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Theorem pntlemg 27507

Description: Lemma for pnt 27523. 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 12937	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5		1red 11116	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
6		pntlem1.y	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
7	6	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
8	7	rpred 12937	. . . . . . . 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 11274	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑋)
12	4, 11	rplogcld 26536	. . . . . 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 27504	. . . . . . . . 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 12937	. . . . . . 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 26536	. . . . . 6 ⊢ (𝜑 → (log‘𝐾) ∈ ℝ⁺)
29	12, 28	rpdivcld 12954	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ⁺)
30	29	rprege0d 12944	. . . 4 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))))
31		flge0nn0 13724	. . . 4 ⊢ ((((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))) → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ₀)
32		nn0p1nn 12423	. . . 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 2832	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
35	34	nnzd 12498	. . 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 27506	. . . . . . . . 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 26530	. . . . . . 7 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
43	42, 28	rerpdivcld 12968	. . . . . 6 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
44	43	rehalfcld 12371	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ)
45	44	flcld 13702	. . . 4 ⊢ (𝜑 → (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) ∈ ℤ)
46	36, 45	eqeltrid 2832	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
47		0red 11118	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
48		4nn 12211	. . . . . 6 ⊢ 4 ∈ ℕ
49		nndivre 12169	. . . . . 6 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
50	43, 48, 49	sylancl 586	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
51	46	zred 12580	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
52	34	nnred 12143	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
53	51, 52	resubcld 11548	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℝ)
54	41	rpred 12937	. . . . . . . . 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 26536	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ⁺)
58	57, 28	rpdivcld 12954	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺)
59		4re 12212	. . . . . . . 8 ⊢ 4 ∈ ℝ
60		4pos 12235	. . . . . . . 8 ⊢ 0 < 4
61	59, 60	elrpii 12896	. . . . . . 7 ⊢ 4 ∈ ℝ⁺
62		rpdivcl 12920	. . . . . . 7 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺ ∧ 4 ∈ ℝ⁺) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ⁺)
63	58, 61, 62	sylancl 586	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ⁺)
64	63	rpge0d 12941	. . . . 5 ⊢ (𝜑 → 0 ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
65	50	recnd 11143	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
66	34	nncnd 12144	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67		1cnd 11110	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
68	65, 66, 67	addassd 11137	. . . . . . . 8 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)))
69	52, 5	readdcld 11144	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
70	50, 69	readdcld 11144	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ∈ ℝ)
71		peano2re 11289	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
72	51, 71	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
73	29	rpred 12937	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ)
74		2re 12202	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℝ)
76	73, 75	readdcld 11144	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ∈ ℝ)
77		reflcl 13700	. . . . . . . . . . . . . . . . 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 11143	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℂ)
80	79, 67, 67	addassd 11137	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1)))
81	1	oveq1i 7359	. . . . . . . . . . . . . 14 ⊢ (𝑀 + 1) = (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1)
82		df-2 12191	. . . . . . . . . . . . . . 15 ⊢ 2 = (1 + 1)
83	82	oveq2i 7360	. . . . . . . . . . . . . 14 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1))
84	80, 81, 83	3eqtr4g 2789	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2))
85		flle 13703	. . . . . . . . . . . . . . 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 11734	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
88	84, 87	eqbrtrd 5114	. . . . . . . . . . . 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 11273	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
92	69, 50, 50, 91	leadd2dd 11735	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
93	43	recnd 11143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
94		2cnd 12206	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
95		2ne0 12232	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
96	95	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 0)
97	93, 94, 94, 96, 96	divdiv1d 11931	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / (2 · 2)))
98		2t2e4 12287	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
99	98	oveq2i 7360	. . . . . . . . . . . . 13 ⊢ (((log‘𝑍) / (log‘𝐾)) / (2 · 2)) = (((log‘𝑍) / (log‘𝐾)) / 4)
100	97, 99	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / 4))
101	100	oveq2d 7365	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (2 · (((log‘𝑍) / (log‘𝐾)) / 4)))
102	44	recnd 11143	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℂ)
103	102, 94, 96	divcan2d 11902	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (((log‘𝑍) / (log‘𝐾)) / 2))
104	65	2timesd 12367	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
105	101, 103, 104	3eqtr3d 2772	. . . . . . . . . 10 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
106	92, 105	breqtrrd 5120	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (((log‘𝑍) / (log‘𝐾)) / 2))
107		fllep1 13705	. . . . . . . . . . 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 7359	. . . . . . . . . 10 ⊢ (𝑁 + 1) = ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1)
110	108, 109	breqtrrdi 5134	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ (𝑁 + 1))
111	70, 44, 72, 106, 110	letrd 11273	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (𝑁 + 1))
112	68, 111	eqbrtrd 5114	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1))
113	50, 52	readdcld 11144	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ∈ ℝ)
114	113, 51, 5	leadd1d 11714	. . . . . . 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 11596	. . . . . . 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 1373	. . . . . 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 11273	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑁 − 𝑀))
120	51, 52	subge0d 11710	. . . 4 ⊢ (𝜑 → (0 ≤ (𝑁 − 𝑀) ↔ 𝑀 ≤ 𝑁))
121	119, 120	mpbid 232	. . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
122		eluz2 12741	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
123	35, 46, 121, 122	syl3anbrc 1344	. 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 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5092 ↦ cmpt 5173 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 +∞cpnf 11146 < clt 11149 ≤ cle 11150 − cmin 11347 / cdiv 11777 ℕcn 12128 2c2 12183 3c3 12184 4c4 12185 ℕ₀cn0 12384 ℤcz 12471 cdc 12591 ℤ_≥cuz 12735 ℝ⁺crp 12893 (,)cioo 13248 [,)cico 13250 ⌊cfl 13694 ↑cexp 13968 √csqrt 15140 expce 15968 eceu 15969 logclog 26461 ψcchp 27001

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-e 15975 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-log 26463

This theorem is referenced by: pntlemh 27508 pntlemq 27510 pntlemr 27511 pntlemj 27512 pntlemf 27514

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