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Theorem lgamgulmlem5 26087

Description: Lemma for lgamgulm 26089. (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 5074	. . 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 5074	. . 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 6756	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (abs‘𝑥) = (abs‘𝑡))
8	7	breq1d 5080	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑡) ≤ 𝑅))
9		fvoveq1 7278	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑡 + 𝑘)))
10	9	breq2d 5082	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
11	10	ralbidv 3120	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
12	8, 11	anbi12d 630	. . . . . 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 3416	. . . . 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 2766	. . . 4 ⊢ 𝑈 = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))}
15		simplrl 773	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑛 ∈ ℕ)
16		simprr 769	. . . . 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 26085	. . 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 26083	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
21	20	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
22	21, 16	sseldd 3918	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
23	22	eldifad 3895	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ ℂ)
24		simprl 767	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℕ)
25	24	peano2nnd 11920	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑛 + 1) ∈ ℕ)
26	25	nnrpd 12699	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑛 + 1) ∈ ℝ⁺)
27	24	nnrpd 12699	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℝ⁺)
28	26, 27	rpdivcld 12718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
29	28	relogcld 25683	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
30	29	recnd 10934	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
31	23, 30	mulcld 10926	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑦 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
32	24	nncnd 11919	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℂ)
33	24	nnne0d 11953	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ≠ 0)
34	23, 32, 33	divcld 11681	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑦 / 𝑛) ∈ ℂ)
35		1cnd 10901	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 1 ∈ ℂ)
36	34, 35	addcld 10925	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 / 𝑛) + 1) ∈ ℂ)
37	22, 24	dmgmdivn0 26082	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 / 𝑛) + 1) ≠ 0)
38	36, 37	logcld 25631	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘((𝑦 / 𝑛) + 1)) ∈ ℂ)
39	31, 38	subcld 11262	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
40	39	abscld 15076	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
41	31	abscld 15076	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ∈ ℝ)
42	38	abscld 15076	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
43	41, 42	readdcld 10935	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
44	4	nnred 11918	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑅 ∈ ℝ)
45	44, 29	remulcld 10936	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
46	4	peano2nnd 11920	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ∈ ℕ)
47	46	nnrpd 12699	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ∈ ℝ⁺)
48	47, 27	rpmulcld 12717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ⁺)
49	48	relogcld 25683	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
50		pire 25520	. . . . . . . 8 ⊢ π ∈ ℝ
51	50	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → π ∈ ℝ)
52	49, 51	readdcld 10935	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
53	45, 52	readdcld 10935	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
54	31, 38	abs2dif2d 15098	. . . . 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 15094	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))))
56	28	rpred 12701	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ)
57	32	mulid2d 10924	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 · 𝑛) = 𝑛)
58	24	nnred 11918	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℝ)
59	58	lep1d 11836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ≤ (𝑛 + 1))
60	57, 59	eqbrtrd 5092	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 · 𝑛) ≤ (𝑛 + 1))
61		1red 10907	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 1 ∈ ℝ)
62	58, 61	readdcld 10935	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑛 + 1) ∈ ℝ)
63	61, 62, 27	lemuldivd 12750	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((1 · 𝑛) ≤ (𝑛 + 1) ↔ 1 ≤ ((𝑛 + 1) / 𝑛)))
64	60, 63	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 1 ≤ ((𝑛 + 1) / 𝑛))
65	56, 64	logge0d 25690	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ (log‘((𝑛 + 1) / 𝑛)))
66	29, 65	absidd 15062	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘((𝑛 + 1) / 𝑛))) = (log‘((𝑛 + 1) / 𝑛)))
67	66	oveq2d 7271	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
68	55, 67	eqtrd 2778	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
69	23	abscld 15076	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘𝑦) ∈ ℝ)
70		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (abs‘𝑥) = (abs‘𝑦))
71	70	breq1d 5080	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
72		fvoveq1 7278	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑦 + 𝑘)))
73	72	breq2d 5082	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
74	73	ralbidv 3120	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
75	71, 74	anbi12d 630	. . . . . . . . . . . 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 3620	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑈 ↔ (𝑦 ∈ ℂ ∧ ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))))
77	76	simprbi 496	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑈 → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
78	77	ad2antll 725	. . . . . . . . 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 11844	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
81	68, 80	eqbrtrd 5092	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
82	36, 37	absrpcld 15088	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ⁺)
83	82	relogcld 25683	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
84	83	recnd 10934	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
85	84	abscld 15076	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
86	85, 51	readdcld 10935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ∈ ℝ)
87		abslogle 25678	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π))
89		1rp 12663	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
90		relogdiv 25653	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ⁺ ∧ ((𝑅 + 1) · 𝑛) ∈ ℝ⁺) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
91	89, 48, 90	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
92		log1 25646	. . . . . . . . . . . . 13 ⊢ (log‘1) = 0
93	92	oveq1i 7265	. . . . . . . . . . . 12 ⊢ ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = (0 − (log‘((𝑅 + 1) · 𝑛)))
94		df-neg 11138	. . . . . . . . . . . 12 ⊢ -(log‘((𝑅 + 1) · 𝑛)) = (0 − (log‘((𝑅 + 1) · 𝑛)))
95	93, 94	eqtr4i 2769	. . . . . . . . . . 11 ⊢ ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = -(log‘((𝑅 + 1) · 𝑛))
96	91, 95	eqtr2di 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) = (log‘(1 / ((𝑅 + 1) · 𝑛))))
97	46	nnrecred 11954	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑅 + 1)) ∈ ℝ)
98	23, 32	addcld 10925	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑦 + 𝑛) ∈ ℂ)
99	98	abscld 15076	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 + 𝑛)) ∈ ℝ)
100	4	nnrecred 11954	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / 𝑅) ∈ ℝ)
101	4	nnrpd 12699	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑅 ∈ ℝ⁺)
102		0le1 11428	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ 1
103	102	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ 1)
104	44	lep1d 11836	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑅 ≤ (𝑅 + 1))
105	101, 47, 61, 103, 104	lediv2ad 12723	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑅 + 1)) ≤ (1 / 𝑅))
106		oveq2 7263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝑦 + 𝑘) = (𝑦 + 𝑛))
107	106	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (abs‘(𝑦 + 𝑘)) = (abs‘(𝑦 + 𝑛)))
108	107	breq2d 5082	. . . . . . . . . . . . . . 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 12223	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑛 ∈ ℕ₀)
111	108, 109, 110	rspcdva 3554	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛)))
112	97, 100, 99, 105, 111	letrd 11062	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑅 + 1)) ≤ (abs‘(𝑦 + 𝑛)))
113	97, 99, 27, 112	lediv1dd 12759	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) ≤ ((abs‘(𝑦 + 𝑛)) / 𝑛))
114	46	nncnd 11919	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ∈ ℂ)
115	46	nnne0d 11953	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ≠ 0)
116	114, 32, 115, 33	recdiv2d 11699	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) = (1 / ((𝑅 + 1) · 𝑛)))
117	23, 32, 32, 33	divdird 11719	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 + 𝑛) / 𝑛) = ((𝑦 / 𝑛) + (𝑛 / 𝑛)))
118	32, 33	dividd 11679	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑛 / 𝑛) = 1)
119	118	oveq2d 7271	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 / 𝑛) + (𝑛 / 𝑛)) = ((𝑦 / 𝑛) + 1))
120	117, 119	eqtr2d 2779	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑦 / 𝑛) + 1) = ((𝑦 + 𝑛) / 𝑛))
121	120	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) = (abs‘((𝑦 + 𝑛) / 𝑛)))
122	98, 32, 33	absdivd 15095	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 + 𝑛) / 𝑛)) = ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)))
123	27	rpge0d 12705	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ 𝑛)
124	58, 123	absidd 15062	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘𝑛) = 𝑛)
125	124	oveq2d 7271	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)) = ((abs‘(𝑦 + 𝑛)) / 𝑛))
126	121, 122, 125	3eqtrrd 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 + 𝑛)) / 𝑛) = (abs‘((𝑦 / 𝑛) + 1)))
127	113, 116, 126	3brtr3d 5101	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)))
128	48	rpreccld 12711	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ∈ ℝ⁺)
129	128, 82	logled 25687	. . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
131	96, 130	eqbrtrd 5092	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
132	36	abscld 15076	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ)
133	44, 61	readdcld 10935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ∈ ℝ)
134	48	rpred 12701	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ)
135	34	abscld 15076	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 / 𝑛)) ∈ ℝ)
136	135, 61	readdcld 10935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ∈ ℝ)
137	34, 35	abstrid 15096	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + (abs‘1)))
138		abs1 14937	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
139	138	oveq2i 7266	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝑦 / 𝑛)) + (abs‘1)) = ((abs‘(𝑦 / 𝑛)) + 1)
140	137, 139	breqtrdi 5111	. . . . . . . . . . . 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 15084	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ (abs‘𝑦))
143	24	nnge1d 11951	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 1 ≤ 𝑛)
144	69, 44, 141, 58, 142, 79, 143	lediv12ad 12760	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) / 𝑛) ≤ (𝑅 / 1))
145	23, 32, 33	absdivd 15095	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 / 𝑛)) = ((abs‘𝑦) / (abs‘𝑛)))
146	124	oveq2d 7271	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) / (abs‘𝑛)) = ((abs‘𝑦) / 𝑛))
147	145, 146	eqtr2d 2779	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘𝑦) / 𝑛) = (abs‘(𝑦 / 𝑛)))
148	4	nncnd 11919	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 𝑅 ∈ ℂ)
149	148	div1d 11673	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 / 1) = 𝑅)
150	144, 147, 149	3brtr3d 5101	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(𝑦 / 𝑛)) ≤ 𝑅)
151	135, 44, 61, 150	leadd1dd 11519	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ≤ (𝑅 + 1))
152	132, 136, 133, 140, 151	letrd 11062	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ (𝑅 + 1))
153	47	rpge0d 12705	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → 0 ≤ (𝑅 + 1))
154	133, 58, 153, 143	lemulge11d 11842	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝑅 + 1) ≤ ((𝑅 + 1) · 𝑛))
155	132, 133, 134, 152, 154	letrd 11062	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛))
156	82, 48	logled 25687	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛)))
158	83, 49	absled 15070	. . . . . . . . 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 709	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ≤ (log‘((𝑅 + 1) · 𝑛)))
160	85, 49, 51, 159	leadd1dd 11519	. . . . . . 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 11062	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
162	41, 42, 45, 52, 81, 161	le2addd 11524	. . . . 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 11062	. . . 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 4494	. 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 7262	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
167		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
168	166, 167	oveq12d 7273	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
169	168	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
170	169	oveq2d 7271	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
171		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
172	171	fvoveq1d 7277	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
173	170, 172	oveq12d 7273	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
174	173	mpteq2dv 5172	. . . . . . 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 10883	. . . . . . . . 9 ⊢ ℂ ∈ V
177	6, 176	rabex2 5253	. . . . . . . 8 ⊢ 𝑈 ∈ V
178	177	mptex 7081	. . . . . . 7 ⊢ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) ∈ V
179	174, 175, 178	fvmpt 6857	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
180	179	ad2antrl 724	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (𝐺‘𝑛) = (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
181	180	fveq1d 6758	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝐺‘𝑛)‘𝑦) = ((𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦))
182		oveq1 7262	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) = (𝑦 · (log‘((𝑛 + 1) / 𝑛))))
183		oveq1 7262	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑧 / 𝑛) = (𝑦 / 𝑛))
184	183	fvoveq1d 7277	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (log‘((𝑧 / 𝑛) + 1)) = (log‘((𝑦 / 𝑛) + 1)))
185	182, 184	oveq12d 7273	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
186		eqid 2738	. . . . . 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 7288	. . . . . 6 ⊢ ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ V
188	185, 186, 187	fvmpt 6857	. . . . 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 725	. . . 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 2778	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → ((𝐺‘𝑛)‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
191	190	fveq2d 6760	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝐺‘𝑛)‘𝑦)) = (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))))
192		breq2 5074	. . . . 5 ⊢ (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
193		oveq1 7262	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
194	193	oveq2d 7271	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
195	194	oveq2d 7271	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
196	169	oveq2d 7271	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
197		oveq2 7263	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
198	197	fveq2d 6760	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
199	198	oveq1d 7270	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
200	196, 199	oveq12d 7273	. . . . 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 4484	. . . 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 7288	. . . . 5 ⊢ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
204		ovex 7288	. . . . 5 ⊢ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
205	203, 204	ifex 4506	. . . 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 6857	. . 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 724	. 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 5102	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 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 {crab 3067 ∖ cdif 3880 ⊆ wss 3883 ifcif 4456 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℝ⁺crp 12659 ↑cexp 13710 abscabs 14873 πcpi 15704 logclog 25615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-tan 15709 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617

This theorem is referenced by: lgamgulmlem6 26088

W3C validator