Theorem pntlemg 26168

Description: Lemma for pnt 26184. 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 497	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ+)
4	3	rpred 12425	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5		1red 10636	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
6		pntlem1.y	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))
7	6	simpld 497	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ+)
8	7	rpred 12425	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
9	6	simprd 498	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑌)
10	2	simprd 498	. . . . . . . 8 ⊢ (𝜑 → 𝑌 < 𝑋)
11	5, 8, 4, 9, 10	lelttrd 10792	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑋)
12	4, 11	rplogcld 25206	. . . . . 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 26165	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)))
24	23	simp2d 1139	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ+)
25	24	rpred 12425	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ)
26	23	simp3d 1140	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))
27	26	simp2d 1139	. . . . . . 7 ⊢ (𝜑 → 1 < 𝐾)
28	25, 27	rplogcld 25206	. . . . . 6 ⊢ (𝜑 → (log‘𝐾) ∈ ℝ+)
29	12, 28	rpdivcld 12442	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ+)
30	29	rprege0d 12432	. . . 4 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))))
31		flge0nn0 13184	. . . 4 ⊢ ((((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))) → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ0)
32		nn0p1nn 11930	. . . 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 2917	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
35	34	nnzd 12080	. . 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 26167	. . . . . . . . 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 1138	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ+)
42	41	relogcld 25200	. . . . . . 7 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
43	42, 28	rerpdivcld 12456	. . . . . 6 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
44	43	rehalfcld 11878	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ)
45	44	flcld 13162	. . . 4 ⊢ (𝜑 → (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) ∈ ℤ)
46	36, 45	eqeltrid 2917	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
47		0red 10638	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
48		4nn 11714	. . . . . 6 ⊢ 4 ∈ ℕ
49		nndivre 11672	. . . . . 6 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
50	43, 48, 49	sylancl 588	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
51	46	zred 12081	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
52	34	nnred 11647	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
53	51, 52	resubcld 11062	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℝ)
54	41	rpred 12425	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ ℝ)
55	40	simp2d 1139	. . . . . . . . . 10 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
56	55	simp1d 1138	. . . . . . . . 9 ⊢ (𝜑 → 1 < 𝑍)
57	54, 56	rplogcld 25206	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ+)
58	57, 28	rpdivcld 12442	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ+)
59		4re 11715	. . . . . . . 8 ⊢ 4 ∈ ℝ
60		4pos 11738	. . . . . . . 8 ⊢ 0 < 4
61	59, 60	elrpii 12386	. . . . . . 7 ⊢ 4 ∈ ℝ+
62		rpdivcl 12408	. . . . . . 7 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ+ ∧ 4 ∈ ℝ+) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ+)
63	58, 61, 62	sylancl 588	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ+)
64	63	rpge0d 12429	. . . . 5 ⊢ (𝜑 → 0 ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
65	50	recnd 10663	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
66	34	nncnd 11648	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67		1cnd 10630	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
68	65, 66, 67	addassd 10657	. . . . . . . 8 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)))
69	52, 5	readdcld 10664	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
70	50, 69	readdcld 10664	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ∈ ℝ)
71		peano2re 10807	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
72	51, 71	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
73	29	rpred 12425	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ)
74		2re 11705	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℝ)
76	73, 75	readdcld 10664	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ∈ ℝ)
77		reflcl 13160	. . . . . . . . . . . . . . . . 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 10663	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℂ)
80	79, 67, 67	addassd 10657	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1)))
81	1	oveq1i 7160	. . . . . . . . . . . . . 14 ⊢ (𝑀 + 1) = (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1)
82		df-2 11694	. . . . . . . . . . . . . . 15 ⊢ 2 = (1 + 1)
83	82	oveq2i 7161	. . . . . . . . . . . . . 14 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1))
84	80, 81, 83	3eqtr4g 2881	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2))
85		flle 13163	. . . . . . . . . . . . . . 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 11248	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
88	84, 87	eqbrtrd 5081	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 + 1) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
89	40	simp3d 1140	. . . . . . . . . . . . 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 1139	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
91	69, 76, 50, 88, 90	letrd 10791	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
92	69, 50, 50, 91	leadd2dd 11249	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
93	43	recnd 10663	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
94		2cnd 11709	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
95		2ne0 11735	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
96	95	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 0)
97	93, 94, 94, 96, 96	divdiv1d 11441	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / (2 · 2)))
98		2t2e4 11795	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
99	98	oveq2i 7161	. . . . . . . . . . . . 13 ⊢ (((log‘𝑍) / (log‘𝐾)) / (2 · 2)) = (((log‘𝑍) / (log‘𝐾)) / 4)
100	97, 99	syl6eq 2872	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / 4))
101	100	oveq2d 7166	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (2 · (((log‘𝑍) / (log‘𝐾)) / 4)))
102	44	recnd 10663	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℂ)
103	102, 94, 96	divcan2d 11412	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (((log‘𝑍) / (log‘𝐾)) / 2))
104	65	2timesd 11874	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
105	101, 103, 104	3eqtr3d 2864	. . . . . . . . . 10 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
106	92, 105	breqtrrd 5087	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (((log‘𝑍) / (log‘𝐾)) / 2))
107		fllep1 13165	. . . . . . . . . . 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 7160	. . . . . . . . . 10 ⊢ (𝑁 + 1) = ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1)
110	108, 109	breqtrrdi 5101	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ (𝑁 + 1))
111	70, 44, 72, 106, 110	letrd 10791	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (𝑁 + 1))
112	68, 111	eqbrtrd 5081	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1))
113	50, 52	readdcld 10664	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ∈ ℝ)
114	113, 51, 5	leadd1d 11228	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1)))
115	112, 114	mpbird 259	. . . . . 6 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁)
116		leaddsub 11110	. . . . . . 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 1367	. . . . . 6 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
118	115, 117	mpbid 234	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀))
119	47, 50, 53, 64, 118	letrd 10791	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑁 − 𝑀))
120	51, 52	subge0d 11224	. . . 4 ⊢ (𝜑 → (0 ≤ (𝑁 − 𝑀) ↔ 𝑀 ≤ 𝑁))
121	119, 120	mpbid 234	. . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
122		eluz2 12243	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
123	35, 46, 121, 122	syl3anbrc 1339	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
124	34, 123, 118	3jca 1124	1 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   class class class wbr 5059   ↦ cmpt 5139  ‘cfv 6350  (class class class)co 7150  ℝcr 10530  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536  +∞cpnf 10666   < clt 10669   ≤ cle 10670   − cmin 10864   / cdiv 11291  ℕcn 11632  2c2 11686  3c3 11687  4c4 11688  ℕ₀cn0 11891  ℤcz 11975  ;cdc 12092  ℤ_≥cuz 12237  ℝ⁺crp 12383  (,)cioo 12732  [,)cico 12734  ⌊cfl 13154  ↑cexp 13423  √csqrt 14586  expce 15409  eceu 15410  logclog 25132  ψcchp 25664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-addf 10610  ax-mulf 10611

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-fi 8869  df-sup 8900  df-inf 8901  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-ioo 12736  df-ioc 12737  df-ico 12738  df-icc 12739  df-fz 12887  df-fzo 13028  df-fl 13156  df-mod 13232  df-seq 13364  df-exp 13424  df-fac 13628  df-bc 13657  df-hash 13685  df-shft 14420  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-limsup 14822  df-clim 14839  df-rlim 14840  df-sum 15037  df-ef 15415  df-e 15416  df-sin 15417  df-cos 15418  df-pi 15420  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-starv 16574  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-unif 16582  df-hom 16583  df-cco 16584  df-rest 16690  df-topn 16691  df-0g 16709  df-gsum 16710  df-topgen 16711  df-pt 16712  df-prds 16715  df-xrs 16769  df-qtop 16774  df-imas 16775  df-xps 16777  df-mre 16851  df-mrc 16852  df-acs 16854  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-submnd 17951  df-mulg 18219  df-cntz 18441  df-cmn 18902  df-psmet 20531  df-xmet 20532  df-met 20533  df-bl 20534  df-mopn 20535  df-fbas 20536  df-fg 20537  df-cnfld 20540  df-top 21496  df-topon 21513  df-topsp 21535  df-bases 21548  df-cld 21621  df-ntr 21622  df-cls 21623  df-nei 21700  df-lp 21738  df-perf 21739  df-cn 21829  df-cnp 21830  df-haus 21917  df-tx 22164  df-hmeo 22357  df-fil 22448  df-fm 22540  df-flim 22541  df-flf 22542  df-xms 22924  df-ms 22925  df-tms 22926  df-cncf 23480  df-limc 24458  df-dv 24459  df-log 25134

This theorem is referenced by:  pntlemh  26169  pntlemq  26171  pntlemr  26172  pntlemj  26173  pntlemf  26175