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Theorem lgamgulmlem3 24757

Description: Lemma for lgamgulm 24761. (Contributed by Mario Carneiro, 3-Jul-2017.)

**Hypotheses**
Ref	Expression
lgamgulm.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
lgamgulm.u	⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}
lgamgulm.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
lgamgulm.a	⊢ (𝜑 → 𝐴 ∈ 𝑈)
lgamgulm.l	⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁)

**Assertion**
Ref	Expression
lgamgulmlem3	⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))

Distinct variable groups: 𝑥,𝑁 𝑥,𝑘,𝑅 𝐴,𝑘,𝑥 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑘) 𝑈(𝑥,𝑘) 𝑁(𝑘)

**Proof of Theorem lgamgulmlem3**
Step	Hyp	Ref	Expression
1		lgamgulm.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℕ)
2		lgamgulm.u	. . . . . . . 8 ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}
3	1, 2	lgamgulmlem1 24755	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4		lgamgulm.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5	3, 4	sseldd 3604	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
6	5	eldifad 3586	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7		lgamgulm.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	peano2nnd 11037	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
9	8	nnrpd 11870	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
10	7	nnrpd 11870	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
11	9, 10	rpdivcld 11889	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
12	11	relogcld 24369	. . . . . 6 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ)
13	12	recnd 10068	. . . . 5 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℂ)
14	6, 13	mulcld 10060	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘((𝑁 + 1) / 𝑁))) ∈ ℂ)
15	7	nncnd 11036	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
16	7	nnne0d 11065	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≠ 0)
17	6, 15, 16	divcld 10801	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑁) ∈ ℂ)
18		1cnd 10056	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
19	17, 18	addcld 10059	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ∈ ℂ)
20	5, 7	dmgmdivn0 24754	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ≠ 0)
21	19, 20	logcld 24317	. . . 4 ⊢ (𝜑 → (log‘((𝐴 / 𝑁) + 1)) ∈ ℂ)
22	14, 21	subcld 10392	. . 3 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
23	22	abscld 14175	. 2 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
24	14, 17	subcld 10392	. . . 4 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) ∈ ℂ)
25	24	abscld 14175	. . 3 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ∈ ℝ)
26	17, 21	subcld 10392	. . . 4 ⊢ (𝜑 → ((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
27	26	abscld 14175	. . 3 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
28	25, 27	readdcld 10069	. 2 ⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ∈ ℝ)
29	1	nnred 11035	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		2re 11090	. . . . . 6 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ)
32		1red 10055	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
33	29, 32	readdcld 10069	. . . . 5 ⊢ (𝜑 → (𝑅 + 1) ∈ ℝ)
34	31, 33	remulcld 10070	. . . 4 ⊢ (𝜑 → (2 · (𝑅 + 1)) ∈ ℝ)
35	7	nnsqcld 13029	. . . 4 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ)
36	34, 35	nndivred 11069	. . 3 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) ∈ ℝ)
37	29, 36	remulcld 10070	. 2 ⊢ (𝜑 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))) ∈ ℝ)
38	14, 21, 17	abs3difd 14199	. 2 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))))
39	7	nnrecred 11066	. . . . . 6 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
40	8	nnrecred 11066	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℝ)
41	39, 40	resubcld 10458	. . . . 5 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℝ)
42	29, 41	remulcld 10070	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) ∈ ℝ)
43	31, 29	remulcld 10070	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ∈ ℝ)
44	7	nnred 11035	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
45	1	nnrpd 11870	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
46	29, 45	ltaddrpd 11905	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 < (𝑅 + 𝑅))
47	1	nncnd 11036	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℂ)
48	47	2timesd 11275	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑅) = (𝑅 + 𝑅))
49	46, 48	breqtrrd 4681	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 < (2 · 𝑅))
50		lgamgulm.l	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁)
51	29, 43, 44, 49, 50	ltletrd 10197	. . . . . . . 8 ⊢ (𝜑 → 𝑅 < 𝑁)
52		difrp 11868	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈ ℝ⁺))
53	29, 44, 52	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈ ℝ⁺))
54	51, 53	mpbid 222	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ⁺)
55	54	rprecred 11883	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℝ)
56	55, 39	resubcld 10458	. . . . 5 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℝ)
57	29, 56	remulcld 10070	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) ∈ ℝ)
58	42, 57	readdcld 10069	. . 3 ⊢ (𝜑 → ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) ∈ ℝ)
59	6, 15, 16	divrecd 10804	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑁) = (𝐴 · (1 / 𝑁)))
60	59	oveq2d 6666	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
61	39	recnd 10068	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝑁) ∈ ℂ)
62	6, 13, 61	subdid 10486	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
63	60, 62	eqtr4d 2659	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
64	63	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
65	13, 61	subcld 10392	. . . . . . 7 ⊢ (𝜑 → ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)) ∈ ℂ)
66	6, 65	absmuld 14193	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
67	64, 66	eqtrd 2656	. . . . 5 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
68	6	abscld 14175	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
69	65	abscld 14175	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ∈ ℝ)
70	6	absge0d 14183	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
71	65	absge0d 14183	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
72		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (abs‘𝑥) = (abs‘𝐴))
73	72	breq1d 4663	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝐴) ≤ 𝑅))
74		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑘) = (𝐴 + 𝑘))
75	74	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝐴 + 𝑘)))
76	75	breq2d 4665	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
77	76	ralbidv 2986	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
78	73, 77	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))))
79	78, 2	elrab2 3366	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑈 ↔ (𝐴 ∈ ℂ ∧ ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))))
80	79	simprbi 480	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑈 → ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
81	4, 80	syl 17	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
82	81	simpld 475	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑅)
83	9, 10	relogdivd 24372	. . . . . . . . 9 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
84		logdifbnd 24720	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → ((log‘(𝑁 + 1)) − (log‘𝑁)) ≤ (1 / 𝑁))
85	10, 84	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((log‘(𝑁 + 1)) − (log‘𝑁)) ≤ (1 / 𝑁))
86	83, 85	eqbrtrd 4675	. . . . . . . 8 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ≤ (1 / 𝑁))
87	12, 39, 86	abssuble0d 14171	. . . . . . 7 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))))
88		logdiflbnd 24721	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
89	10, 88	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
90	89, 83	breqtrrd 4681	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)))
91	40, 12, 39, 90	lesub2dd 10644	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
92	87, 91	eqbrtrd 4675	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
93	68, 29, 69, 41, 70, 71, 82, 92	lemul12ad 10966	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
94	67, 93	eqbrtrd 4675	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
95	1, 2, 7, 4, 50	lgamgulmlem2 24756	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
96	25, 27, 42, 57, 94, 95	le2addd 10646	. . 3 ⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ≤ ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))))
97	15, 47	subcld 10392	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℂ)
98	15, 18	addcld 10059	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
99	29, 51	gtned 10172	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≠ 𝑅)
100	15, 47, 99	subne0d 10401	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ≠ 0)
101	8	nnne0d 11065	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
102	97, 98, 100, 101	subrecd 10856	. . . . . . 7 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))))
103	15, 18, 47	pnncand 10431	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (1 + 𝑅))
104	18, 47	addcomd 10238	. . . . . . . . 9 ⊢ (𝜑 → (1 + 𝑅) = (𝑅 + 1))
105	103, 104	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (𝑅 + 1))
106	105	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))))
107	102, 106	eqtr2d 2657	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
108	107	oveq2d 6666	. . . . 5 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) = (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))))
109	98, 101	reccld 10794	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℂ)
110	97, 100	reccld 10794	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℂ)
111	61, 109, 110	npncan3d 10428	. . . . . . 7 ⊢ (𝜑 → (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
112	111	eqcomd 2628	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
113	112	oveq2d 6666	. . . . 5 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))) = (𝑅 · (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))))
114	41	recnd 10068	. . . . . 6 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℂ)
115	56	recnd 10068	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℂ)
116	47, 114, 115	adddid 10064	. . . . 5 ⊢ (𝜑 → (𝑅 · (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) = ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))))
117	108, 113, 116	3eqtrd 2660	. . . 4 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) = ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))))
118	54, 9	rpmulcld 11888	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 𝑅) · (𝑁 + 1)) ∈ ℝ⁺)
119	33, 118	rerpdivcld 11903	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ∈ ℝ)
120	45	rpge0d 11876	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝑅)
121		2z 11409	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
122	121	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℤ)
123	10, 122	rpexpcld 13032	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
124	123	rphalfcld 11884	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ∈ ℝ⁺)
125		0le1 10551	. . . . . . . . 9 ⊢ 0 ≤ 1
126	125	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 1)
127	29, 32, 120, 126	addge0d 10603	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑅 + 1))
128	15	sqvald 13005	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) = (𝑁 · 𝑁))
129	128	oveq1d 6665	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 · 𝑁) / 2))
130	31	recnd 10068	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
131		2ne0 11113	. . . . . . . . . . 11 ⊢ 2 ≠ 0
132	131	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
133	15, 15, 130, 132	div23d 10838	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 · 𝑁) / 2) = ((𝑁 / 2) · 𝑁))
134	129, 133	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 / 2) · 𝑁))
135	44	rehalfcld 11279	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ∈ ℝ)
136	44, 29	resubcld 10458	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ)
137	44, 32	readdcld 10069	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
138		2rp 11837	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
139	138	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℝ⁺)
140	10	rpge0d 11876	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝑁)
141	44, 139, 140	divge0d 11912	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑁 / 2))
142	29, 44, 139	lemuldiv2d 11922	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 · 𝑅) ≤ 𝑁 ↔ 𝑅 ≤ (𝑁 / 2)))
143	50, 142	mpbid 222	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ≤ (𝑁 / 2))
144	15	2halvesd 11278	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 / 2) + (𝑁 / 2)) = 𝑁)
145	135	recnd 10068	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 / 2) ∈ ℂ)
146	15, 145, 145	subaddd 10410	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − (𝑁 / 2)) = (𝑁 / 2) ↔ ((𝑁 / 2) + (𝑁 / 2)) = 𝑁))
147	144, 146	mpbird 247	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − (𝑁 / 2)) = (𝑁 / 2))
148	143, 147	breqtrrd 4681	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ≤ (𝑁 − (𝑁 / 2)))
149	29, 44, 135, 148	lesubd 10631	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ≤ (𝑁 − 𝑅))
150	44	lep1d 10955	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≤ (𝑁 + 1))
151	135, 136, 44, 137, 141, 140, 149, 150	lemul12ad 10966	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 / 2) · 𝑁) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
152	134, 151	eqbrtrd 4675	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
153	124, 118, 33, 127, 152	lediv2ad 11894	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((𝑅 + 1) / ((𝑁↑2) / 2)))
154	1	peano2nnd 11037	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 + 1) ∈ ℕ)
155	154	nncnd 11036	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + 1) ∈ ℂ)
156	35	nncnd 11036	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
157	35	nnne0d 11065	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ≠ 0)
158	155, 156, 130, 157, 132	divdiv2d 10833	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁↑2) / 2)) = (((𝑅 + 1) · 2) / (𝑁↑2)))
159	155, 130	mulcomd 10061	. . . . . . . 8 ⊢ (𝜑 → ((𝑅 + 1) · 2) = (2 · (𝑅 + 1)))
160	159	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → (((𝑅 + 1) · 2) / (𝑁↑2)) = ((2 · (𝑅 + 1)) / (𝑁↑2)))
161	158, 160	eqtr2d 2657	. . . . . 6 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) = ((𝑅 + 1) / ((𝑁↑2) / 2)))
162	153, 161	breqtrrd 4681	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((2 · (𝑅 + 1)) / (𝑁↑2)))
163	119, 36, 29, 120, 162	lemul2ad 10964	. . . 4 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
164	117, 163	eqbrtrrd 4677	. . 3 ⊢ (𝜑 → ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
165	28, 58, 37, 96, 164	letrd 10194	. 2 ⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
166	23, 28, 37, 38, 165	letrd 10194	1 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ∖ cdif 3571 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℝ⁺crp 11832 ↑cexp 12860 abscabs 13974 logclog 24301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303

This theorem is referenced by: lgamgulmlem5 24759

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