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Theorem lgamgulmlem3 27232

Description: Lemma for lgamgulm 27236. (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 27230	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4		lgamgulm.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5	3, 4	sseldd 3941	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
6	5	eldifad 3920	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7		lgamgulm.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	peano2nnd 12268	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
9	8	nnrpd 13076	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
10	7	nnrpd 13076	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
11	9, 10	rpdivcld 13095	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
12	11	relogcld 26825	. . . . . 6 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ)
13	12	recnd 11255	. . . . 5 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℂ)
14	6, 13	mulcld 11247	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘((𝑁 + 1) / 𝑁))) ∈ ℂ)
15	7	nncnd 12267	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
16	7	nnne0d 12304	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≠ 0)
17	6, 15, 16	divcld 12009	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑁) ∈ ℂ)
18		1cnd 11220	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
19	17, 18	addcld 11246	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ∈ ℂ)
20	5, 7	dmgmdivn0 27229	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ≠ 0)
21	19, 20	logcld 26772	. . . 4 ⊢ (𝜑 → (log‘((𝐴 / 𝑁) + 1)) ∈ ℂ)
22	14, 21	subcld 11587	. . 3 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
23	22	abscld 15516	. 2 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
24	14, 17	subcld 11587	. . . 4 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) ∈ ℂ)
25	24	abscld 15516	. . 3 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ∈ ℝ)
26	17, 21	subcld 11587	. . . 4 ⊢ (𝜑 → ((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
27	26	abscld 15516	. . 3 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
28	25, 27	readdcld 11256	. 2 ⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ∈ ℝ)
29	1	nnred 12266	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		2re 12333	. . . . . 6 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ)
32		1red 11227	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
33	29, 32	readdcld 11256	. . . . 5 ⊢ (𝜑 → (𝑅 + 1) ∈ ℝ)
34	31, 33	remulcld 11257	. . . 4 ⊢ (𝜑 → (2 · (𝑅 + 1)) ∈ ℝ)
35	7	nnsqcld 14300	. . . 4 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ)
36	34, 35	nndivred 12308	. . 3 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) ∈ ℝ)
37	29, 36	remulcld 11257	. 2 ⊢ (𝜑 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))) ∈ ℝ)
38	14, 21, 17	abs3difd 15540	. 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 12305	. . . . . 6 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
40	8	nnrecred 12305	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℝ)
41	39, 40	resubcld 11660	. . . . 5 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℝ)
42	29, 41	remulcld 11257	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) ∈ ℝ)
43	31, 29	remulcld 11257	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ∈ ℝ)
44	7	nnred 12266	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
45	1	nnrpd 13076	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
46	29, 45	ltaddrpd 13111	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 < (𝑅 + 𝑅))
47	1	nncnd 12267	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℂ)
48	47	2timesd 12505	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑅) = (𝑅 + 𝑅))
49	46, 48	breqtrrd 5144	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 < (2 · 𝑅))
50		lgamgulm.l	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁)
51	29, 43, 44, 49, 50	ltletrd 11388	. . . . . . . 8 ⊢ (𝜑 → 𝑅 < 𝑁)
52		difrp 13074	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈ ℝ⁺))
53	29, 44, 52	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈ ℝ⁺))
54	51, 53	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ⁺)
55	54	rprecred 13089	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℝ)
56	55, 39	resubcld 11660	. . . . 5 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℝ)
57	29, 56	remulcld 11257	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) ∈ ℝ)
58	42, 57	readdcld 11256	. . 3 ⊢ (𝜑 → ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) ∈ ℝ)
59	6, 15, 16	divrecd 12012	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑁) = (𝐴 · (1 / 𝑁)))
60	59	oveq2d 7439	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
61	39	recnd 11255	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝑁) ∈ ℂ)
62	6, 13, 61	subdid 11688	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
63	60, 62	eqtr4d 2804	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
64	63	fveq2d 6892	. . . . . 6 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
65	13, 61	subcld 11587	. . . . . . 7 ⊢ (𝜑 → ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)) ∈ ℂ)
66	6, 65	absmuld 15534	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
67	64, 66	eqtrd 2801	. . . . 5 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
68	6	abscld 15516	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
69	65	abscld 15516	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ∈ ℝ)
70	6	absge0d 15524	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
71	65	absge0d 15524	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
72		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (abs‘𝑥) = (abs‘𝐴))
73	72	breq1d 5124	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝐴) ≤ 𝑅))
74		fvoveq1 7446	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝐴 + 𝑘)))
75	74	breq2d 5126	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
76	75	ralbidv 3191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
77	73, 76	anbi12d 644	. . . . . . . . . 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 3657	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑈 ↔ (𝐴 ∈ ℂ ∧ ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))))
79	78	simprbi 503	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑈 → ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
80	4, 79	syl 18	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
81	80	simpld 500	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑅)
82	9, 10	relogdivd 26828	. . . . . . . . 9 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
83		logdifbnd 27195	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → ((log‘(𝑁 + 1)) − (log‘𝑁)) ≤ (1 / 𝑁))
84	10, 83	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((log‘(𝑁 + 1)) − (log‘𝑁)) ≤ (1 / 𝑁))
85	82, 84	eqbrtrd 5138	. . . . . . . 8 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ≤ (1 / 𝑁))
86	12, 39, 85	abssuble0d 15512	. . . . . . 7 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))))
87		logdiflbnd 27196	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
88	10, 87	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ ((log‘(𝑁 + 1)) − (log‘𝑁)))
89	88, 82	breqtrrd 5144	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)))
90	40, 12, 39, 89	lesub2dd 11849	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
91	86, 90	eqbrtrd 5138	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
92	68, 29, 69, 41, 70, 71, 81, 91	lemul12ad 12175	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
93	67, 92	eqbrtrd 5138	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
94	1, 2, 7, 4, 50	lgamgulmlem2 27231	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
95	25, 27, 42, 57, 93, 94	le2addd 11851	. . 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 11587	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℂ)
97	15, 18	addcld 11246	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
98	29, 51	gtned 11363	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≠ 𝑅)
99	15, 47, 98	subne0d 11596	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ≠ 0)
100	8	nnne0d 12304	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
101	96, 97, 99, 100	subrecd 12062	. . . . . . 7 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))))
102	15, 18, 47	pnncand 11626	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (1 + 𝑅))
103	18, 47, 102	comraddd 11442	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (𝑅 + 1))
104	103	oveq1d 7438	. . . . . . 7 ⊢ (𝜑 → (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))))
105	101, 104	eqtr2d 2802	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
106	105	oveq2d 7439	. . . . 5 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) = (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))))
107	97, 100	reccld 12002	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℂ)
108	96, 99	reccld 12002	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℂ)
109	61, 107, 108	npncan3d 11623	. . . . . . 7 ⊢ (𝜑 → (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
110	109	eqcomd 2772	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
111	110	oveq2d 7439	. . . . 5 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))) = (𝑅 · (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))))
112	41	recnd 11255	. . . . . 6 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℂ)
113	56	recnd 11255	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℂ)
114	47, 112, 113	adddid 11251	. . . . 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 2805	. . . 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 13094	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 𝑅) · (𝑁 + 1)) ∈ ℝ⁺)
117	33, 116	rerpdivcld 13109	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ∈ ℝ)
118	45	rpge0d 13082	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝑅)
119		2z 12644	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
120	119	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℤ)
121	10, 120	rpexpcld 14303	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
122	121	rphalfcld 13090	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ∈ ℝ⁺)
123		0le1 11755	. . . . . . . . 9 ⊢ 0 ≤ 1
124	123	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 1)
125	29, 32, 118, 124	addge0d 11808	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑅 + 1))
126	15	sqvald 14199	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) = (𝑁 · 𝑁))
127	126	oveq1d 7438	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 · 𝑁) / 2))
128	31	recnd 11255	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
129		2ne0 12365	. . . . . . . . . . 11 ⊢ 2 ≠ 0
130	129	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
131	15, 15, 128, 130	div23d 12046	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 · 𝑁) / 2) = ((𝑁 / 2) · 𝑁))
132	127, 131	eqtrd 2801	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 / 2) · 𝑁))
133	44	rehalfcld 12509	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ∈ ℝ)
134	44, 29	resubcld 11660	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ)
135	44, 32	readdcld 11256	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
136		2rp 13039	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
137	136	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℝ⁺)
138	10	rpge0d 13082	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝑁)
139	44, 137, 138	divge0d 13118	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑁 / 2))
140	29, 44, 137	lemuldiv2d 13128	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 · 𝑅) ≤ 𝑁 ↔ 𝑅 ≤ (𝑁 / 2)))
141	50, 140	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ≤ (𝑁 / 2))
142	15	2halvesd 12508	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 / 2) + (𝑁 / 2)) = 𝑁)
143	133	recnd 11255	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 / 2) ∈ ℂ)
144	15, 143, 143	subaddd 11605	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − (𝑁 / 2)) = (𝑁 / 2) ↔ ((𝑁 / 2) + (𝑁 / 2)) = 𝑁))
145	142, 144	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − (𝑁 / 2)) = (𝑁 / 2))
146	141, 145	breqtrrd 5144	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ≤ (𝑁 − (𝑁 / 2)))
147	29, 44, 133, 146	lesubd 11836	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ≤ (𝑁 − 𝑅))
148	44	lep1d 12164	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≤ (𝑁 + 1))
149	133, 134, 44, 135, 139, 138, 147, 148	lemul12ad 12175	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 / 2) · 𝑁) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
150	132, 149	eqbrtrd 5138	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
151	122, 116, 33, 125, 150	lediv2ad 13100	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((𝑅 + 1) / ((𝑁↑2) / 2)))
152	1	peano2nnd 12268	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 + 1) ∈ ℕ)
153	152	nncnd 12267	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + 1) ∈ ℂ)
154	35	nncnd 12267	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
155	35	nnne0d 12304	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ≠ 0)
156	153, 154, 128, 155, 130	divdiv2d 12041	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁↑2) / 2)) = (((𝑅 + 1) · 2) / (𝑁↑2)))
157	153, 128	mulcomd 11248	. . . . . . . 8 ⊢ (𝜑 → ((𝑅 + 1) · 2) = (2 · (𝑅 + 1)))
158	157	oveq1d 7438	. . . . . . 7 ⊢ (𝜑 → (((𝑅 + 1) · 2) / (𝑁↑2)) = ((2 · (𝑅 + 1)) / (𝑁↑2)))
159	156, 158	eqtr2d 2802	. . . . . 6 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) = ((𝑅 + 1) / ((𝑁↑2) / 2)))
160	151, 159	breqtrrd 5144	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((2 · (𝑅 + 1)) / (𝑁↑2)))
161	117, 36, 29, 118, 160	lemul2ad 12173	. . . 4 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
162	115, 161	eqbrtrrd 5140	. . 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 11385	. 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 11385	1 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 {crab 3419 ∖ cdif 3905 class class class wbr 5114 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 < clt 11261 ≤ cle 11262 − cmin 11459 / cdiv 11889 ℕcn 12251 2c2 12313 ℕ₀cn0 12522 ℤcz 12609 ℝ⁺crp 13034 ↑cexp 14117 abscabs 15311 logclog 26756

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 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 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-ef 16146 df-sin 16148 df-cos 16149 df-tan 16150 df-pi 16151 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-mulg 19165 df-cntz 19418 df-cmn 19883 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-fbas 21556 df-fg 21557 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cld 23213 df-ntr 23214 df-cls 23215 df-nei 23292 df-lp 23330 df-perf 23331 df-cn 23421 df-cnp 23422 df-haus 23509 df-cmp 23581 df-tx 23756 df-hmeo 23949 df-fil 24040 df-fm 24132 df-flim 24133 df-flf 24134 df-xms 24514 df-ms 24515 df-tms 24516 df-cncf 25074 df-limc 26062 df-dv 26063 df-log 26758

This theorem is used by: lgamgulmlem5 27234

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