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Theorem lgamgulmlem3 26971

Description: Lemma for lgamgulm 26975. (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 26969	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4		lgamgulm.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5	3, 4	sseldd 3931	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
6	5	eldifad 3910	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7		lgamgulm.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	peano2nnd 12151	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
9	8	nnrpd 12936	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
10	7	nnrpd 12936	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
11	9, 10	rpdivcld 12955	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
12	11	relogcld 26562	. . . . . 6 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ)
13	12	recnd 11149	. . . . 5 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℂ)
14	6, 13	mulcld 11141	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘((𝑁 + 1) / 𝑁))) ∈ ℂ)
15	7	nncnd 12150	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
16	7	nnne0d 12184	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≠ 0)
17	6, 15, 16	divcld 11906	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑁) ∈ ℂ)
18		1cnd 11116	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
19	17, 18	addcld 11140	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ∈ ℂ)
20	5, 7	dmgmdivn0 26968	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ≠ 0)
21	19, 20	logcld 26509	. . . 4 ⊢ (𝜑 → (log‘((𝐴 / 𝑁) + 1)) ∈ ℂ)
22	14, 21	subcld 11481	. . 3 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
23	22	abscld 15350	. 2 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
24	14, 17	subcld 11481	. . . 4 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) ∈ ℂ)
25	24	abscld 15350	. . 3 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ∈ ℝ)
26	17, 21	subcld 11481	. . . 4 ⊢ (𝜑 → ((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
27	26	abscld 15350	. . 3 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
28	25, 27	readdcld 11150	. 2 ⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ∈ ℝ)
29	1	nnred 12149	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		2re 12208	. . . . . 6 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ)
32		1red 11122	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
33	29, 32	readdcld 11150	. . . . 5 ⊢ (𝜑 → (𝑅 + 1) ∈ ℝ)
34	31, 33	remulcld 11151	. . . 4 ⊢ (𝜑 → (2 · (𝑅 + 1)) ∈ ℝ)
35	7	nnsqcld 14155	. . . 4 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ)
36	34, 35	nndivred 12188	. . 3 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) ∈ ℝ)
37	29, 36	remulcld 11151	. 2 ⊢ (𝜑 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))) ∈ ℝ)
38	14, 21, 17	abs3difd 15374	. 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 12185	. . . . . 6 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
40	8	nnrecred 12185	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℝ)
41	39, 40	resubcld 11554	. . . . 5 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℝ)
42	29, 41	remulcld 11151	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) ∈ ℝ)
43	31, 29	remulcld 11151	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ∈ ℝ)
44	7	nnred 12149	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
45	1	nnrpd 12936	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
46	29, 45	ltaddrpd 12971	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 < (𝑅 + 𝑅))
47	1	nncnd 12150	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℂ)
48	47	2timesd 12373	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑅) = (𝑅 + 𝑅))
49	46, 48	breqtrrd 5123	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 < (2 · 𝑅))
50		lgamgulm.l	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁)
51	29, 43, 44, 49, 50	ltletrd 11282	. . . . . . . 8 ⊢ (𝜑 → 𝑅 < 𝑁)
52		difrp 12934	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈ ℝ⁺))
53	29, 44, 52	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑅 < 𝑁 ↔ (𝑁 − 𝑅) ∈ ℝ⁺))
54	51, 53	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ⁺)
55	54	rprecred 12949	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℝ)
56	55, 39	resubcld 11554	. . . . 5 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℝ)
57	29, 56	remulcld 11151	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) ∈ ℝ)
58	42, 57	readdcld 11150	. . 3 ⊢ (𝜑 → ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) ∈ ℝ)
59	6, 15, 16	divrecd 11909	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑁) = (𝐴 · (1 / 𝑁)))
60	59	oveq2d 7370	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
61	39	recnd 11149	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝑁) ∈ ℂ)
62	6, 13, 61	subdid 11582	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
63	60, 62	eqtr4d 2771	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
64	63	fveq2d 6834	. . . . . 6 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
65	13, 61	subcld 11481	. . . . . . 7 ⊢ (𝜑 → ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)) ∈ ℂ)
66	6, 65	absmuld 15368	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
67	64, 66	eqtrd 2768	. . . . 5 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
68	6	abscld 15350	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
69	65	abscld 15350	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ∈ ℝ)
70	6	absge0d 15358	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
71	65	absge0d 15358	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
72		fveq2 6830	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (abs‘𝑥) = (abs‘𝐴))
73	72	breq1d 5105	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝐴) ≤ 𝑅))
74		fvoveq1 7377	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝐴 + 𝑘)))
75	74	breq2d 5107	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
76	75	ralbidv 3156	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
77	73, 76	anbi12d 632	. . . . . . . . . 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 3646	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑈 ↔ (𝐴 ∈ ℂ ∧ ((abs‘𝐴) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘)))))
79	78	simprbi 496	. . . . . . . 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 26565	. . . . . . . . 9 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
83		logdifbnd 26934	. . . . . . . . . 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 5117	. . . . . . . 8 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ≤ (1 / 𝑁))
86	12, 39, 85	abssuble0d 15346	. . . . . . 7 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))))
87		logdiflbnd 26935	. . . . . . . . . 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 5123	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)))
90	40, 12, 39, 89	lesub2dd 11743	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
91	86, 90	eqbrtrd 5117	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
92	68, 29, 69, 41, 70, 71, 81, 91	lemul12ad 12073	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
93	67, 92	eqbrtrd 5117	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
94	1, 2, 7, 4, 50	lgamgulmlem2 26970	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
95	25, 27, 42, 57, 93, 94	le2addd 11745	. . 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 11481	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℂ)
97	15, 18	addcld 11140	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
98	29, 51	gtned 11257	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≠ 𝑅)
99	15, 47, 98	subne0d 11490	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ≠ 0)
100	8	nnne0d 12184	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
101	96, 97, 99, 100	subrecd 11959	. . . . . . 7 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))))
102	15, 18, 47	pnncand 11520	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (1 + 𝑅))
103	18, 47, 102	comraddd 11336	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (𝑅 + 1))
104	103	oveq1d 7369	. . . . . . 7 ⊢ (𝜑 → (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))))
105	101, 104	eqtr2d 2769	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
106	105	oveq2d 7370	. . . . 5 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) = (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))))
107	97, 100	reccld 11899	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℂ)
108	96, 99	reccld 11899	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℂ)
109	61, 107, 108	npncan3d 11517	. . . . . . 7 ⊢ (𝜑 → (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
110	109	eqcomd 2739	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
111	110	oveq2d 7370	. . . . 5 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))) = (𝑅 · (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))))
112	41	recnd 11149	. . . . . 6 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℂ)
113	56	recnd 11149	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℂ)
114	47, 112, 113	adddid 11145	. . . . 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 2772	. . . 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 12954	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 𝑅) · (𝑁 + 1)) ∈ ℝ⁺)
117	33, 116	rerpdivcld 12969	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ∈ ℝ)
118	45	rpge0d 12942	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝑅)
119		2z 12512	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
120	119	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℤ)
121	10, 120	rpexpcld 14158	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
122	121	rphalfcld 12950	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ∈ ℝ⁺)
123		0le1 11649	. . . . . . . . 9 ⊢ 0 ≤ 1
124	123	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 1)
125	29, 32, 118, 124	addge0d 11702	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑅 + 1))
126	15	sqvald 14054	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) = (𝑁 · 𝑁))
127	126	oveq1d 7369	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 · 𝑁) / 2))
128	31	recnd 11149	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
129		2ne0 12238	. . . . . . . . . . 11 ⊢ 2 ≠ 0
130	129	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
131	15, 15, 128, 130	div23d 11943	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 · 𝑁) / 2) = ((𝑁 / 2) · 𝑁))
132	127, 131	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 / 2) · 𝑁))
133	44	rehalfcld 12377	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ∈ ℝ)
134	44, 29	resubcld 11554	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ)
135	44, 32	readdcld 11150	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
136		2rp 12899	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
137	136	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℝ⁺)
138	10	rpge0d 12942	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝑁)
139	44, 137, 138	divge0d 12978	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑁 / 2))
140	29, 44, 137	lemuldiv2d 12988	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 · 𝑅) ≤ 𝑁 ↔ 𝑅 ≤ (𝑁 / 2)))
141	50, 140	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ≤ (𝑁 / 2))
142	15	2halvesd 12376	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 / 2) + (𝑁 / 2)) = 𝑁)
143	133	recnd 11149	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 / 2) ∈ ℂ)
144	15, 143, 143	subaddd 11499	. . . . . . . . . . . 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 5123	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ≤ (𝑁 − (𝑁 / 2)))
147	29, 44, 133, 146	lesubd 11730	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ≤ (𝑁 − 𝑅))
148	44	lep1d 12062	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≤ (𝑁 + 1))
149	133, 134, 44, 135, 139, 138, 147, 148	lemul12ad 12073	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 / 2) · 𝑁) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
150	132, 149	eqbrtrd 5117	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
151	122, 116, 33, 125, 150	lediv2ad 12960	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((𝑅 + 1) / ((𝑁↑2) / 2)))
152	1	peano2nnd 12151	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 + 1) ∈ ℕ)
153	152	nncnd 12150	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + 1) ∈ ℂ)
154	35	nncnd 12150	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
155	35	nnne0d 12184	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ≠ 0)
156	153, 154, 128, 155, 130	divdiv2d 11938	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁↑2) / 2)) = (((𝑅 + 1) · 2) / (𝑁↑2)))
157	153, 128	mulcomd 11142	. . . . . . . 8 ⊢ (𝜑 → ((𝑅 + 1) · 2) = (2 · (𝑅 + 1)))
158	157	oveq1d 7369	. . . . . . 7 ⊢ (𝜑 → (((𝑅 + 1) · 2) / (𝑁↑2)) = ((2 · (𝑅 + 1)) / (𝑁↑2)))
159	156, 158	eqtr2d 2769	. . . . . 6 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) = ((𝑅 + 1) / ((𝑁↑2) / 2)))
160	151, 159	breqtrrd 5123	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((2 · (𝑅 + 1)) / (𝑁↑2)))
161	117, 36, 29, 118, 160	lemul2ad 12071	. . . 4 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
162	115, 161	eqbrtrrd 5119	. . 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 11279	. 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 11279	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 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 {crab 3396 ∖ cdif 3895 class class class wbr 5095 ‘cfv 6488 (class class class)co 7354 ℂcc 11013 ℝcr 11014 0cc0 11015 1c1 11016 + caddc 11018 · cmul 11020 < clt 11155 ≤ cle 11156 − cmin 11353 / cdiv 11783 ℕcn 12134 2c2 12189 ℕ₀cn0 12390 ℤcz 12477 ℝ⁺crp 12894 ↑cexp 13972 abscabs 15145 logclog 26493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-inf2 9540 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092 ax-pre-sup 11093 ax-addf 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7618 df-om 7805 df-1st 7929 df-2nd 7930 df-supp 8099 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-2o 8394 df-er 8630 df-map 8760 df-pm 8761 df-ixp 8830 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-fsupp 9255 df-fi 9304 df-sup 9335 df-inf 9336 df-oi 9405 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-div 11784 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-5 12200 df-6 12201 df-7 12202 df-8 12203 df-9 12204 df-n0 12391 df-z 12478 df-dec 12597 df-uz 12741 df-q 12851 df-rp 12895 df-xneg 13015 df-xadd 13016 df-xmul 13017 df-ioo 13253 df-ioc 13254 df-ico 13255 df-icc 13256 df-fz 13412 df-fzo 13559 df-fl 13700 df-mod 13778 df-seq 13913 df-exp 13973 df-fac 14185 df-bc 14214 df-hash 14242 df-shft 14978 df-cj 15010 df-re 15011 df-im 15012 df-sqrt 15146 df-abs 15147 df-limsup 15382 df-clim 15399 df-rlim 15400 df-sum 15598 df-ef 15978 df-sin 15980 df-cos 15981 df-tan 15982 df-pi 15983 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17125 df-ress 17146 df-plusg 17178 df-mulr 17179 df-starv 17180 df-sca 17181 df-vsca 17182 df-ip 17183 df-tset 17184 df-ple 17185 df-ds 17187 df-unif 17188 df-hom 17189 df-cco 17190 df-rest 17330 df-topn 17331 df-0g 17349 df-gsum 17350 df-topgen 17351 df-pt 17352 df-prds 17355 df-xrs 17410 df-qtop 17415 df-imas 17416 df-xps 17418 df-mre 17492 df-mrc 17493 df-acs 17495 df-mgm 18552 df-sgrp 18631 df-mnd 18647 df-submnd 18696 df-mulg 18985 df-cntz 19233 df-cmn 19698 df-psmet 21287 df-xmet 21288 df-met 21289 df-bl 21290 df-mopn 21291 df-fbas 21292 df-fg 21293 df-cnfld 21296 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22864 df-cld 22937 df-ntr 22938 df-cls 22939 df-nei 23016 df-lp 23054 df-perf 23055 df-cn 23145 df-cnp 23146 df-haus 23233 df-cmp 23305 df-tx 23480 df-hmeo 23673 df-fil 23764 df-fm 23856 df-flim 23857 df-flf 23858 df-xms 24238 df-ms 24239 df-tms 24240 df-cncf 24801 df-limc 25797 df-dv 25798 df-log 26495

This theorem is referenced by: lgamgulmlem5 26973

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