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Theorem lgambdd 26974

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 26973	. . . 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 2731	. . . . 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 26971	. . . 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 12932	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑅 ∈ ℝ⁺)
12	11	relogcld 26559	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (log‘𝑅) ∈ ℝ)
13		pire 26393	. . . . . . 7 ⊢ π ∈ ℝ
14	13	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → π ∈ ℝ)
15	12, 14	readdcld 11141	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((log‘𝑅) + π) ∈ ℝ)
16		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
17	15, 16	readdcld 11141	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
18	17	adantrr 717	. . 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 3224	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) ∈ ℂ)
22	21	abscld 15346	. . . . . . . 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 26559	. . . . . . . . . 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 11141	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log‘𝑅) + π) ∈ ℝ)
28	1, 2	lgamgulmlem1 26966	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
29	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
30	29	sselda 3929	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
31	30	eldifad 3909	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℂ)
32	30	dmgmn0 26963	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ 0)
33	31, 32	logcld 26506	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘𝑧) ∈ ℂ)
34	21, 33	addcld 11131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) ∈ ℂ)
35	34	abscld 15346	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
36	27, 35	readdcld 11141	. . . . . . . 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 726	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
39	33	abscld 15346	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ∈ ℝ)
40	39, 35	readdcld 11141	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
41	33	negcld 11459	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑧) ∈ ℂ)
42	21, 41	abs2difd 15367	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) − -(log‘𝑧))))
43	33	absnegd 15359	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘-(log‘𝑧)) = (abs‘(log‘𝑧)))
44	43	oveq2d 7362	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) = ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))))
45	21, 33	subnegd 11479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) − -(log‘𝑧)) = ((log Γ‘𝑧) + (log‘𝑧)))
46	45	fveq2d 6826	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘((log Γ‘𝑧) − -(log‘𝑧))) = (abs‘((log Γ‘𝑧) + (log‘𝑧))))
47	42, 44, 46	3brtr3d 5120	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))))
48	22, 39, 35	lesubadd2d 11716	. . . . . . . . . 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 15358	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘𝑧) ∈ ℝ⁺)
51	50	relogcld 26559	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ∈ ℝ)
52	51	recnd 11140	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ∈ ℂ)
53	52	abscld 15346	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘(abs‘𝑧))) ∈ ℝ)
54	53, 26	readdcld 11141	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ∈ ℝ)
55		abslogle 26554	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
56	31, 32, 55	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
57		df-neg 11347	. . . . . . . . . . . . . . . 16 ⊢ -(log‘𝑅) = (0 − (log‘𝑅))
58		log1 26521	. . . . . . . . . . . . . . . . 17 ⊢ (log‘1) = 0
59	58	oveq1i 7356	. . . . . . . . . . . . . . . 16 ⊢ ((log‘1) − (log‘𝑅)) = (0 − (log‘𝑅))
60	57, 59	eqtr4i 2757	. . . . . . . . . . . . . . 15 ⊢ -(log‘𝑅) = ((log‘1) − (log‘𝑅))
61		1rp 12894	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
62		relogdiv 26529	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑅 ∈ ℝ⁺) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
63	61, 24, 62	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
64	60, 63	eqtr4id 2785	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑅) = (log‘(1 / 𝑅)))
65		oveq2 7354	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝑧 + 𝑘) = (𝑧 + 0))
66	65	fveq2d 6826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (abs‘(𝑧 + 𝑘)) = (abs‘(𝑧 + 0)))
67	66	breq2d 5101	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → ((1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
68		fveq2 6822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (abs‘𝑥) = (abs‘𝑧))
69	68	breq1d 5099	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑧) ≤ 𝑅))
70		fvoveq1 7369	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑧 + 𝑘)))
71	70	breq2d 5101	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
72	71	ralbidv 3155	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
73	69, 72	anbi12d 632	. . . . . . . . . . . . . . . . . . . . 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 3645	. . . . . . . . . . . . . . . . . . . 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 12396	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ₀
79	78	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 0 ∈ ℕ₀)
80	67, 77, 79	rspcdva 3573	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0)))
81	31	addridd 11313	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (𝑧 + 0) = 𝑧)
82	81	fveq2d 6826	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(𝑧 + 0)) = (abs‘𝑧))
83	80, 82	breqtrd 5115	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ≤ (abs‘𝑧))
84	24	rpreccld 12944	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ∈ ℝ⁺)
85	84, 50	logled 26563	. . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . 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 26563	. . . . . . . . . . . . . 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 15340	. . . . . . . . . . . . 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 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅))
93	53, 25, 26, 92	leadd1dd 11731	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ≤ ((log‘𝑅) + π))
94	39, 54, 27, 56, 93	letrd 11270	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ≤ ((log‘𝑅) + π))
95	39, 27, 35, 94	leadd1dd 11731	. . . . . . . . 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 11270	. . . . . . . 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 775	. . . . . . . 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 11732	. . . . . . 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 11270	. . . . . 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 3144	. . . 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 5149	. . 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 584	. 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 3139	1 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 {crab 3395 ∖ cdif 3894 ⊆ wss 3897 ifcif 4472 class class class wbr 5089 ↦ cmpt 5170 dom cdm 5614 ‘cfv 6481 (class class class)co 7346 ∘_f cof 7608 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 ≤ cle 11147 − cmin 11344 -cneg 11345 / cdiv 11774 ℕcn 12125 2c2 12180 ℕ₀cn0 12381 ℤcz 12468 ℝ⁺crp 12890 seqcseq 13908 ↑cexp 13968 abscabs 15141 πcpi 15973 ⇝_𝑢culm 26312 logclog 26490 log Γclgam 26953

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-tan 15978 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-cmp 23302 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-ulm 26313 df-log 26492 df-cxp 26493 df-lgam 26956

This theorem is referenced by: (None)

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