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Theorem pntlemg 27660

Description: Lemma for pnt 27676. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑀 is j^* and 𝑁 is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)

**Hypotheses**
Ref	Expression
pntlem1.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
pntlem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
pntlem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pntlem1.l	⊢ (𝜑 → 𝐿 ∈ (0(,)1))
pntlem1.d	⊢ 𝐷 = (𝐴 + 1)
pntlem1.f	⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
pntlem1.u	⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
pntlem1.u2	⊢ (𝜑 → 𝑈 ≤ 𝐴)
pntlem1.e	⊢ 𝐸 = (𝑈 / 𝐷)
pntlem1.k	⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
pntlem1.y	⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
pntlem1.x	⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
pntlem1.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
pntlem1.w	⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
pntlem1.z	⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞))
pntlem1.m	⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
pntlem1.n	⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))

**Assertion**
Ref	Expression
pntlemg	⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))

Distinct variable group: 𝐸,𝑎 Allowed substitution hints: 𝜑(𝑎) 𝐴(𝑎) 𝐵(𝑎) 𝐶(𝑎) 𝐷(𝑎) 𝑅(𝑎) 𝑈(𝑎) 𝐹(𝑎) 𝐾(𝑎) 𝐿(𝑎) 𝑀(𝑎) 𝑁(𝑎) 𝑊(𝑎) 𝑋(𝑎) 𝑌(𝑎) 𝑍(𝑎)

**Proof of Theorem pntlemg**
Step	Hyp	Ref	Expression
1		pntlem1.m	. . 3 ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
2		pntlem1.x	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
3	2	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
4	3	rpred 13099	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5		1red 11291	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
6		pntlem1.y	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
7	6	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
8	7	rpred 13099	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
9	6	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑌)
10	2	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑌 < 𝑋)
11	5, 8, 4, 9, 10	lelttrd 11448	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑋)
12	4, 11	rplogcld 26689	. . . . . 6 ⊢ (𝜑 → (log‘𝑋) ∈ ℝ⁺)
13		pntlem1.r	. . . . . . . . . 10 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
14		pntlem1.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
15		pntlem1.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
16		pntlem1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (0(,)1))
17		pntlem1.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝐴 + 1)
18		pntlem1.f	. . . . . . . . . 10 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
19		pntlem1.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
20		pntlem1.u2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
21		pntlem1.e	. . . . . . . . . 10 ⊢ 𝐸 = (𝑈 / 𝐷)
22		pntlem1.k	. . . . . . . . . 10 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
23	13, 14, 15, 16, 17, 18, 19, 20, 21, 22	pntlemc 27657	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
24	23	simp2d 1143	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
25	24	rpred 13099	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ)
26	23	simp3d 1144	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
27	26	simp2d 1143	. . . . . . 7 ⊢ (𝜑 → 1 < 𝐾)
28	25, 27	rplogcld 26689	. . . . . 6 ⊢ (𝜑 → (log‘𝐾) ∈ ℝ⁺)
29	12, 28	rpdivcld 13116	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ⁺)
30	29	rprege0d 13106	. . . 4 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))))
31		flge0nn0 13871	. . . 4 ⊢ ((((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))) → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ₀)
32		nn0p1nn 12592	. . . 4 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ₀ → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) ∈ ℕ)
33	30, 31, 32	3syl 18	. . 3 ⊢ (𝜑 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) ∈ ℕ)
34	1, 33	eqeltrid 2848	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
35	34	nnzd 12666	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36		pntlem1.n	. . . 4 ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
37		pntlem1.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
38		pntlem1.w	. . . . . . . . . 10 ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
39		pntlem1.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞))
40	13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 6, 2, 37, 38, 39	pntlemb 27659	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 ∈ ℝ⁺ ∧ (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)) ∧ ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))))
41	40	simp1d 1142	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
42	41	relogcld 26683	. . . . . . 7 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
43	42, 28	rerpdivcld 13130	. . . . . 6 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
44	43	rehalfcld 12540	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ)
45	44	flcld 13849	. . . 4 ⊢ (𝜑 → (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) ∈ ℤ)
46	36, 45	eqeltrid 2848	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
47		0red 11293	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
48		4nn 12376	. . . . . 6 ⊢ 4 ∈ ℕ
49		nndivre 12334	. . . . . 6 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
50	43, 48, 49	sylancl 585	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
51	46	zred 12747	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
52	34	nnred 12308	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
53	51, 52	resubcld 11718	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℝ)
54	41	rpred 13099	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ ℝ)
55	40	simp2d 1143	. . . . . . . . . 10 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
56	55	simp1d 1142	. . . . . . . . 9 ⊢ (𝜑 → 1 < 𝑍)
57	54, 56	rplogcld 26689	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ⁺)
58	57, 28	rpdivcld 13116	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺)
59		4re 12377	. . . . . . . 8 ⊢ 4 ∈ ℝ
60		4pos 12400	. . . . . . . 8 ⊢ 0 < 4
61	59, 60	elrpii 13060	. . . . . . 7 ⊢ 4 ∈ ℝ⁺
62		rpdivcl 13082	. . . . . . 7 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺ ∧ 4 ∈ ℝ⁺) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ⁺)
63	58, 61, 62	sylancl 585	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ⁺)
64	63	rpge0d 13103	. . . . 5 ⊢ (𝜑 → 0 ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
65	50	recnd 11318	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
66	34	nncnd 12309	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67		1cnd 11285	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
68	65, 66, 67	addassd 11312	. . . . . . . 8 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)))
69	52, 5	readdcld 11319	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
70	50, 69	readdcld 11319	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ∈ ℝ)
71		peano2re 11463	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
72	51, 71	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
73	29	rpred 13099	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ)
74		2re 12367	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℝ)
76	73, 75	readdcld 11319	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ∈ ℝ)
77		reflcl 13847	. . . . . . . . . . . . . . . . 17 ⊢ (((log‘𝑋) / (log‘𝐾)) ∈ ℝ → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℝ)
78	73, 77	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℝ)
79	78	recnd 11318	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℂ)
80	79, 67, 67	addassd 11312	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1)))
81	1	oveq1i 7458	. . . . . . . . . . . . . 14 ⊢ (𝑀 + 1) = (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1)
82		df-2 12356	. . . . . . . . . . . . . . 15 ⊢ 2 = (1 + 1)
83	82	oveq2i 7459	. . . . . . . . . . . . . 14 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1))
84	80, 81, 83	3eqtr4g 2805	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2))
85		flle 13850	. . . . . . . . . . . . . . 15 ⊢ (((log‘𝑋) / (log‘𝐾)) ∈ ℝ → (⌊‘((log‘𝑋) / (log‘𝐾))) ≤ ((log‘𝑋) / (log‘𝐾)))
86	73, 85	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ≤ ((log‘𝑋) / (log‘𝐾)))
87	78, 73, 75, 86	leadd1dd 11904	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
88	84, 87	eqbrtrd 5188	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 + 1) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
89	40	simp3d 1144	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))
90	89	simp2d 1143	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
91	69, 76, 50, 88, 90	letrd 11447	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
92	69, 50, 50, 91	leadd2dd 11905	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
93	43	recnd 11318	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
94		2cnd 12371	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
95		2ne0 12397	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
96	95	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 0)
97	93, 94, 94, 96, 96	divdiv1d 12101	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / (2 · 2)))
98		2t2e4 12457	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
99	98	oveq2i 7459	. . . . . . . . . . . . 13 ⊢ (((log‘𝑍) / (log‘𝐾)) / (2 · 2)) = (((log‘𝑍) / (log‘𝐾)) / 4)
100	97, 99	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / 4))
101	100	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (2 · (((log‘𝑍) / (log‘𝐾)) / 4)))
102	44	recnd 11318	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℂ)
103	102, 94, 96	divcan2d 12072	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (((log‘𝑍) / (log‘𝐾)) / 2))
104	65	2timesd 12536	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
105	101, 103, 104	3eqtr3d 2788	. . . . . . . . . 10 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
106	92, 105	breqtrrd 5194	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (((log‘𝑍) / (log‘𝐾)) / 2))
107		fllep1 13852	. . . . . . . . . . 11 ⊢ ((((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1))
108	44, 107	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1))
109	36	oveq1i 7458	. . . . . . . . . 10 ⊢ (𝑁 + 1) = ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1)
110	108, 109	breqtrrdi 5208	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ (𝑁 + 1))
111	70, 44, 72, 106, 110	letrd 11447	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (𝑁 + 1))
112	68, 111	eqbrtrd 5188	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1))
113	50, 52	readdcld 11319	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ∈ ℝ)
114	113, 51, 5	leadd1d 11884	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1)))
115	112, 114	mpbird 257	. . . . . 6 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁)
116		leaddsub 11766	. . . . . . 7 ⊢ (((((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
117	50, 52, 51, 116	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
118	115, 117	mpbid 232	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀))
119	47, 50, 53, 64, 118	letrd 11447	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑁 − 𝑀))
120	51, 52	subge0d 11880	. . . 4 ⊢ (𝜑 → (0 ≤ (𝑁 − 𝑀) ↔ 𝑀 ≤ 𝑁))
121	119, 120	mpbid 232	. . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
122		eluz2 12909	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
123	35, 46, 121, 122	syl3anbrc 1343	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
124	34, 123, 118	3jca 1128	1 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 +∞cpnf 11321 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 3c3 12349 4c4 12350 ℕ₀cn0 12553 ℤcz 12639 cdc 12758 ℤ_≥cuz 12903 ℝ⁺crp 13057 (,)cioo 13407 [,)cico 13409 ⌊cfl 13841 ↑cexp 14112 √csqrt 15282 expce 16109 eceu 16110 logclog 26614 ψcchp 27154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-e 16116 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616

This theorem is referenced by: pntlemh 27661 pntlemq 27663 pntlemr 27664 pntlemj 27665 pntlemf 27667

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