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Theorem pntlemg 27934

Description: Lemma for pnt 27950. 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 500	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
4	3	rpred 13171	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5		1red 11316	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
6		pntlem1.y	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
7	6	simpld 500	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
8	7	rpred 13171	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
9	6	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑌)
10	2	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 𝑌 < 𝑋)
11	5, 8, 4, 9, 10	lelttrd 11475	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑋)
12	4, 11	rplogcld 26966	. . . . . 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 27931	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
24	23	simp2d 1161	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
25	24	rpred 13171	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ)
26	23	simp3d 1162	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
27	26	simp2d 1161	. . . . . . 7 ⊢ (𝜑 → 1 < 𝐾)
28	25, 27	rplogcld 26966	. . . . . 6 ⊢ (𝜑 → (log‘𝐾) ∈ ℝ⁺)
29	12, 28	rpdivcld 13188	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ⁺)
30	29	rprege0d 13178	. . . 4 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))))
31		flge0nn0 13967	. . . 4 ⊢ ((((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))) → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ₀)
32		nn0p1nn 12652	. . . 4 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ₀ → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) ∈ ℕ)
33	30, 31, 32	3syl 19	. . 3 ⊢ (𝜑 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) ∈ ℕ)
34	1, 33	eqeltrid 2864	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
35	34	nnzd 12726	. . 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 27933	. . . . . . . . 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 1160	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
42	41	relogcld 26960	. . . . . . 7 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
43	42, 28	rerpdivcld 13202	. . . . . 6 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
44	43	rehalfcld 12600	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ)
45	44	flcld 13945	. . . 4 ⊢ (𝜑 → (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) ∈ ℤ)
46	36, 45	eqeltrid 2864	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
47		0red 11318	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
48		4nn 12433	. . . . . 6 ⊢ 4 ∈ ℕ
49		nndivre 12386	. . . . . 6 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
50	43, 48, 49	sylancl 598	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
51	46	zred 12810	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
52	34	nnred 12357	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
53	51, 52	resubcld 11751	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℝ)
54	41	rpred 13171	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ ℝ)
55	40	simp2d 1161	. . . . . . . . . 10 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
56	55	simp1d 1160	. . . . . . . . 9 ⊢ (𝜑 → 1 < 𝑍)
57	54, 56	rplogcld 26966	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ⁺)
58	57, 28	rpdivcld 13188	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺)
59		4re 12434	. . . . . . . 8 ⊢ 4 ∈ ℝ
60		4pos 12460	. . . . . . . 8 ⊢ 0 < 4
61	59, 60	elrpii 13130	. . . . . . 7 ⊢ 4 ∈ ℝ⁺
62		rpdivcl 13154	. . . . . . 7 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺ ∧ 4 ∈ ℝ⁺) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ⁺)
63	58, 61, 62	sylancl 598	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ⁺)
64	63	rpge0d 13175	. . . . 5 ⊢ (𝜑 → 0 ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
65	50	recnd 11344	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
66	34	nncnd 12358	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67		1cnd 11309	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
68	65, 66, 67	addassd 11338	. . . . . . . 8 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)))
69	52, 5	readdcld 11345	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
70	50, 69	readdcld 11345	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ∈ ℝ)
71		peano2re 11490	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
72	51, 71	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
73	29	rpred 13171	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ)
74		2re 12424	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℝ)
76	73, 75	readdcld 11345	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ∈ ℝ)
77		reflcl 13943	. . . . . . . . . . . . . . . . 17 ⊢ (((log‘𝑋) / (log‘𝐾)) ∈ ℝ → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℝ)
78	73, 77	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℝ)
79	78	recnd 11344	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℂ)
80	79, 67, 67	addassd 11338	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1)))
81	1	oveq1i 7434	. . . . . . . . . . . . . 14 ⊢ (𝑀 + 1) = (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1)
82		df-2 12412	. . . . . . . . . . . . . . 15 ⊢ 2 = (1 + 1)
83	82	oveq2i 7435	. . . . . . . . . . . . . 14 ⊢ ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1))
84	80, 81, 83	3eqtr4g 2820	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2))
85		flle 13946	. . . . . . . . . . . . . . 15 ⊢ (((log‘𝑋) / (log‘𝐾)) ∈ ℝ → (⌊‘((log‘𝑋) / (log‘𝐾))) ≤ ((log‘𝑋) / (log‘𝐾)))
86	73, 85	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ≤ ((log‘𝑋) / (log‘𝐾)))
87	78, 73, 75, 86	leadd1dd 11937	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
88	84, 87	eqbrtrd 5126	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 + 1) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
89	40	simp3d 1162	. . . . . . . . . . . . 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 1161	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
91	69, 76, 50, 88, 90	letrd 11474	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
92	69, 50, 50, 91	leadd2dd 11938	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
93	43	recnd 11344	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
94		2cnd 12428	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
95		2ne0 12456	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
96	95	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 0)
97	93, 94, 94, 96, 96	divdiv1d 12131	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / (2 · 2)))
98		2t2e4 12513	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
99	98	oveq2i 7435	. . . . . . . . . . . . 13 ⊢ (((log‘𝑍) / (log‘𝐾)) / (2 · 2)) = (((log‘𝑍) / (log‘𝐾)) / 4)
100	97, 99	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / 4))
101	100	oveq2d 7440	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (2 · (((log‘𝑍) / (log‘𝐾)) / 4)))
102	44	recnd 11344	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℂ)
103	102, 94, 96	divcan2d 12102	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (((log‘𝑍) / (log‘𝐾)) / 2))
104	65	2timesd 12596	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
105	101, 103, 104	3eqtr3d 2803	. . . . . . . . . 10 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
106	92, 105	breqtrrd 5132	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (((log‘𝑍) / (log‘𝐾)) / 2))
107		fllep1 13948	. . . . . . . . . . 11 ⊢ ((((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1))
108	44, 107	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1))
109	36	oveq1i 7434	. . . . . . . . . 10 ⊢ (𝑁 + 1) = ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1)
110	108, 109	breqtrrdi 5146	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ (𝑁 + 1))
111	70, 44, 72, 106, 110	letrd 11474	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (𝑁 + 1))
112	68, 111	eqbrtrd 5126	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1))
113	50, 52	readdcld 11345	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ∈ ℝ)
114	113, 51, 5	leadd1d 11917	. . . . . . 7 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1)))
115	112, 114	mpbird 260	. . . . . 6 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁)
116		leaddsub 11799	. . . . . . 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 1398	. . . . . 6 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
118	115, 117	mpbid 235	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀))
119	47, 50, 53, 64, 118	letrd 11474	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑁 − 𝑀))
120	51, 52	subge0d 11913	. . . 4 ⊢ (𝜑 → (0 ≤ (𝑁 − 𝑀) ↔ 𝑀 ≤ 𝑁))
121	119, 120	mpbid 235	. . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
122		eluz2 12978	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
123	35, 46, 121, 122	syl3anbrc 1362	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
124	34, 123, 118	3jca 1146	1 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6543 (class class class)co 7424 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 · cmul 11212 +∞cpnf 11347 < clt 11350 ≤ cle 11351 − cmin 11548 / cdiv 11980 ℕcn 12342 2c2 12404 3c3 12405 4c4 12406 ℕ₀cn0 12613 ℤcz 12700 cdc 12821 ℤ_≥cuz 12972 ℝ⁺crp 13127 (,)cioo 13483 [,)cico 13485 ⌊cfl 13937 ↑cexp 14211 √csqrt 15409 expce 16236 eceu 16237 logclog 26891 ψcchp 27429

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 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 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8184 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-2o 8484 df-er 8724 df-map 8856 df-pm 8857 df-ixp 8933 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-fsupp 9361 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ioc 13488 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-fac 14425 df-bc 14454 df-hash 14482 df-shft 15229 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-limsup 15647 df-clim 15664 df-rlim 15665 df-sum 15863 df-ef 16242 df-e 16243 df-sin 16244 df-cos 16245 df-pi 16247 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196 df-log 26893

This theorem is used by: pntlemh 27935 pntlemq 27937 pntlemr 27938 pntlemj 27939 pntlemf 27941

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