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Theorem lgambdd 25306

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 25305	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘_𝑓 + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))
5	4	simprd 488	. . 3 ⊢ (𝜑 → seq1( ∘_𝑓 + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
6		eqid 2772	. . . . 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 25303	. . . 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 488	. . 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 12239	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
11	10	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑅 ∈ ℝ⁺)
12	11	relogcld 24897	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (log‘𝑅) ∈ ℝ)
13		pire 24737	. . . . . . 7 ⊢ π ∈ ℝ
14	13	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → π ∈ ℝ)
15	12, 14	readdcld 10461	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((log‘𝑅) + π) ∈ ℝ)
16		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
17	15, 16	readdcld 10461	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
18	17	adantrr 704	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧 ∈ 𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
19	4	simpld 487	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
20	19	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
21	20	r19.21bi 3152	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) ∈ ℂ)
22	21	abscld 14647	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
23	22	adantr 473	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
24	11	adantr 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑅 ∈ ℝ⁺)
25	24	relogcld 24897	. . . . . . . . . 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 10461	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log‘𝑅) + π) ∈ ℝ)
28	1, 2	lgamgulmlem1 25298	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
29	28	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
30	29	sselda 3854	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
31	30	eldifad 3837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℂ)
32	30	dmgmn0 25295	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ 0)
33	31, 32	logcld 24845	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘𝑧) ∈ ℂ)
34	21, 33	addcld 10451	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) ∈ ℂ)
35	34	abscld 14647	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
36	27, 35	readdcld 10461	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
37	36	adantr 473	. . . . . . 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 713	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
39	33	abscld 14647	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ∈ ℝ)
40	39, 35	readdcld 10461	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
41	33	negcld 10777	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑧) ∈ ℂ)
42	21, 41	abs2difd 14668	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) − -(log‘𝑧))))
43	33	absnegd 14660	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘-(log‘𝑧)) = (abs‘(log‘𝑧)))
44	43	oveq2d 6986	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) = ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))))
45	21, 33	subnegd 10797	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) − -(log‘𝑧)) = ((log Γ‘𝑧) + (log‘𝑧)))
46	45	fveq2d 6497	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘((log Γ‘𝑧) − -(log‘𝑧))) = (abs‘((log Γ‘𝑧) + (log‘𝑧))))
47	42, 44, 46	3brtr3d 4954	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))))
48	22, 39, 35	lesubadd2d 11032	. . . . . . . . . 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 224	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log Γ‘𝑧)) ≤ ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
50	31, 32	absrpcld 14659	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘𝑧) ∈ ℝ⁺)
51	50	relogcld 24897	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ∈ ℝ)
52	51	recnd 10460	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ∈ ℂ)
53	52	abscld 14647	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘(abs‘𝑧))) ∈ ℝ)
54	53, 26	readdcld 10461	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ∈ ℝ)
55		abslogle 24892	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
56	31, 32, 55	syl2anc 576	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
57		1rp 12201	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
58		relogdiv 24867	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑅 ∈ ℝ⁺) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
59	57, 24, 58	sylancr 578	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
60		df-neg 10665	. . . . . . . . . . . . . . . 16 ⊢ -(log‘𝑅) = (0 − (log‘𝑅))
61		log1 24860	. . . . . . . . . . . . . . . . 17 ⊢ (log‘1) = 0
62	61	oveq1i 6980	. . . . . . . . . . . . . . . 16 ⊢ ((log‘1) − (log‘𝑅)) = (0 − (log‘𝑅))
63	60, 62	eqtr4i 2799	. . . . . . . . . . . . . . 15 ⊢ -(log‘𝑅) = ((log‘1) − (log‘𝑅))
64	59, 63	syl6reqr 2827	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑅) = (log‘(1 / 𝑅)))
65		oveq2 6978	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝑧 + 𝑘) = (𝑧 + 0))
66	65	fveq2d 6497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (abs‘(𝑧 + 𝑘)) = (abs‘(𝑧 + 0)))
67	66	breq2d 4935	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → ((1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
68		fveq2 6493	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (abs‘𝑥) = (abs‘𝑧))
69	68	breq1d 4933	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑧) ≤ 𝑅))
70		fvoveq1 6993	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑧 + 𝑘)))
71	70	breq2d 4935	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
72	71	ralbidv 3141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
73	69, 72	anbi12d 621	. . . . . . . . . . . . . . . . . . . . 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 3593	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑈 ↔ (𝑧 ∈ ℂ ∧ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
75	74	simprbi 489	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑈 → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
76	75	adantl 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
77	76	simprd 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))
78		0nn0 11717	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ₀
79	78	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 0 ∈ ℕ₀)
80	67, 77, 79	rspcdva 3535	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0)))
81	31	addid1d 10632	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (𝑧 + 0) = 𝑧)
82	81	fveq2d 6497	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(𝑧 + 0)) = (abs‘𝑧))
83	80, 82	breqtrd 4949	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ≤ (abs‘𝑧))
84	24	rpreccld 12251	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ∈ ℝ⁺)
85	84, 50	logled 24901	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((1 / 𝑅) ≤ (abs‘𝑧) ↔ (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧))))
86	83, 85	mpbid 224	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧)))
87	64, 86	eqbrtrd 4945	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑅) ≤ (log‘(abs‘𝑧)))
88	76	simpld 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘𝑧) ≤ 𝑅)
89	50, 24	logled 24901	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘𝑧) ≤ 𝑅 ↔ (log‘(abs‘𝑧)) ≤ (log‘𝑅)))
90	88, 89	mpbid 224	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ≤ (log‘𝑅))
91	51, 25	absled 14641	. . . . . . . . . . . . 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 700	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅))
93	53, 25, 26, 92	leadd1dd 11047	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ≤ ((log‘𝑅) + π))
94	39, 54, 27, 56, 93	letrd 10589	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ≤ ((log‘𝑅) + π))
95	39, 27, 35, 94	leadd1dd 11047	. . . . . . . . 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 10589	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
97	96	adantr 473	. . . . . . 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 473	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
99		simpllr 763	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → 𝑦 ∈ ℝ)
100	27	adantr 473	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → ((log‘𝑅) + π) ∈ ℝ)
101		simpr 477	. . . . . . . 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 11048	. . . . . . 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 10589	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
104	103	ex 405	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
105	104	ralimdva 3121	. . . 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 447	. . 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 4983	. . 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 576	. 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 3230	1 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ∀wral 3082 ∃wrex 3083 {crab 3086 ∖ cdif 3822 ⊆ wss 3825 ifcif 4344 class class class wbr 4923 ↦ cmpt 5002 dom cdm 5400 ‘cfv 6182 (class class class)co 6970 ∘_𝑓 cof 7219 ℂcc 10325 ℝcr 10326 0cc0 10327 1c1 10328 + caddc 10330 · cmul 10332 ≤ cle 10467 − cmin 10662 -cneg 10663 / cdiv 11090 ℕcn 11431 2c2 11488 ℕ₀cn0 11700 ℤcz 11786 ℝ⁺crp 12197 seqcseq 13177 ↑cexp 13237 abscabs 14444 πcpi 15270 ⇝_𝑢culm 24657 logclog 24829 log Γclgam 25285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 ax-addf 10406 ax-mulf 10407

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-fi 8662 df-sup 8693 df-inf 8694 df-oi 8761 df-dju 9116 df-card 9154 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-ioo 12551 df-ioc 12552 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-fl 12970 df-mod 13046 df-seq 13178 df-exp 13238 df-fac 13442 df-bc 13471 df-hash 13499 df-shft 14277 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-limsup 14679 df-clim 14696 df-rlim 14697 df-sum 14894 df-ef 15271 df-sin 15273 df-cos 15274 df-tan 15275 df-pi 15276 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-hom 16435 df-cco 16436 df-rest 16542 df-topn 16543 df-0g 16561 df-gsum 16562 df-topgen 16563 df-pt 16564 df-prds 16567 df-xrs 16621 df-qtop 16626 df-imas 16627 df-xps 16629 df-mre 16705 df-mrc 16706 df-acs 16708 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-mulg 18002 df-cntz 18208 df-cmn 18658 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-fbas 20234 df-fg 20235 df-cnfld 20238 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cld 21321 df-ntr 21322 df-cls 21323 df-nei 21400 df-lp 21438 df-perf 21439 df-cn 21529 df-cnp 21530 df-haus 21617 df-cmp 21689 df-tx 21864 df-hmeo 22057 df-fil 22148 df-fm 22240 df-flim 22241 df-flf 22242 df-xms 22623 df-ms 22624 df-tms 22625 df-cncf 23179 df-limc 24157 df-dv 24158 df-ulm 24658 df-log 24831 df-cxp 24832 df-lgam 25288

This theorem is referenced by: (None)

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