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Theorem pntlemg 27925

Description: Lemma for pnt 27941. 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 13164	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
5		1red 11309	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
6		pntlem1.y	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
7	6	simpld 500	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
8	7	rpred 13164	. . . . . . . 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 11468	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑋)
12	4, 11	rplogcld 26957	. . . . . 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 27922	. . . . . . . . 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 13164	. . . . . . 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 26957	. . . . . 6 ⊢ (𝜑 → (log‘𝐾) ∈ ℝ⁺)
29	12, 28	rpdivcld 13181	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ⁺)
30	29	rprege0d 13171	. . . 4 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))))
31		flge0nn0 13960	. . . 4 ⊢ ((((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))) → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ₀)
32		nn0p1nn 12645	. . . 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 12719	. . 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 27924	. . . . . . . . 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 26951	. . . . . . 7 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
43	42, 28	rerpdivcld 13195	. . . . . 6 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
44	43	rehalfcld 12593	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ)
45	44	flcld 13938	. . . 4 ⊢ (𝜑 → (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) ∈ ℤ)
46	36, 45	eqeltrid 2864	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
47		0red 11311	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
48		4nn 12426	. . . . . 6 ⊢ 4 ∈ ℕ
49		nndivre 12379	. . . . . 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 12803	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
52	34	nnred 12350	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
53	51, 52	resubcld 11744	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℝ)
54	41	rpred 13164	. . . . . . . . 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 26957	. . . . . . . 8 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ⁺)
58	57, 28	rpdivcld 13181	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺)
59		4re 12427	. . . . . . . 8 ⊢ 4 ∈ ℝ
60		4pos 12453	. . . . . . . 8 ⊢ 0 < 4
61	59, 60	elrpii 13123	. . . . . . 7 ⊢ 4 ∈ ℝ⁺
62		rpdivcl 13147	. . . . . . 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 13168	. . . . 5 ⊢ (𝜑 → 0 ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
65	50	recnd 11337	. . . . . . . . 9 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
66	34	nncnd 12351	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67		1cnd 11302	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
68	65, 66, 67	addassd 11331	. . . . . . . 8 ⊢ (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)))
69	52, 5	readdcld 11338	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
70	50, 69	readdcld 11338	. . . . . . . . 9 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ∈ ℝ)
71		peano2re 11483	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
72	51, 71	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
73	29	rpred 13164	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ)
74		2re 12417	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℝ)
76	73, 75	readdcld 11338	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ∈ ℝ)
77		reflcl 13936	. . . . . . . . . . . . . . . . 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 11337	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℂ)
80	79, 67, 67	addassd 11331	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1)))
81	1	oveq1i 7430	. . . . . . . . . . . . . 14 ⊢ (𝑀 + 1) = (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1)
82		df-2 12405	. . . . . . . . . . . . . . 15 ⊢ 2 = (1 + 1)
83	82	oveq2i 7431	. . . . . . . . . . . . . 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 13939	. . . . . . . . . . . . . . 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 11930	. . . . . . . . . . . . 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 11467	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
92	69, 50, 50, 91	leadd2dd 11931	. . . . . . . . . 10 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
93	43	recnd 11337	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
94		2cnd 12421	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
95		2ne0 12449	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
96	95	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 0)
97	93, 94, 94, 96, 96	divdiv1d 12124	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / (2 · 2)))
98		2t2e4 12506	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
99	98	oveq2i 7431	. . . . . . . . . . . . 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 7436	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (2 · (((log‘𝑍) / (log‘𝐾)) / 4)))
102	44	recnd 11337	. . . . . . . . . . . 12 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℂ)
103	102, 94, 96	divcan2d 12095	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (((log‘𝑍) / (log‘𝐾)) / 2))
104	65	2timesd 12589	. . . . . . . . . . 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 13941	. . . . . . . . . . 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 7430	. . . . . . . . . 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 11467	. . . . . . . 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 11338	. . . . . . . 8 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ∈ ℝ)
114	113, 51, 5	leadd1d 11910	. . . . . . 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 11792	. . . . . . 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 11467	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑁 − 𝑀))
120	51, 52	subge0d 11906	. . . 4 ⊢ (𝜑 → (0 ≤ (𝑁 − 𝑀) ↔ 𝑀 ≤ 𝑁))
121	119, 120	mpbid 235	. . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
122		eluz2 12971	. . 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 6538 (class class class)co 7420 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 +∞cpnf 11340 < clt 11343 ≤ cle 11344 − cmin 11541 / cdiv 11973 ℕcn 12335 2c2 12397 3c3 12398 4c4 12399 ℕ₀cn0 12606 ℤcz 12693 cdc 12814 ℤ_≥cuz 12965 ℝ⁺crp 13120 (,)cioo 13476 [,)cico 13478 ⌊cfl 13930 ↑cexp 14204 √csqrt 15400 expce 16227 eceu 16228 logclog 26882 ψcchp 27420

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 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

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 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 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 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-e 16234 df-sin 16235 df-cos 16236 df-pi 16238 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-log 26884

This theorem is used by: pntlemh 27926 pntlemq 27928 pntlemr 27929 pntlemj 27930 pntlemf 27932

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