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Theorem lgamgulmlem3 26993

Description: Lemma for lgamgulm 26997. (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 26991	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4		lgamgulm.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5	3, 4	sseldd 3959	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
6	5	eldifad 3938	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7		lgamgulm.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	peano2nnd 12257	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
9	8	nnrpd 13049	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
10	7	nnrpd 13049	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
11	9, 10	rpdivcld 13068	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) / 𝑁) ∈ ℝ⁺)
12	11	relogcld 26584	. . . . . 6 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℝ)
13	12	recnd 11263	. . . . 5 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ∈ ℂ)
14	6, 13	mulcld 11255	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘((𝑁 + 1) / 𝑁))) ∈ ℂ)
15	7	nncnd 12256	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
16	7	nnne0d 12290	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≠ 0)
17	6, 15, 16	divcld 12017	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑁) ∈ ℂ)
18		1cnd 11230	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
19	17, 18	addcld 11254	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ∈ ℂ)
20	5, 7	dmgmdivn0 26990	. . . . 5 ⊢ (𝜑 → ((𝐴 / 𝑁) + 1) ≠ 0)
21	19, 20	logcld 26531	. . . 4 ⊢ (𝜑 → (log‘((𝐴 / 𝑁) + 1)) ∈ ℂ)
22	14, 21	subcld 11594	. . 3 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
23	22	abscld 15455	. 2 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
24	14, 17	subcld 11594	. . . 4 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) ∈ ℂ)
25	24	abscld 15455	. . 3 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ∈ ℝ)
26	17, 21	subcld 11594	. . . 4 ⊢ (𝜑 → ((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))) ∈ ℂ)
27	26	abscld 15455	. . 3 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ∈ ℝ)
28	25, 27	readdcld 11264	. 2 ⊢ (𝜑 → ((abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) + (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1))))) ∈ ℝ)
29	1	nnred 12255	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		2re 12314	. . . . . 6 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ)
32		1red 11236	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
33	29, 32	readdcld 11264	. . . . 5 ⊢ (𝜑 → (𝑅 + 1) ∈ ℝ)
34	31, 33	remulcld 11265	. . . 4 ⊢ (𝜑 → (2 · (𝑅 + 1)) ∈ ℝ)
35	7	nnsqcld 14262	. . . 4 ⊢ (𝜑 → (𝑁↑2) ∈ ℕ)
36	34, 35	nndivred 12294	. . 3 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) ∈ ℝ)
37	29, 36	remulcld 11265	. 2 ⊢ (𝜑 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))) ∈ ℝ)
38	14, 21, 17	abs3difd 15479	. 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 12291	. . . . . 6 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
40	8	nnrecred 12291	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℝ)
41	39, 40	resubcld 11665	. . . . 5 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℝ)
42	29, 41	remulcld 11265	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) ∈ ℝ)
43	31, 29	remulcld 11265	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ∈ ℝ)
44	7	nnred 12255	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
45	1	nnrpd 13049	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
46	29, 45	ltaddrpd 13084	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 < (𝑅 + 𝑅))
47	1	nncnd 12256	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℂ)
48	47	2timesd 12484	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑅) = (𝑅 + 𝑅))
49	46, 48	breqtrrd 5147	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 < (2 · 𝑅))
50		lgamgulm.l	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁)
51	29, 43, 44, 49, 50	ltletrd 11395	. . . . . . . 8 ⊢ (𝜑 → 𝑅 < 𝑁)
52		difrp 13047	. . . . . . . . 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 13062	. . . . . 6 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℝ)
56	55, 39	resubcld 11665	. . . . 5 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℝ)
57	29, 56	remulcld 11265	. . . 4 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) ∈ ℝ)
58	42, 57	readdcld 11264	. . 3 ⊢ (𝜑 → ((𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))) + (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) ∈ ℝ)
59	6, 15, 16	divrecd 12020	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑁) = (𝐴 · (1 / 𝑁)))
60	59	oveq2d 7421	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
61	39	recnd 11263	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝑁) ∈ ℂ)
62	6, 13, 61	subdid 11693	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 · (1 / 𝑁))))
63	60, 62	eqtr4d 2773	. . . . . . 7 ⊢ (𝜑 → ((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁)) = (𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
64	63	fveq2d 6880	. . . . . 6 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
65	13, 61	subcld 11594	. . . . . . 7 ⊢ (𝜑 → ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)) ∈ ℂ)
66	6, 65	absmuld 15473	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴 · ((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
67	64, 66	eqtrd 2770	. . . . 5 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) = ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))))
68	6	abscld 15455	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
69	65	abscld 15455	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ∈ ℝ)
70	6	absge0d 15463	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
71	65	absge0d 15463	. . . . . 6 ⊢ (𝜑 → 0 ≤ (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))))
72		fveq2 6876	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (abs‘𝑥) = (abs‘𝐴))
73	72	breq1d 5129	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝐴) ≤ 𝑅))
74		fvoveq1 7428	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝐴 + 𝑘)))
75	74	breq2d 5131	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝐴 + 𝑘))))
76	75	ralbidv 3163	. . . . . . . . . . 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 3674	. . . . . . . . 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 26587	. . . . . . . . 9 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) = ((log‘(𝑁 + 1)) − (log‘𝑁)))
83		logdifbnd 26956	. . . . . . . . . 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 5141	. . . . . . . 8 ⊢ (𝜑 → (log‘((𝑁 + 1) / 𝑁)) ≤ (1 / 𝑁))
86	12, 39, 85	abssuble0d 15451	. . . . . . 7 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) = ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))))
87		logdiflbnd 26957	. . . . . . . . . 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 5147	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ≤ (log‘((𝑁 + 1) / 𝑁)))
90	40, 12, 39, 89	lesub2dd 11854	. . . . . . 7 ⊢ (𝜑 → ((1 / 𝑁) − (log‘((𝑁 + 1) / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
91	86, 90	eqbrtrd 5141	. . . . . 6 ⊢ (𝜑 → (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁))) ≤ ((1 / 𝑁) − (1 / (𝑁 + 1))))
92	68, 29, 69, 41, 70, 71, 81, 91	lemul12ad 12184	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴) · (abs‘((log‘((𝑁 + 1) / 𝑁)) − (1 / 𝑁)))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
93	67, 92	eqbrtrd 5141	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (𝐴 / 𝑁))) ≤ (𝑅 · ((1 / 𝑁) − (1 / (𝑁 + 1)))))
94	1, 2, 7, 4, 50	lgamgulmlem2 26992	. . . 4 ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
95	25, 27, 42, 57, 93, 94	le2addd 11856	. . 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 11594	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℂ)
97	15, 18	addcld 11254	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
98	29, 51	gtned 11370	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≠ 𝑅)
99	15, 47, 98	subne0d 11603	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑅) ≠ 0)
100	8	nnne0d 12290	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
101	96, 97, 99, 100	subrecd 12070	. . . . . . 7 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))))
102	15, 18, 47	pnncand 11633	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (1 + 𝑅))
103	18, 47, 102	comraddd 11449	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + 1) − (𝑁 − 𝑅)) = (𝑅 + 1))
104	103	oveq1d 7420	. . . . . . 7 ⊢ (𝜑 → (((𝑁 + 1) − (𝑁 − 𝑅)) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))))
105	101, 104	eqtr2d 2771	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
106	105	oveq2d 7421	. . . . 5 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) = (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))))
107	97, 100	reccld 12010	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 + 1)) ∈ ℂ)
108	96, 99	reccld 12010	. . . . . . . 8 ⊢ (𝜑 → (1 / (𝑁 − 𝑅)) ∈ ℂ)
109	61, 107, 108	npncan3d 11630	. . . . . . 7 ⊢ (𝜑 → (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))) = ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))))
110	109	eqcomd 2741	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1))) = (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁))))
111	110	oveq2d 7421	. . . . 5 ⊢ (𝜑 → (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / (𝑁 + 1)))) = (𝑅 · (((1 / 𝑁) − (1 / (𝑁 + 1))) + ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))))
112	41	recnd 11263	. . . . . 6 ⊢ (𝜑 → ((1 / 𝑁) − (1 / (𝑁 + 1))) ∈ ℂ)
113	56	recnd 11263	. . . . . 6 ⊢ (𝜑 → ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)) ∈ ℂ)
114	47, 112, 113	adddid 11259	. . . . 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 2774	. . . 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 13067	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 𝑅) · (𝑁 + 1)) ∈ ℝ⁺)
117	33, 116	rerpdivcld 13082	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ∈ ℝ)
118	45	rpge0d 13055	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝑅)
119		2z 12624	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
120	119	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℤ)
121	10, 120	rpexpcld 14265	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℝ⁺)
122	121	rphalfcld 13063	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ∈ ℝ⁺)
123		0le1 11760	. . . . . . . . 9 ⊢ 0 ≤ 1
124	123	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 1)
125	29, 32, 118, 124	addge0d 11813	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑅 + 1))
126	15	sqvald 14161	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) = (𝑁 · 𝑁))
127	126	oveq1d 7420	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 · 𝑁) / 2))
128	31	recnd 11263	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
129		2ne0 12344	. . . . . . . . . . 11 ⊢ 2 ≠ 0
130	129	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
131	15, 15, 128, 130	div23d 12054	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 · 𝑁) / 2) = ((𝑁 / 2) · 𝑁))
132	127, 131	eqtrd 2770	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑2) / 2) = ((𝑁 / 2) · 𝑁))
133	44	rehalfcld 12488	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ∈ ℝ)
134	44, 29	resubcld 11665	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 𝑅) ∈ ℝ)
135	44, 32	readdcld 11264	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
136		2rp 13013	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
137	136	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℝ⁺)
138	10	rpge0d 13055	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝑁)
139	44, 137, 138	divge0d 13091	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑁 / 2))
140	29, 44, 137	lemuldiv2d 13101	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 · 𝑅) ≤ 𝑁 ↔ 𝑅 ≤ (𝑁 / 2)))
141	50, 140	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ≤ (𝑁 / 2))
142	15	2halvesd 12487	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 / 2) + (𝑁 / 2)) = 𝑁)
143	133	recnd 11263	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 / 2) ∈ ℂ)
144	15, 143, 143	subaddd 11612	. . . . . . . . . . . 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 5147	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ≤ (𝑁 − (𝑁 / 2)))
147	29, 44, 133, 146	lesubd 11841	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 / 2) ≤ (𝑁 − 𝑅))
148	44	lep1d 12173	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≤ (𝑁 + 1))
149	133, 134, 44, 135, 139, 138, 147, 148	lemul12ad 12184	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 / 2) · 𝑁) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
150	132, 149	eqbrtrd 5141	. . . . . . 7 ⊢ (𝜑 → ((𝑁↑2) / 2) ≤ ((𝑁 − 𝑅) · (𝑁 + 1)))
151	122, 116, 33, 125, 150	lediv2ad 13073	. . . . . 6 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((𝑅 + 1) / ((𝑁↑2) / 2)))
152	1	peano2nnd 12257	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 + 1) ∈ ℕ)
153	152	nncnd 12256	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + 1) ∈ ℂ)
154	35	nncnd 12256	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
155	35	nnne0d 12290	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑2) ≠ 0)
156	153, 154, 128, 155, 130	divdiv2d 12049	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁↑2) / 2)) = (((𝑅 + 1) · 2) / (𝑁↑2)))
157	153, 128	mulcomd 11256	. . . . . . . 8 ⊢ (𝜑 → ((𝑅 + 1) · 2) = (2 · (𝑅 + 1)))
158	157	oveq1d 7420	. . . . . . 7 ⊢ (𝜑 → (((𝑅 + 1) · 2) / (𝑁↑2)) = ((2 · (𝑅 + 1)) / (𝑁↑2)))
159	156, 158	eqtr2d 2771	. . . . . 6 ⊢ (𝜑 → ((2 · (𝑅 + 1)) / (𝑁↑2)) = ((𝑅 + 1) / ((𝑁↑2) / 2)))
160	151, 159	breqtrrd 5147	. . . . 5 ⊢ (𝜑 → ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1))) ≤ ((2 · (𝑅 + 1)) / (𝑁↑2)))
161	117, 36, 29, 118, 160	lemul2ad 12182	. . . 4 ⊢ (𝜑 → (𝑅 · ((𝑅 + 1) / ((𝑁 − 𝑅) · (𝑁 + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))
162	115, 161	eqbrtrrd 5143	. . 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 11392	. 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 11392	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 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 {crab 3415 ∖ cdif 3923 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 < clt 11269 ≤ cle 11270 − cmin 11466 / cdiv 11894 ℕcn 12240 2c2 12295 ℕ₀cn0 12501 ℤcz 12588 ℝ⁺crp 13008 ↑cexp 14079 abscabs 15253 logclog 26515

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-tan 16087 df-pi 16088 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-cmp 23325 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820 df-log 26517

This theorem is referenced by: lgamgulmlem5 26995

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