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Theorem lgamgulmlem5 27011

Description: Lemma for lgamgulm 27013. (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.g	⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
lgamgulm.t	⊢ 𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))

**Assertion**
Ref	Expression
lgamgulmlem5	⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝐺‘𝑛)‘𝑦)) ≤ (𝑇‘𝑛))

Distinct variable groups: 𝑦,𝑛,𝐺 𝑥,𝑦 𝑘,𝑚,𝑛,𝑥,𝑦,𝑧,𝑅 𝑈,𝑚,𝑛,𝑦,𝑧 𝜑,𝑚,𝑛,𝑥,𝑦,𝑧 𝑇,𝑛,𝑦 Allowed substitution hints: 𝜑(𝑘) 𝑇(𝑥,𝑧,𝑘,𝑚) 𝑈(𝑥,𝑘) 𝐺(𝑥,𝑧,𝑘,𝑚)

Proof of Theorem lgamgulmlem5 Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5104	. . 3 ⊢ ((𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) → ((abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ↔ (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))))
2		breq2 5104	. . 3 ⊢ (((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) → ((abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ↔ (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))))
3		lgamgulm.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℕ)
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑅 ∈ ℕ)
5	4	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑅 ∈ ℕ)
6		lgamgulm.u	. . . . 5 ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}
7		fveq2 6842	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (abs‘𝑥) = (abs‘𝑡))
8	7	breq1d 5110	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑡) ≤ 𝑅))
9		fvoveq1 7391	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑡 + 𝑘)))
10	9	breq2d 5112	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
11	10	ralbidv 3161	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
12	8, 11	anbi12d 633	. . . . . 6 ⊢ (𝑥 = 𝑡 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))))
13	12	cbvrabv 3411	. . . . 5 ⊢ {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))}
14	6, 13	eqtri 2760	. . . 4 ⊢ 𝑈 = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))}
15		simplrl 777	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑛 ∈ ℕ)
16		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
17	16	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑦 ∈ 𝑈)
18		simpr 484	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → (2 · 𝑅) ≤ 𝑛)
19	5, 14, 15, 17, 18	lgamgulmlem3 27009	. . 3 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
20	3, 6	lgamgulmlem1 27007	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
21	20	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
22	21, 16	sseldd 3936	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
23	22	eldifad 3915	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ ℂ)
24		simprl 771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℕ)
25	24	peano2nnd 12174	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑛 + 1) ∈ ℕ)
26	25	nnrpd 12959	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑛 + 1) ∈ ℝ⁺)
27	24	nnrpd 12959	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℝ⁺)
28	26, 27	rpdivcld 12978	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
29	28	relogcld 26600	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
30	29	recnd 11172	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
31	23, 30	mulcld 11164	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑦 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
32	24	nncnd 12173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℂ)
33	24	nnne0d 12207	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ≠ 0)
34	23, 32, 33	divcld 11929	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑦 / 𝑛) ∈ ℂ)
35		1cnd 11139	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 1 ∈ ℂ)
36	34, 35	addcld 11163	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 / 𝑛) + 1) ∈ ℂ)
37	22, 24	dmgmdivn0 27006	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 / 𝑛) + 1) ≠ 0)
38	36, 37	logcld 26547	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘((𝑦 / 𝑛) + 1)) ∈ ℂ)
39	31, 38	subcld 11504	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
40	39	abscld 15374	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
41	31	abscld 15374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ∈ ℝ)
42	38	abscld 15374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
43	41, 42	readdcld 11173	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
44	4	nnred 12172	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑅 ∈ ℝ)
45	44, 29	remulcld 11174	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
46	4	peano2nnd 12174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ∈ ℕ)
47	46	nnrpd 12959	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ∈ ℝ⁺)
48	47, 27	rpmulcld 12977	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ⁺)
49	48	relogcld 26600	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
50		pire 26434	. . . . . . . 8 ⊢ π ∈ ℝ
51	50	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → π ∈ ℝ)
52	49, 51	readdcld 11173	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
53	45, 52	readdcld 11173	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
54	31, 38	abs2dif2d 15396	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))))
55	23, 30	absmuld 15392	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))))
56	28	rpred 12961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ)
57	32	mullidd 11162	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 · 𝑛) = 𝑛)
58	24	nnred 12172	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℝ)
59	58	lep1d 12085	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ≤ (𝑛 + 1))
60	57, 59	eqbrtrd 5122	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 · 𝑛) ≤ (𝑛 + 1))
61		1red 11145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 1 ∈ ℝ)
62	58, 61	readdcld 11173	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑛 + 1) ∈ ℝ)
63	61, 62, 27	lemuldivd 13010	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((1 · 𝑛) ≤ (𝑛 + 1) ↔ 1 ≤ ((𝑛 + 1) / 𝑛)))
64	60, 63	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 1 ≤ ((𝑛 + 1) / 𝑛))
65	56, 64	logge0d 26607	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ (log‘((𝑛 + 1) / 𝑛)))
66	29, 65	absidd 15358	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘((𝑛 + 1) / 𝑛))) = (log‘((𝑛 + 1) / 𝑛)))
67	66	oveq2d 7384	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
68	55, 67	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
69	23	abscld 15374	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘𝑦) ∈ ℝ)
70		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (abs‘𝑥) = (abs‘𝑦))
71	70	breq1d 5110	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
72		fvoveq1 7391	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑦 + 𝑘)))
73	72	breq2d 5112	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
74	73	ralbidv 3161	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
75	71, 74	anbi12d 633	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))))
76	75, 6	elrab2 3651	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑈 ↔ (𝑦 ∈ ℂ ∧ ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))))
77	76	simprbi 497	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑈 → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
78	77	ad2antll 730	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
79	78	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘𝑦) ≤ 𝑅)
80	69, 44, 29, 65, 79	lemul1ad 12093	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
81	68, 80	eqbrtrd 5122	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
82	36, 37	absrpcld 15386	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ⁺)
83	82	relogcld 26600	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
84	83	recnd 11172	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
85	84	abscld 15374	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
86	85, 51	readdcld 11173	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ∈ ℝ)
87		abslogle 26595	. . . . . . . 8 ⊢ ((((𝑦 / 𝑛) + 1) ∈ ℂ ∧ ((𝑦 / 𝑛) + 1) ≠ 0) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π))
88	36, 37, 87	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π))
89		1rp 12921	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
90		relogdiv 26570	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ⁺ ∧ ((𝑅 + 1) · 𝑛) ∈ ℝ⁺) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
91	89, 48, 90	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
92		log1 26562	. . . . . . . . . . . . 13 ⊢ (log‘1) = 0
93	92	oveq1i 7378	. . . . . . . . . . . 12 ⊢ ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = (0 − (log‘((𝑅 + 1) · 𝑛)))
94		df-neg 11379	. . . . . . . . . . . 12 ⊢ -(log‘((𝑅 + 1) · 𝑛)) = (0 − (log‘((𝑅 + 1) · 𝑛)))
95	93, 94	eqtr4i 2763	. . . . . . . . . . 11 ⊢ ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = -(log‘((𝑅 + 1) · 𝑛))
96	91, 95	eqtr2di 2789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) = (log‘(1 / ((𝑅 + 1) · 𝑛))))
97	46	nnrecred 12208	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑅 + 1)) ∈ ℝ)
98	23, 32	addcld 11163	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑦 + 𝑛) ∈ ℂ)
99	98	abscld 15374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 + 𝑛)) ∈ ℝ)
100	4	nnrecred 12208	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / 𝑅) ∈ ℝ)
101	4	nnrpd 12959	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑅 ∈ ℝ⁺)
102		0le1 11672	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ 1
103	102	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ 1)
104	44	lep1d 12085	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑅 ≤ (𝑅 + 1))
105	101, 47, 61, 103, 104	lediv2ad 12983	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑅 + 1)) ≤ (1 / 𝑅))
106		oveq2 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝑦 + 𝑘) = (𝑦 + 𝑛))
107	106	fveq2d 6846	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (abs‘(𝑦 + 𝑘)) = (abs‘(𝑦 + 𝑛)))
108	107	breq2d 5112	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → ((1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛))))
109	78	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))
110	24	nnnn0d 12474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℕ₀)
111	108, 109, 110	rspcdva 3579	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛)))
112	97, 100, 99, 105, 111	letrd 11302	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑅 + 1)) ≤ (abs‘(𝑦 + 𝑛)))
113	97, 99, 27, 112	lediv1dd 13019	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) ≤ ((abs‘(𝑦 + 𝑛)) / 𝑛))
114	46	nncnd 12173	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ∈ ℂ)
115	46	nnne0d 12207	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ≠ 0)
116	114, 32, 115, 33	recdiv2d 11947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) = (1 / ((𝑅 + 1) · 𝑛)))
117	23, 32, 32, 33	divdird 11967	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 + 𝑛) / 𝑛) = ((𝑦 / 𝑛) + (𝑛 / 𝑛)))
118	32, 33	dividd 11927	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑛 / 𝑛) = 1)
119	118	oveq2d 7384	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 / 𝑛) + (𝑛 / 𝑛)) = ((𝑦 / 𝑛) + 1))
120	117, 119	eqtr2d 2773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 / 𝑛) + 1) = ((𝑦 + 𝑛) / 𝑛))
121	120	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) = (abs‘((𝑦 + 𝑛) / 𝑛)))
122	98, 32, 33	absdivd 15393	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 + 𝑛) / 𝑛)) = ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)))
123	27	rpge0d 12965	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ 𝑛)
124	58, 123	absidd 15358	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘𝑛) = 𝑛)
125	124	oveq2d 7384	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)) = ((abs‘(𝑦 + 𝑛)) / 𝑛))
126	121, 122, 125	3eqtrrd 2777	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 + 𝑛)) / 𝑛) = (abs‘((𝑦 / 𝑛) + 1)))
127	113, 116, 126	3brtr3d 5131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)))
128	48	rpreccld 12971	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ∈ ℝ⁺)
129	128, 82	logled 26604	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)) ↔ (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1)))))
130	127, 129	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
131	96, 130	eqbrtrd 5122	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
132	36	abscld 15374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ)
133	44, 61	readdcld 11173	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ∈ ℝ)
134	48	rpred 12961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ)
135	34	abscld 15374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 / 𝑛)) ∈ ℝ)
136	135, 61	readdcld 11173	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ∈ ℝ)
137	34, 35	abstrid 15394	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + (abs‘1)))
138		abs1 15232	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
139	138	oveq2i 7379	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝑦 / 𝑛)) + (abs‘1)) = ((abs‘(𝑦 / 𝑛)) + 1)
140	137, 139	breqtrdi 5141	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + 1))
141	89	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 1 ∈ ℝ⁺)
142	23	absge0d 15382	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ (abs‘𝑦))
143	24	nnge1d 12205	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 1 ≤ 𝑛)
144	69, 44, 141, 58, 142, 79, 143	lediv12ad 13020	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) / 𝑛) ≤ (𝑅 / 1))
145	23, 32, 33	absdivd 15393	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 / 𝑛)) = ((abs‘𝑦) / (abs‘𝑛)))
146	124	oveq2d 7384	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) / (abs‘𝑛)) = ((abs‘𝑦) / 𝑛))
147	145, 146	eqtr2d 2773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) / 𝑛) = (abs‘(𝑦 / 𝑛)))
148	4	nncnd 12173	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑅 ∈ ℂ)
149	148	div1d 11921	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 / 1) = 𝑅)
150	144, 147, 149	3brtr3d 5131	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 / 𝑛)) ≤ 𝑅)
151	135, 44, 61, 150	leadd1dd 11763	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ≤ (𝑅 + 1))
152	132, 136, 133, 140, 151	letrd 11302	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ (𝑅 + 1))
153	47	rpge0d 12965	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ (𝑅 + 1))
154	133, 58, 153, 143	lemulge11d 12091	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ≤ ((𝑅 + 1) · 𝑛))
155	132, 133, 134, 152, 154	letrd 11302	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛))
156	82, 48	logled 26604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛) ↔ (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛))))
157	155, 156	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛)))
158	83, 49	absled 15368	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ≤ (log‘((𝑅 + 1) · 𝑛)) ↔ (-(log‘((𝑅 + 1) · 𝑛)) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))) ∧ (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛)))))
159	131, 157, 158	mpbir2and 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ≤ (log‘((𝑅 + 1) · 𝑛)))
160	85, 49, 51, 159	leadd1dd 11763	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
161	42, 86, 52, 88, 160	letrd 11302	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
162	41, 42, 45, 52, 81, 161	le2addd 11768	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))) ≤ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
163	40, 43, 53, 54, 162	letrd 11302	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
164	163	adantr 480	. . 3 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) ∧ ¬ (2 · 𝑅) ≤ 𝑛) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
165	1, 2, 19, 164	ifbothda 4520	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
166		oveq1 7375	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
167		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
168	166, 167	oveq12d 7386	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
169	168	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
170	169	oveq2d 7384	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
171		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
172	171	fvoveq1d 7390	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
173	170, 172	oveq12d 7386	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
174	173	mpteq2dv 5194	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))) = (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
175		lgamgulm.g	. . . . . . 7 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
176		cnex 11119	. . . . . . . . 9 ⊢ ℂ ∈ V
177	6, 176	rabex2 5288	. . . . . . . 8 ⊢ 𝑈 ∈ V
178	177	mptex 7179	. . . . . . 7 ⊢ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) ∈ V
179	174, 175, 178	fvmpt 6949	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
180	179	ad2antrl 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝐺‘𝑛) = (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
181	180	fveq1d 6844	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝐺‘𝑛)‘𝑦) = ((𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦))
182		oveq1 7375	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) = (𝑦 · (log‘((𝑛 + 1) / 𝑛))))
183		oveq1 7375	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑧 / 𝑛) = (𝑦 / 𝑛))
184	183	fvoveq1d 7390	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (log‘((𝑧 / 𝑛) + 1)) = (log‘((𝑦 / 𝑛) + 1)))
185	182, 184	oveq12d 7386	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
186		eqid 2737	. . . . . 6 ⊢ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) = (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
187		ovex 7401	. . . . . 6 ⊢ ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ V
188	185, 186, 187	fvmpt 6949	. . . . 5 ⊢ (𝑦 ∈ 𝑈 → ((𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
189	188	ad2antll 730	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
190	181, 189	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝐺‘𝑛)‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
191	190	fveq2d 6846	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝐺‘𝑛)‘𝑦)) = (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))))
192		breq2 5104	. . . . 5 ⊢ (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
193		oveq1 7375	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
194	193	oveq2d 7384	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
195	194	oveq2d 7384	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
196	169	oveq2d 7384	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
197		oveq2 7376	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
198	197	fveq2d 6846	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
199	198	oveq1d 7383	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
200	196, 199	oveq12d 7386	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π)) = ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
201	192, 195, 200	ifbieq12d 4510	. . . 4 ⊢ (𝑚 = 𝑛 → if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
202		lgamgulm.t	. . . 4 ⊢ 𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))
203		ovex 7401	. . . . 5 ⊢ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
204		ovex 7401	. . . . 5 ⊢ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
205	203, 204	ifex 4532	. . . 4 ⊢ if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) ∈ V
206	201, 202, 205	fvmpt 6949	. . 3 ⊢ (𝑛 ∈ ℕ → (𝑇‘𝑛) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
207	206	ad2antrl 729	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑇‘𝑛) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
208	165, 191, 207	3brtr4d 5132	1 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝐺‘𝑛)‘𝑦)) ≤ (𝑇‘𝑛))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 {crab 3401 ∖ cdif 3900 ⊆ wss 3903 ifcif 4481 class class class wbr 5100 ↦ cmpt 5181 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ≤ cle 11179 − cmin 11376 -cneg 11377 / cdiv 11806 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ℤcz 12500 ℝ⁺crp 12917 ↑cexp 13996 abscabs 15169 πcpi 16001 logclog 26531

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-tan 16006 df-pi 16007 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-cmp 23343 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836 df-log 26533

This theorem is referenced by: lgamgulmlem6 27012

W3C validator