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Theorem lgambdd 27005

Description: The log-Gamma function is bounded on the region 𝑈. (Contributed by Mario Carneiro, 9-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)))))

**Assertion**
Ref	Expression
lgambdd	⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)

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

Proof of Theorem lgambdd Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lgamgulm.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℕ)
2		lgamgulm.u	. . . . 5 ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}
3		lgamgulm.g	. . . . 5 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
4	1, 2, 3	lgamgulm2 27004	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))
5	4	simprd 495	. . 3 ⊢ (𝜑 → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
6		eqid 2735	. . . . 5 ⊢ (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π)))) = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))
7	1, 2, 3, 6	lgamgulmlem6 27002	. . . 4 ⊢ (𝜑 → (seq1( ∘_f + , 𝐺) ∈ dom (⇝_𝑢‘𝑈) ∧ (seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)))
8	7	simprd 495	. . 3 ⊢ (𝜑 → (seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦))
9	5, 8	mpd 15	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)
10	1	nnrpd 12949	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑅 ∈ ℝ⁺)
12	11	relogcld 26590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (log‘𝑅) ∈ ℝ)
13		pire 26424	. . . . . . 7 ⊢ π ∈ ℝ
14	13	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → π ∈ ℝ)
15	12, 14	readdcld 11163	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((log‘𝑅) + π) ∈ ℝ)
16		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
17	15, 16	readdcld 11163	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
18	17	adantrr 718	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧 ∈ 𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
19	4	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
20	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
21	20	r19.21bi 3227	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) ∈ ℂ)
22	21	abscld 15364	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
23	22	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
24	11	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑅 ∈ ℝ⁺)
25	24	relogcld 26590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘𝑅) ∈ ℝ)
26	13	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → π ∈ ℝ)
27	25, 26	readdcld 11163	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log‘𝑅) + π) ∈ ℝ)
28	1, 2	lgamgulmlem1 26997	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
29	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
30	29	sselda 3932	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
31	30	eldifad 3912	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℂ)
32	30	dmgmn0 26994	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ 0)
33	31, 32	logcld 26537	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘𝑧) ∈ ℂ)
34	21, 33	addcld 11153	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) ∈ ℂ)
35	34	abscld 15364	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
36	27, 35	readdcld 11163	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
37	36	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
38	17	ad2antrr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
39	33	abscld 15364	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ∈ ℝ)
40	39, 35	readdcld 11163	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
41	33	negcld 11481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑧) ∈ ℂ)
42	21, 41	abs2difd 15385	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) − -(log‘𝑧))))
43	33	absnegd 15377	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘-(log‘𝑧)) = (abs‘(log‘𝑧)))
44	43	oveq2d 7374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) = ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))))
45	21, 33	subnegd 11501	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) − -(log‘𝑧)) = ((log Γ‘𝑧) + (log‘𝑧)))
46	45	fveq2d 6837	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘((log Γ‘𝑧) − -(log‘𝑧))) = (abs‘((log Γ‘𝑧) + (log‘𝑧))))
47	42, 44, 46	3brtr3d 5128	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))))
48	22, 39, 35	lesubadd2d 11738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ↔ (abs‘(log Γ‘𝑧)) ≤ ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧))))))
49	47, 48	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log Γ‘𝑧)) ≤ ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
50	31, 32	absrpcld 15376	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘𝑧) ∈ ℝ⁺)
51	50	relogcld 26590	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ∈ ℝ)
52	51	recnd 11162	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ∈ ℂ)
53	52	abscld 15364	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘(abs‘𝑧))) ∈ ℝ)
54	53, 26	readdcld 11163	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ∈ ℝ)
55		abslogle 26585	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
56	31, 32, 55	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
57		df-neg 11369	. . . . . . . . . . . . . . . 16 ⊢ -(log‘𝑅) = (0 − (log‘𝑅))
58		log1 26552	. . . . . . . . . . . . . . . . 17 ⊢ (log‘1) = 0
59	58	oveq1i 7368	. . . . . . . . . . . . . . . 16 ⊢ ((log‘1) − (log‘𝑅)) = (0 − (log‘𝑅))
60	57, 59	eqtr4i 2761	. . . . . . . . . . . . . . 15 ⊢ -(log‘𝑅) = ((log‘1) − (log‘𝑅))
61		1rp 12911	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
62		relogdiv 26560	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑅 ∈ ℝ⁺) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
63	61, 24, 62	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
64	60, 63	eqtr4id 2789	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑅) = (log‘(1 / 𝑅)))
65		oveq2 7366	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝑧 + 𝑘) = (𝑧 + 0))
66	65	fveq2d 6837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (abs‘(𝑧 + 𝑘)) = (abs‘(𝑧 + 0)))
67	66	breq2d 5109	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → ((1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
68		fveq2 6833	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (abs‘𝑥) = (abs‘𝑧))
69	68	breq1d 5107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑧) ≤ 𝑅))
70		fvoveq1 7381	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑧 + 𝑘)))
71	70	breq2d 5109	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
72	71	ralbidv 3158	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
73	69, 72	anbi12d 633	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
74	73, 2	elrab2 3648	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑈 ↔ (𝑧 ∈ ℂ ∧ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
75	74	simprbi 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑈 → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
76	75	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
77	76	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))
78		0nn0 12418	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ₀
79	78	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 0 ∈ ℕ₀)
80	67, 77, 79	rspcdva 3576	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0)))
81	31	addridd 11335	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (𝑧 + 0) = 𝑧)
82	81	fveq2d 6837	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(𝑧 + 0)) = (abs‘𝑧))
83	80, 82	breqtrd 5123	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ≤ (abs‘𝑧))
84	24	rpreccld 12961	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ∈ ℝ⁺)
85	84, 50	logled 26594	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((1 / 𝑅) ≤ (abs‘𝑧) ↔ (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧))))
86	83, 85	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧)))
87	64, 86	eqbrtrd 5119	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑅) ≤ (log‘(abs‘𝑧)))
88	76	simpld 494	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘𝑧) ≤ 𝑅)
89	50, 24	logled 26594	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘𝑧) ≤ 𝑅 ↔ (log‘(abs‘𝑧)) ≤ (log‘𝑅)))
90	88, 89	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ≤ (log‘𝑅))
91	51, 25	absled 15358	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅) ↔ (-(log‘𝑅) ≤ (log‘(abs‘𝑧)) ∧ (log‘(abs‘𝑧)) ≤ (log‘𝑅))))
92	87, 90, 91	mpbir2and 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅))
93	53, 25, 26, 92	leadd1dd 11753	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ≤ ((log‘𝑅) + π))
94	39, 54, 27, 56, 93	letrd 11292	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ≤ ((log‘𝑅) + π))
95	39, 27, 35, 94	leadd1dd 11753	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
96	22, 40, 36, 49, 95	letrd 11292	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
97	96	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
98	35	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
99		simpllr 776	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → 𝑦 ∈ ℝ)
100	27	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → ((log‘𝑅) + π) ∈ ℝ)
101		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)
102	98, 99, 100, 101	leadd2dd 11754	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ≤ (((log‘𝑅) + π) + 𝑦))
103	23, 37, 38, 97, 102	letrd 11292	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
104	103	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
105	104	ralimdva 3147	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑧 ∈ 𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
106	105	impr 454	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧 ∈ 𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
107		brralrspcev 5157	. . 3 ⊢ (((((log‘𝑅) + π) + 𝑦) ∈ ℝ ∧ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)) → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
108	18, 106, 107	syl2anc 585	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧 ∈ 𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
109	9, 108	rexlimddv 3142	1 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2931 ∀wral 3050 ∃wrex 3059 {crab 3398 ∖ cdif 3897 ⊆ wss 3900 ifcif 4478 class class class wbr 5097 ↦ cmpt 5178 dom cdm 5623 ‘cfv 6491 (class class class)co 7358 ∘_f cof 7620 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 ≤ cle 11169 − cmin 11366 -cneg 11367 / cdiv 11796 ℕcn 12147 2c2 12202 ℕ₀cn0 12403 ℤcz 12490 ℝ⁺crp 12907 seqcseq 13926 ↑cexp 13986 abscabs 15159 πcpi 15991 ⇝_𝑢culm 26343 logclog 26521 log Γclgam 26984

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 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-dju 9815 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-ioc 13268 df-ico 13269 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-tan 15996 df-pi 15997 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19248 df-cmn 19713 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cld 22965 df-ntr 22966 df-cls 22967 df-nei 23044 df-lp 23082 df-perf 23083 df-cn 23173 df-cnp 23174 df-haus 23261 df-cmp 23333 df-tx 23508 df-hmeo 23701 df-fil 23792 df-fm 23884 df-flim 23885 df-flf 23886 df-xms 24266 df-ms 24267 df-tms 24268 df-cncf 24829 df-limc 25825 df-dv 25826 df-ulm 26344 df-log 26523 df-cxp 26524 df-lgam 26987

This theorem is referenced by: (None)

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