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Theorem lgamgulmlem3 26994

Description: Lemma for lgamgulm 26998. (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 26992	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4		lgamgulm.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5	3, 4	sseldd 3922	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
6	5	eldifad 3901	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7		lgamgulm.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	peano2nnd 12191	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
9	8	nnrpd 12984	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
10	7	nnrpd 12984	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
11	9, 10	rpdivcld 13003	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
12	11	relogcld 26587	. . . . . 6 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ)
13	12	recnd 11173	. . . . 5 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℂ)
14	6, 13	mulcld 11165	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘((𝑁 + 1) / 𝑁))) ∈ ℂ)
15	7	nncnd 12190	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
16	7	nnne0d 12227	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≠ 0)
17	6, 15, 16	divcld 11931	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑁) ∈ ℂ)
18		1cnd 11139	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
19	17, 18	addcld 11164	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ∈ ℂ)
20	5, 7	dmgmdivn0 26991	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ≠ 0)
21	19, 20	logcld 26534	. . . 4 ⊢ (𝜑 → (log‘((𝐴 / 𝑁) + 1)) ∈ ℂ)
22	14, 21	subcld 11505	. . 3 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
23	22	abscld 15401	. 2 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
24	14, 17	subcld 11505	. . . 4 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) ∈ ℂ)
25	24	abscld 15401	. . 3 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ∈ ℝ)
26	17, 21	subcld 11505	. . . 4 ⊢ (𝜑 → ((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
27	26	abscld 15401	. . 3 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
28	25, 27	readdcld 11174	. 2 ⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ∈ ℝ)
29	1	nnred 12189	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		2re 12255	. . . . . 6 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ)
32		1red 11145	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
33	29, 32	readdcld 11174	. . . . 5 ⊢ (𝜑 → (𝑅 + 1) ∈ ℝ)
34	31, 33	remulcld 11175	. . . 4 ⊢ (𝜑 → (2 · (𝑅 + 1)) ∈ ℝ)
35	7	nnsqcld 14206	. . . 4 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ)
36	34, 35	nndivred 12231	. . 3 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) ∈ ℝ)
37	29, 36	remulcld 11175	. 2 ⊢ (𝜑 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))) ∈ ℝ)
38	14, 21, 17	abs3difd 15425	. 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 12228	. . . . . 6 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
40	8	nnrecred 12228	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℝ)
41	39, 40	resubcld 11578	. . . . 5 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℝ)
42	29, 41	remulcld 11175	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) ∈ ℝ)
43	31, 29	remulcld 11175	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ∈ ℝ)
44	7	nnred 12189	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
45	1	nnrpd 12984	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
46	29, 45	ltaddrpd 13019	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 < (𝑅 + 𝑅))
47	1	nncnd 12190	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℂ)
48	47	2timesd 12420	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑅) = (𝑅 + 𝑅))
49	46, 48	breqtrrd 5113	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 < (2 · 𝑅))
50		lgamgulm.l	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁)
51	29, 43, 44, 49, 50	ltletrd 11306	. . . . . . . 8 ⊢ (𝜑 → 𝑅 < 𝑁)
52		difrp 12982	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈ ℝ⁺))
53	29, 44, 52	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈ ℝ⁺))
54	51, 53	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ⁺)
55	54	rprecred 12997	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℝ)
56	55, 39	resubcld 11578	. . . . 5 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℝ)
57	29, 56	remulcld 11175	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) ∈ ℝ)
58	42, 57	readdcld 11174	. . 3 ⊢ (𝜑 → ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) ∈ ℝ)
59	6, 15, 16	divrecd 11934	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑁) = (𝐴 · (1 / 𝑁)))
60	59	oveq2d 7383	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
61	39	recnd 11173	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝑁) ∈ ℂ)
62	6, 13, 61	subdid 11606	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
63	60, 62	eqtr4d 2774	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
64	63	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
65	13, 61	subcld 11505	. . . . . . 7 ⊢ (𝜑 → ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)) ∈ ℂ)
66	6, 65	absmuld 15419	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
67	64, 66	eqtrd 2771	. . . . 5 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
68	6	abscld 15401	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
69	65	abscld 15401	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ∈ ℝ)
70	6	absge0d 15409	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
71	65	absge0d 15409	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
72		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (abs‘𝑥) = (abs‘𝐴))
73	72	breq1d 5095	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝐴) ≤ 𝑅))
74		fvoveq1 7390	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝐴 + 𝑘)))
75	74	breq2d 5097	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
76	75	ralbidv 3160	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
77	73, 76	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))))
78	77, 2	elrab2 3637	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑈 ↔ (𝐴 ∈ ℂ ∧ ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))))
79	78	simprbi 497	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑈 → ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
80	4, 79	syl 17	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
81	80	simpld 494	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑅)
82	9, 10	relogdivd 26590	. . . . . . . . 9 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
83		logdifbnd 26957	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → ((log‘(𝑁 + 1)) − (log‘𝑁)) ≤ (1 / 𝑁))
84	10, 83	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((log‘(𝑁 + 1)) − (log‘𝑁)) ≤ (1 / 𝑁))
85	82, 84	eqbrtrd 5107	. . . . . . . 8 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ≤ (1 / 𝑁))
86	12, 39, 85	abssuble0d 15397	. . . . . . 7 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))))
87		logdiflbnd 26958	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
88	10, 87	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
89	88, 82	breqtrrd 5113	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)))
90	40, 12, 39, 89	lesub2dd 11767	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
91	86, 90	eqbrtrd 5107	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
92	68, 29, 69, 41, 70, 71, 81, 91	lemul12ad 12098	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
93	67, 92	eqbrtrd 5107	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
94	1, 2, 7, 4, 50	lgamgulmlem2 26993	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
95	25, 27, 42, 57, 93, 94	le2addd 11769	. . 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)))))
96	15, 47	subcld 11505	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℂ)
97	15, 18	addcld 11164	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
98	29, 51	gtned 11281	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≠ 𝑅)
99	15, 47, 98	subne0d 11514	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ≠ 0)
100	8	nnne0d 12227	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
101	96, 97, 99, 100	subrecd 11984	. . . . . . 7 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))))
102	15, 18, 47	pnncand 11544	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (1 + 𝑅))
103	18, 47, 102	comraddd 11360	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (𝑅 + 1))
104	103	oveq1d 7382	. . . . . . 7 ⊢ (𝜑 → (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))))
105	101, 104	eqtr2d 2772	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
106	105	oveq2d 7383	. . . . 5 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) = (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))))
107	97, 100	reccld 11924	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℂ)
108	96, 99	reccld 11924	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℂ)
109	61, 107, 108	npncan3d 11541	. . . . . . 7 ⊢ (𝜑 → (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
110	109	eqcomd 2742	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
111	110	oveq2d 7383	. . . . 5 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))) = (𝑅 · (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))))
112	41	recnd 11173	. . . . . 6 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℂ)
113	56	recnd 11173	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℂ)
114	47, 112, 113	adddid 11169	. . . . 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)))))
115	106, 111, 114	3eqtrd 2775	. . . 4 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) = ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))))
116	54, 9	rpmulcld 13002	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 𝑅) · (𝑁 + 1)) ∈ ℝ⁺)
117	33, 116	rerpdivcld 13017	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ∈ ℝ)
118	45	rpge0d 12990	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝑅)
119		2z 12559	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
120	119	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℤ)
121	10, 120	rpexpcld 14209	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
122	121	rphalfcld 12998	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ∈ ℝ⁺)
123		0le1 11673	. . . . . . . . 9 ⊢ 0 ≤ 1
124	123	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 1)
125	29, 32, 118, 124	addge0d 11726	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑅 + 1))
126	15	sqvald 14105	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) = (𝑁 · 𝑁))
127	126	oveq1d 7382	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 · 𝑁) / 2))
128	31	recnd 11173	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
129		2ne0 12285	. . . . . . . . . . 11 ⊢ 2 ≠ 0
130	129	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
131	15, 15, 128, 130	div23d 11968	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 · 𝑁) / 2) = ((𝑁 / 2) · 𝑁))
132	127, 131	eqtrd 2771	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 / 2) · 𝑁))
133	44	rehalfcld 12424	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ∈ ℝ)
134	44, 29	resubcld 11578	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ)
135	44, 32	readdcld 11174	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
136		2rp 12947	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
137	136	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℝ⁺)
138	10	rpge0d 12990	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝑁)
139	44, 137, 138	divge0d 13026	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑁 / 2))
140	29, 44, 137	lemuldiv2d 13036	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 · 𝑅) ≤ 𝑁 ↔ 𝑅 ≤ (𝑁 / 2)))
141	50, 140	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ≤ (𝑁 / 2))
142	15	2halvesd 12423	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 / 2) + (𝑁 / 2)) = 𝑁)
143	133	recnd 11173	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 / 2) ∈ ℂ)
144	15, 143, 143	subaddd 11523	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − (𝑁 / 2)) = (𝑁 / 2) ↔ ((𝑁 / 2) + (𝑁 / 2)) = 𝑁))
145	142, 144	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − (𝑁 / 2)) = (𝑁 / 2))
146	141, 145	breqtrrd 5113	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ≤ (𝑁 − (𝑁 / 2)))
147	29, 44, 133, 146	lesubd 11754	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ≤ (𝑁 − 𝑅))
148	44	lep1d 12087	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≤ (𝑁 + 1))
149	133, 134, 44, 135, 139, 138, 147, 148	lemul12ad 12098	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 / 2) · 𝑁) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
150	132, 149	eqbrtrd 5107	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
151	122, 116, 33, 125, 150	lediv2ad 13008	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((𝑅 + 1) / ((𝑁↑2) / 2)))
152	1	peano2nnd 12191	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 + 1) ∈ ℕ)
153	152	nncnd 12190	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + 1) ∈ ℂ)
154	35	nncnd 12190	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
155	35	nnne0d 12227	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ≠ 0)
156	153, 154, 128, 155, 130	divdiv2d 11963	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁↑2) / 2)) = (((𝑅 + 1) · 2) / (𝑁↑2)))
157	153, 128	mulcomd 11166	. . . . . . . 8 ⊢ (𝜑 → ((𝑅 + 1) · 2) = (2 · (𝑅 + 1)))
158	157	oveq1d 7382	. . . . . . 7 ⊢ (𝜑 → (((𝑅 + 1) · 2) / (𝑁↑2)) = ((2 · (𝑅 + 1)) / (𝑁↑2)))
159	156, 158	eqtr2d 2772	. . . . . 6 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) = ((𝑅 + 1) / ((𝑁↑2) / 2)))
160	151, 159	breqtrrd 5113	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((2 · (𝑅 + 1)) / (𝑁↑2)))
161	117, 36, 29, 118, 160	lemul2ad 12096	. . . 4 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
162	115, 161	eqbrtrrd 5109	. . 3 ⊢ (𝜑 → ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
163	28, 58, 37, 95, 162	letrd 11303	. 2 ⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
164	23, 28, 37, 38, 163	letrd 11303	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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 {crab 3389 ∖ cdif 3886 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 2c2 12236 ℕ₀cn0 12437 ℤcz 12524 ℝ⁺crp 12942 ↑cexp 14023 abscabs 15196 logclog 26518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-tan 16036 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520

This theorem is referenced by: lgamgulmlem5 26996

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