𝑚))
95		ovex 7445	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵↑𝑘) ∈ V
96	77, 94, 95	fvmpt 6998	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → ((𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐵↑𝑚))‘𝑘) = (𝐵↑𝑘))
97	96	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐵↑𝑚))‘𝑘) = (𝐵↑𝑘))
98	92, 93, 75, 97	geolim2 15824	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → seq𝑗( + , (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐵↑𝑚))) ⇝ ((𝐵↑𝑗) / (1 − 𝐵)))
99	87	recnd 11249	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐵↑𝑘) ∈ ℂ)
100	97, 99	eqeltrd 2832	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐵↑𝑚))‘𝑘) ∈ ℂ)
101	97	oveq2d 7428	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐴 · ((𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐵↑𝑚))‘𝑘)) = (𝐴 · (𝐵↑𝑘)))
102	82, 101	eqtr4d 2774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))‘𝑘) = (𝐴 · ((𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐵↑𝑚))‘𝑘)))
103	73, 76, 91, 98, 100, 102	isermulc2 15611	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → seq𝑗( + , (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))) ⇝ (𝐴 · ((𝐵↑𝑗) / (1 − 𝐵))))
104	3	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝐵 ∈ ℝ⁺)
105	104, 76	rpexpcld 14217	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝐵↑𝑗) ∈ ℝ⁺)
106	105	rpcnd 13025	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝐵↑𝑗) ∈ ℂ)
107	14	recnd 11249	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 − 𝐵) ∈ ℂ)
108	107	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (1 − 𝐵) ∈ ℂ)
109	18	rpne0d 13028	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 − 𝐵) ≠ 0)
110	109	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (1 − 𝐵) ≠ 0)
111	91, 106, 108, 110	div12d 12033	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝐴 · ((𝐵↑𝑗) / (1 − 𝐵))) = ((𝐵↑𝑗) · (𝐴 / (1 − 𝐵))))
112	103, 111	breqtrd 5174	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → seq𝑗( + , (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))) ⇝ ((𝐵↑𝑗) · (𝐴 / (1 − 𝐵))))
113	73, 76, 82, 89, 112	isumclim 15710	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (ℤ_≥‘𝑗)(𝐴 · (𝐵↑𝑘)) = ((𝐵↑𝑗) · (𝐴 / (1 − 𝐵))))
114		seqex 13975	. . . . . . . . . . . . . . 15 ⊢ seq𝑗( + , (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))) ∈ V
115		ovex 7445	. . . . . . . . . . . . . . 15 ⊢ (𝐴 · ((𝐵↑𝑗) / (1 − 𝐵))) ∈ V
116	114, 115	breldm 5908	. . . . . . . . . . . . . 14 ⊢ (seq𝑗( + , (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))) ⇝ (𝐴 · ((𝐵↑𝑗) / (1 − 𝐵))) → seq𝑗( + , (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))) ∈ dom ⇝ )
117	103, 116	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → seq𝑗( + , (𝑚 ∈ (ℤ_≥‘𝑗) ↦ (𝐴 · (𝐵↑𝑚)))) ∈ dom ⇝ )
118	73, 76, 82, 88, 117	isumrecl 15718	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (ℤ_≥‘𝑗)(𝐴 · (𝐵↑𝑘)) ∈ ℝ)
119	113, 118	eqeltrrd 2833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐵↑𝑗) · (𝐴 / (1 − 𝐵))) ∈ ℝ)
120	119	recnd 11249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐵↑𝑗) · (𝐴 / (1 − 𝐵))) ∈ ℂ)
121	120	abscld 15390	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) ∈ ℝ)
122		fzfid 13945	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗...(𝑛 − 1)) ∈ Fin)
123		simpll 764	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑗...(𝑛 − 1))) → 𝜑)
124		elfzuz 13504	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝑗...(𝑛 − 1)) → 𝑘 ∈ (ℤ_≥‘𝑗))
125		simprl 768	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℕ)
126		eluznn 12909	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ ℕ)
127	125, 126	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ ℕ)
128	124, 127	sylan2 592	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑗...(𝑛 − 1))) → 𝑘 ∈ ℕ)
129	62	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (Met‘𝑋))
130	64	ffvelcdmda 7086	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑋)
131		peano2nn 12231	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
132		ffvelcdm 7083	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶𝑋 ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ 𝑋)
133	64, 131, 132	syl2an 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ 𝑋)
134		metcl 24159	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘(𝑘 + 1)) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
135	129, 130, 133, 134	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
136	123, 128, 135	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑗...(𝑛 − 1))) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
137	122, 136	fsumrecl 15687	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (𝑗...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
138		simprr 770	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ (ℤ_≥‘𝑗))
139		elfzuz 13504	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑗...𝑛) → 𝑘 ∈ (ℤ_≥‘𝑗))
140		simpll 764	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
141	140, 127, 130	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ 𝑋)
142	139, 141	sylan2 592	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑗...𝑛)) → (𝐹‘𝑘) ∈ 𝑋)
143	63, 138, 142	mettrifi 37092	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑗...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
144	124, 88	sylan2 592	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑗...(𝑛 − 1))) → (𝐴 · (𝐵↑𝑘)) ∈ ℝ)
145	122, 144	fsumrecl 15687	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (𝑗...(𝑛 − 1))(𝐴 · (𝐵↑𝑘)) ∈ ℝ)
146		geomcau.7	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (𝐵↑𝑘)))
147	123, 128, 146	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑗...(𝑛 − 1))) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (𝐵↑𝑘)))
148	122, 136, 144, 147	fsumle 15752	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (𝑗...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ≤ Σ𝑘 ∈ (𝑗...(𝑛 − 1))(𝐴 · (𝐵↑𝑘)))
149		fzssuz 13549	. . . . . . . . . . . . . . . 16 ⊢ (𝑗...(𝑛 − 1)) ⊆ (ℤ_≥‘𝑗)
150	149	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗...(𝑛 − 1)) ⊆ (ℤ_≥‘𝑗))
151		0red 11224	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ∈ ℝ)
152		nnz 12586	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
153		rpexpcl 14053	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑘 ∈ ℤ) → (𝐵↑𝑘) ∈ ℝ⁺)
154	3, 152, 153	syl2an 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵↑𝑘) ∈ ℝ⁺)
155	135, 154	rerpdivcld 13054	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) / (𝐵↑𝑘)) ∈ ℝ)
156	11	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
157		metge0 24172	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘(𝑘 + 1)) ∈ 𝑋) → 0 ≤ ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
158	129, 130, 133, 157	syl3anc 1370	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
159	135, 154, 158	divge0d 13063	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) / (𝐵↑𝑘)))
160	135, 156, 154	ledivmul2d 13077	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) / (𝐵↑𝑘)) ≤ 𝐴 ↔ ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (𝐵↑𝑘))))
161	146, 160	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) / (𝐵↑𝑘)) ≤ 𝐴)
162	151, 155, 156, 159, 161	letrd 11378	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
163	140, 127, 162	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 0 ≤ 𝐴)
164	140, 127, 154	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐵↑𝑘) ∈ ℝ⁺)
165	164	rpge0d 13027	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 0 ≤ (𝐵↑𝑘))
166	83, 87, 163, 165	mulge0d 11798	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 0 ≤ (𝐴 · (𝐵↑𝑘)))
167	73, 76, 122, 150, 82, 88, 166, 117	isumless 15798	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (𝑗...(𝑛 − 1))(𝐴 · (𝐵↑𝑘)) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑗)(𝐴 · (𝐵↑𝑘)))
168	137, 145, 118, 148, 167	letrd 11378	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (𝑗...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑗)(𝐴 · (𝐵↑𝑘)))
169	72, 137, 118, 143, 168	letrd 11378	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (ℤ_≥‘𝑗)(𝐴 · (𝐵↑𝑘)))
170	169, 113	breqtrd 5174	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ≤ ((𝐵↑𝑗) · (𝐴 / (1 − 𝐵))))
171	119	leabsd 15368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐵↑𝑗) · (𝐴 / (1 − 𝐵))) ≤ (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))))
172	72, 119, 121, 170, 171	letrd 11378	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ≤ (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))))
173	172	adantlr 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ≤ (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))))
174	72	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ∈ ℝ)
175	121	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) ∈ ℝ)
176		rpre 12989	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
177	176	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑥 ∈ ℝ)
178		lelttr 11311	. . . . . . . . . 10 ⊢ ((((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ∈ ℝ ∧ (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ≤ (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) ∧ (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) < 𝑥) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥))
179	174, 175, 177, 178	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝐹‘𝑗)𝐷(𝐹‘𝑛)) ≤ (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) ∧ (abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) < 𝑥) → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥))
180	173, 179	mpand 692	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) < 𝑥 → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥))
181	180	anassrs 467	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → ((abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) < 𝑥 → ((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥))
182	181	ralrimdva 3153	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℕ) → ((abs‘((𝐵↑𝑗) · (𝐴 / (1 − 𝐵)))) < 𝑥 → ∀𝑛 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥))
183	61, 182	syld 47	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℕ) → (∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘((𝐵↑𝑛) · (𝐴 / (1 − 𝐵)))) < 𝑥 → ∀𝑛 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥))
184	183	reximdva 3167	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑗 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘((𝐵↑𝑛) · (𝐴 / (1 − 𝐵)))) < 𝑥 → ∃𝑗 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥))
185	184	ralimdva 3166	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘((𝐵↑𝑛) · (𝐴 / (1 − 𝐵)))) < 𝑥 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥))
186	53, 185	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥)
187		metxmet 24161	. . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
188	62, 187	syl 17	. . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
189		eqidd 2732	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = (𝐹‘𝑛))
190		eqidd 2732	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = (𝐹‘𝑗))
191	1, 188, 2, 189, 190, 64	iscauf 25129	. 2 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑛)) < 𝑥))
192	186, 191	mpbird 257	1 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 Vcvv 3473 ⊆ wss 3948 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 < clt 11255 ≤ cle 11256 − cmin 11451 / cdiv 11878 ℕcn 12219 ℕ₀cn0 12479 ℤcz 12565 ℤ_≥cuz 12829 ℝ⁺crp 12981 ...cfz 13491 seqcseq 13973 ↑cexp 14034 abscabs 15188 ⇝ cli 15435 Σcsu 15639 ∞Metcxmet 21219 Metcmet 21220 Cauccau 25102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-sum 15640 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-cau 25105

This theorem is referenced by: bfplem1 37157

W3C validator