hashnzfzclim - Mathbox for Steve Rodriguez

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Theorem hashnzfzclim 43672

Description: As the upper bound 𝐾 of the constraint interval (𝐽...𝐾) in hashnzfz 43670 increases, the resulting count of multiples tends to (𝐾 / 𝑀) —that is, there are approximately (𝐾 / 𝑀) multiples of 𝑀 in a finite interval of integers. (Contributed by Steve Rodriguez, 20-Jan-2020.)

**Hypotheses**
Ref	Expression
hashnzfzclim.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
hashnzfzclim.j	⊢ (𝜑 → 𝐽 ∈ ℤ)

**Assertion**
Ref	Expression
hashnzfzclim	⊢ (𝜑 → (𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ ((♯‘(( ∥ “ {𝑀}) ∩ (𝐽...𝑘))) / 𝑘)) ⇝ (1 / 𝑀))

Distinct variable groups: 𝑘,𝐽 𝑘,𝑀 𝜑,𝑘

Proof of Theorem hashnzfzclim Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashnzfzclim.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐽 − 1))) → 𝑀 ∈ ℕ)
3		hashnzfzclim.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ)
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐽 − 1))) → 𝐽 ∈ ℤ)
5		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐽 − 1))) → 𝑘 ∈ (ℤ_≥‘(𝐽 − 1)))
6	2, 4, 5	hashnzfz 43670	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐽 − 1))) → (♯‘(( ∥ “ {𝑀}) ∩ (𝐽...𝑘))) = ((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))))
7	6	oveq1d 7429	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐽 − 1))) → ((♯‘(( ∥ “ {𝑀}) ∩ (𝐽...𝑘))) / 𝑘) = (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘))
8	7	mpteq2dva 5242	. 2 ⊢ (𝜑 → (𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ ((♯‘(( ∥ “ {𝑀}) ∩ (𝐽...𝑘))) / 𝑘)) = (𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)))
9		nnuz 12881	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
10		1z 12608	. . . . . 6 ⊢ 1 ∈ ℤ
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
12	1	nncnd 12244	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
13	1	nnne0d 12278	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
14	12, 13	reccld 11999	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝑀) ∈ ℂ)
15	9	eqimss2i 4039	. . . . . . . . . 10 ⊢ (ℤ_≥‘1) ⊆ ℕ
16		nnex 12234	. . . . . . . . . 10 ⊢ ℕ ∈ V
17	15, 16	climconst2 15510	. . . . . . . . 9 ⊢ (((1 / 𝑀) ∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {(1 / 𝑀)}) ⇝ (1 / 𝑀))
18	14, 10, 17	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → (ℕ × {(1 / 𝑀)}) ⇝ (1 / 𝑀))
19	16	mptex 7229	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘))) ∈ V
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘))) ∈ V)
21		ax-1cn 11182	. . . . . . . . 9 ⊢ 1 ∈ ℂ
22		divcnv 15817	. . . . . . . . 9 ⊢ (1 ∈ ℂ → (𝑘 ∈ ℕ ↦ (1 / 𝑘)) ⇝ 0)
23	21, 22	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (1 / 𝑘)) ⇝ 0)
24		ovex 7447	. . . . . . . . . . 11 ⊢ (1 / 𝑀) ∈ V
25	24	fvconst2 7210	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → ((ℕ × {(1 / 𝑀)})‘𝑥) = (1 / 𝑀))
26	25	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((ℕ × {(1 / 𝑀)})‘𝑥) = (1 / 𝑀))
27	14	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1 / 𝑀) ∈ ℂ)
28	26, 27	eqeltrd 2828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((ℕ × {(1 / 𝑀)})‘𝑥) ∈ ℂ)
29		eqidd 2728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑘 ∈ ℕ ↦ (1 / 𝑘)) = (𝑘 ∈ ℕ ↦ (1 / 𝑘)))
30		oveq2 7422	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑥 → (1 / 𝑘) = (1 / 𝑥))
31	30	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 = 𝑥) → (1 / 𝑘) = (1 / 𝑥))
32		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
33		ovexd 7449	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1 / 𝑥) ∈ V)
34	29, 31, 32, 33	fvmptd 7006	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (1 / 𝑘))‘𝑥) = (1 / 𝑥))
35	32	nnrecred 12279	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1 / 𝑥) ∈ ℝ)
36	34, 35	eqeltrd 2828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (1 / 𝑘))‘𝑥) ∈ ℝ)
37	36	recnd 11258	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (1 / 𝑘))‘𝑥) ∈ ℂ)
38		eqidd 2728	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘))) = (𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘))))
39	30	oveq2d 7430	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → ((1 / 𝑀) − (1 / 𝑘)) = ((1 / 𝑀) − (1 / 𝑥)))
40	39	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 = 𝑥) → ((1 / 𝑀) − (1 / 𝑘)) = ((1 / 𝑀) − (1 / 𝑥)))
41		ovexd 7449	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1 / 𝑀) − (1 / 𝑥)) ∈ V)
42	38, 40, 32, 41	fvmptd 7006	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘)))‘𝑥) = ((1 / 𝑀) − (1 / 𝑥)))
43	26, 34	oveq12d 7432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((ℕ × {(1 / 𝑀)})‘𝑥) − ((𝑘 ∈ ℕ ↦ (1 / 𝑘))‘𝑥)) = ((1 / 𝑀) − (1 / 𝑥)))
44	42, 43	eqtr4d 2770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘)))‘𝑥) = (((ℕ × {(1 / 𝑀)})‘𝑥) − ((𝑘 ∈ ℕ ↦ (1 / 𝑘))‘𝑥)))
45	9, 11, 18, 20, 23, 28, 37, 44	climsub 15596	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘))) ⇝ ((1 / 𝑀) − 0))
46	14	subid1d 11576	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝑀) − 0) = (1 / 𝑀))
47	45, 46	breqtrd 5168	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘))) ⇝ (1 / 𝑀))
48	16	mptex 7229	. . . . . . 7 ⊢ (𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘)) ∈ V
49	48	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘)) ∈ V)
50	1	nnrecred 12279	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝑀) ∈ ℝ)
51	50	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1 / 𝑀) ∈ ℝ)
52		nnre 12235	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
53	52	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ)
54		nnne0 12262	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
55	54	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ≠ 0)
56	53, 55	rereccld 12057	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1 / 𝑥) ∈ ℝ)
57	51, 56	resubcld 11658	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1 / 𝑀) − (1 / 𝑥)) ∈ ℝ)
58	42, 57	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘)))‘𝑥) ∈ ℝ)
59		eqidd 2728	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘)) = (𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘)))
60		fvoveq1 7437	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → (⌊‘(𝑘 / 𝑀)) = (⌊‘(𝑥 / 𝑀)))
61		id 22	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → 𝑘 = 𝑥)
62	60, 61	oveq12d 7432	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → ((⌊‘(𝑘 / 𝑀)) / 𝑘) = ((⌊‘(𝑥 / 𝑀)) / 𝑥))
63	62	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 = 𝑥) → ((⌊‘(𝑘 / 𝑀)) / 𝑘) = ((⌊‘(𝑥 / 𝑀)) / 𝑥))
64		ovexd 7449	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((⌊‘(𝑥 / 𝑀)) / 𝑥) ∈ V)
65	59, 63, 32, 64	fvmptd 7006	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘))‘𝑥) = ((⌊‘(𝑥 / 𝑀)) / 𝑥))
66	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑀 ∈ ℕ)
67	53, 66	nndivred 12282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑥 / 𝑀) ∈ ℝ)
68		reflcl 13779	. . . . . . . . 9 ⊢ ((𝑥 / 𝑀) ∈ ℝ → (⌊‘(𝑥 / 𝑀)) ∈ ℝ)
69	67, 68	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (⌊‘(𝑥 / 𝑀)) ∈ ℝ)
70	69, 53, 55	redivcld 12058	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((⌊‘(𝑥 / 𝑀)) / 𝑥) ∈ ℝ)
71	65, 70	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘))‘𝑥) ∈ ℝ)
72	67	recnd 11258	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑥 / 𝑀) ∈ ℂ)
73		1cnd 11225	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 1 ∈ ℂ)
74		nncn 12236	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
75	74	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℂ)
76	72, 73, 75, 55	divsubdird 12045	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((𝑥 / 𝑀) − 1) / 𝑥) = (((𝑥 / 𝑀) / 𝑥) − (1 / 𝑥)))
77	12	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑀 ∈ ℂ)
78	13	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑀 ≠ 0)
79	75, 77, 78	divrecd 12009	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑥 / 𝑀) = (𝑥 · (1 / 𝑀)))
80	79	oveq1d 7429	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑥 / 𝑀) / 𝑥) = ((𝑥 · (1 / 𝑀)) / 𝑥))
81	27, 75, 55	divcan3d 12011	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑥 · (1 / 𝑀)) / 𝑥) = (1 / 𝑀))
82	80, 81	eqtrd 2767	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑥 / 𝑀) / 𝑥) = (1 / 𝑀))
83	82	oveq1d 7429	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((𝑥 / 𝑀) / 𝑥) − (1 / 𝑥)) = ((1 / 𝑀) − (1 / 𝑥)))
84	76, 83	eqtrd 2767	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((𝑥 / 𝑀) − 1) / 𝑥) = ((1 / 𝑀) − (1 / 𝑥)))
85		1red 11231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 1 ∈ ℝ)
86	67, 85	resubcld 11658	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑥 / 𝑀) − 1) ∈ ℝ)
87		nnrp 13003	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ⁺)
88	87	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ⁺)
89	69, 85	readdcld 11259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((⌊‘(𝑥 / 𝑀)) + 1) ∈ ℝ)
90		flle 13782	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / 𝑀) ∈ ℝ → (⌊‘(𝑥 / 𝑀)) ≤ (𝑥 / 𝑀))
91	67, 90	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (⌊‘(𝑥 / 𝑀)) ≤ (𝑥 / 𝑀))
92		flflp1 13790	. . . . . . . . . . . . . 14 ⊢ (((𝑥 / 𝑀) ∈ ℝ ∧ (𝑥 / 𝑀) ∈ ℝ) → ((⌊‘(𝑥 / 𝑀)) ≤ (𝑥 / 𝑀) ↔ (𝑥 / 𝑀) < ((⌊‘(𝑥 / 𝑀)) + 1)))
93	67, 67, 92	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((⌊‘(𝑥 / 𝑀)) ≤ (𝑥 / 𝑀) ↔ (𝑥 / 𝑀) < ((⌊‘(𝑥 / 𝑀)) + 1)))
94	91, 93	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑥 / 𝑀) < ((⌊‘(𝑥 / 𝑀)) + 1))
95	67, 89, 85, 94	ltsub1dd 11842	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑥 / 𝑀) − 1) < (((⌊‘(𝑥 / 𝑀)) + 1) − 1))
96	69	recnd 11258	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (⌊‘(𝑥 / 𝑀)) ∈ ℂ)
97	96, 73	pncand 11588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((⌊‘(𝑥 / 𝑀)) + 1) − 1) = (⌊‘(𝑥 / 𝑀)))
98	95, 97	breqtrd 5168	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑥 / 𝑀) − 1) < (⌊‘(𝑥 / 𝑀)))
99	86, 69, 88, 98	ltdiv1dd 13091	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((𝑥 / 𝑀) − 1) / 𝑥) < ((⌊‘(𝑥 / 𝑀)) / 𝑥))
100	84, 99	eqbrtrrd 5166	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1 / 𝑀) − (1 / 𝑥)) < ((⌊‘(𝑥 / 𝑀)) / 𝑥))
101	57, 70, 100	ltled 11378	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1 / 𝑀) − (1 / 𝑥)) ≤ ((⌊‘(𝑥 / 𝑀)) / 𝑥))
102		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 = 𝑥) → 𝑘 = 𝑥)
103	102	fvoveq1d 7436	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 = 𝑥) → (⌊‘(𝑘 / 𝑀)) = (⌊‘(𝑥 / 𝑀)))
104	103, 102	oveq12d 7432	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 = 𝑥) → ((⌊‘(𝑘 / 𝑀)) / 𝑘) = ((⌊‘(𝑥 / 𝑀)) / 𝑥))
105	59, 104, 32, 64	fvmptd 7006	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘))‘𝑥) = ((⌊‘(𝑥 / 𝑀)) / 𝑥))
106	101, 42, 105	3brtr4d 5174	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((1 / 𝑀) − (1 / 𝑘)))‘𝑥) ≤ ((𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘))‘𝑥))
107	69, 67, 88, 91	lediv1dd 13092	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((⌊‘(𝑥 / 𝑀)) / 𝑥) ≤ ((𝑥 / 𝑀) / 𝑥))
108	107, 82	breqtrd 5168	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((⌊‘(𝑥 / 𝑀)) / 𝑥) ≤ (1 / 𝑀))
109	105, 108	eqbrtrd 5164	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘))‘𝑥) ≤ (1 / 𝑀))
110	9, 11, 47, 49, 58, 71, 106, 109	climsqz 15603	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘)) ⇝ (1 / 𝑀))
111	16	mptex 7229	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ∈ V
112	111	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ∈ V)
113	3	zred 12682	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℝ)
114		1red 11231	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
115	113, 114	resubcld 11658	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 − 1) ∈ ℝ)
116	115, 1	nndivred 12282	. . . . . . . 8 ⊢ (𝜑 → ((𝐽 − 1) / 𝑀) ∈ ℝ)
117	116	flcld 13781	. . . . . . 7 ⊢ (𝜑 → (⌊‘((𝐽 − 1) / 𝑀)) ∈ ℤ)
118	117	zcnd 12683	. . . . . 6 ⊢ (𝜑 → (⌊‘((𝐽 − 1) / 𝑀)) ∈ ℂ)
119		divcnv 15817	. . . . . 6 ⊢ ((⌊‘((𝐽 − 1) / 𝑀)) ∈ ℂ → (𝑘 ∈ ℕ ↦ ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘)) ⇝ 0)
120	118, 119	syl 17	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘)) ⇝ 0)
121	71	recnd 11258	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘))‘𝑥) ∈ ℂ)
122		eqidd 2728	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑘 ∈ ℕ ↦ ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘)) = (𝑘 ∈ ℕ ↦ ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘)))
123		oveq2 7422	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘) = ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑥))
124	123	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 = 𝑥) → ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘) = ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑥))
125		ovexd 7449	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑥) ∈ V)
126	122, 124, 32, 125	fvmptd 7006	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘))‘𝑥) = ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑥))
127	118	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (⌊‘((𝐽 − 1) / 𝑀)) ∈ ℂ)
128	127, 75, 55	divcld 12006	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑥) ∈ ℂ)
129	126, 128	eqeltrd 2828	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘))‘𝑥) ∈ ℂ)
130	96, 127, 75, 55	divsubdird 12045	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((⌊‘(𝑥 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑥) = (((⌊‘(𝑥 / 𝑀)) / 𝑥) − ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑥)))
131		eqidd 2728	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) = (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)))
132	60	oveq1d 7429	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → ((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) = ((⌊‘(𝑥 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))))
133	132, 61	oveq12d 7432	. . . . . . . 8 ⊢ (𝑘 = 𝑥 → (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘) = (((⌊‘(𝑥 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑥))
134	133	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 = 𝑥) → (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘) = (((⌊‘(𝑥 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑥))
135		ovexd 7449	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((⌊‘(𝑥 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑥) ∈ V)
136	131, 134, 32, 135	fvmptd 7006	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘))‘𝑥) = (((⌊‘(𝑥 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑥))
137	65, 126	oveq12d 7432	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘))‘𝑥) − ((𝑘 ∈ ℕ ↦ ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘))‘𝑥)) = (((⌊‘(𝑥 / 𝑀)) / 𝑥) − ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑥)))
138	130, 136, 137	3eqtr4d 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘))‘𝑥) = (((𝑘 ∈ ℕ ↦ ((⌊‘(𝑘 / 𝑀)) / 𝑘))‘𝑥) − ((𝑘 ∈ ℕ ↦ ((⌊‘((𝐽 − 1) / 𝑀)) / 𝑘))‘𝑥)))
139	9, 11, 110, 112, 120, 121, 129, 138	climsub 15596	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ ((1 / 𝑀) − 0))
140	139, 46	breqtrd 5168	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀))
141		uzssz 12859	. . . . . . 7 ⊢ (ℤ_≥‘(𝐽 − 1)) ⊆ ℤ
142		resmpt 6035	. . . . . . 7 ⊢ ((ℤ_≥‘(𝐽 − 1)) ⊆ ℤ → ((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ (ℤ_≥‘(𝐽 − 1))) = (𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)))
143	141, 142	ax-mp 5	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ (ℤ_≥‘(𝐽 − 1))) = (𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘))
144	143	breq1i 5149	. . . . 5 ⊢ (((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ (ℤ_≥‘(𝐽 − 1))) ⇝ (1 / 𝑀) ↔ (𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀))
145	3, 11	zsubcld 12687	. . . . . 6 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
146		zex 12583	. . . . . . 7 ⊢ ℤ ∈ V
147	146	mptex 7229	. . . . . 6 ⊢ (𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ∈ V
148		climres 15537	. . . . . 6 ⊢ (((𝐽 − 1) ∈ ℤ ∧ (𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ∈ V) → (((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ (ℤ_≥‘(𝐽 − 1))) ⇝ (1 / 𝑀) ↔ (𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀)))
149	145, 147, 148	sylancl 585	. . . . 5 ⊢ (𝜑 → (((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ (ℤ_≥‘(𝐽 − 1))) ⇝ (1 / 𝑀) ↔ (𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀)))
150	144, 149	bitr3id 285	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀) ↔ (𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀)))
151	9	reseq2i 5976	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ ℕ) = ((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ (ℤ_≥‘1))
152	151	breq1i 5149	. . . . 5 ⊢ (((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ ℕ) ⇝ (1 / 𝑀) ↔ ((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ (ℤ_≥‘1)) ⇝ (1 / 𝑀))
153		nnssz 12596	. . . . . . 7 ⊢ ℕ ⊆ ℤ
154		resmpt 6035	. . . . . . 7 ⊢ (ℕ ⊆ ℤ → ((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ ℕ) = (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)))
155	153, 154	ax-mp 5	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ ℕ) = (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘))
156	155	breq1i 5149	. . . . 5 ⊢ (((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ ℕ) ⇝ (1 / 𝑀) ↔ (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀))
157		climres 15537	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ (𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ∈ V) → (((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ (ℤ_≥‘1)) ⇝ (1 / 𝑀) ↔ (𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀)))
158	10, 147, 157	mp2an 691	. . . . 5 ⊢ (((𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ↾ (ℤ_≥‘1)) ⇝ (1 / 𝑀) ↔ (𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀))
159	152, 156, 158	3bitr3i 301	. . . 4 ⊢ ((𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀) ↔ (𝑘 ∈ ℤ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀))
160	150, 159	bitr4di 289	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀) ↔ (𝑘 ∈ ℕ ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀)))
161	140, 160	mpbird 257	. 2 ⊢ (𝜑 → (𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ (((⌊‘(𝑘 / 𝑀)) − (⌊‘((𝐽 − 1) / 𝑀))) / 𝑘)) ⇝ (1 / 𝑀))
162	8, 161	eqbrtrd 5164	1 ⊢ (𝜑 → (𝑘 ∈ (ℤ_≥‘(𝐽 − 1)) ↦ ((♯‘(( ∥ “ {𝑀}) ∩ (𝐽...𝑘))) / 𝑘)) ⇝ (1 / 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 Vcvv 3469 ∩ cin 3943 ⊆ wss 3944 {csn 4624 class class class wbr 5142 ↦ cmpt 5225 × cxp 5670 ↾ cres 5674 “ cima 5675 ‘cfv 6542 (class class class)co 7414 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 < clt 11264 ≤ cle 11265 − cmin 11460 / cdiv 11887 ℕcn 12228 ℤcz 12574 ℤ_≥cuz 12838 ℝ⁺crp 12992 ...cfz 13502 ⌊cfl 13773 ♯chash 14307 ⇝ cli 15446 ∥ cdvds 16216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-pm 8837 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-fz 13503 df-fl 13775 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-rlim 15451 df-dvds 16217

This theorem is referenced by: (None)

W3C validator