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Theorem lgambdd 26530

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 26529	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))
5	4	simprd 496	. . 3 ⊢ (𝜑 → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
6		eqid 2732	. . . . 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 26527	. . . 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 496	. . 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 13010	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
11	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑅 ∈ ℝ⁺)
12	11	relogcld 26122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (log‘𝑅) ∈ ℝ)
13		pire 25959	. . . . . . 7 ⊢ π ∈ ℝ
14	13	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → π ∈ ℝ)
15	12, 14	readdcld 11239	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((log‘𝑅) + π) ∈ ℝ)
16		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
17	15, 16	readdcld 11239	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
18	17	adantrr 715	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧 ∈ 𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
19	4	simpld 495	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
20	19	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
21	20	r19.21bi 3248	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) ∈ ℂ)
22	21	abscld 15379	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
23	22	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
24	11	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑅 ∈ ℝ⁺)
25	24	relogcld 26122	. . . . . . . . . 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 11239	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log‘𝑅) + π) ∈ ℝ)
28	1, 2	lgamgulmlem1 26522	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
29	28	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
30	29	sselda 3981	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
31	30	eldifad 3959	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℂ)
32	30	dmgmn0 26519	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ 0)
33	31, 32	logcld 26070	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘𝑧) ∈ ℂ)
34	21, 33	addcld 11229	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) ∈ ℂ)
35	34	abscld 15379	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
36	27, 35	readdcld 11239	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
37	36	adantr 481	. . . . . . 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 724	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
39	33	abscld 15379	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ∈ ℝ)
40	39, 35	readdcld 11239	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
41	33	negcld 11554	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑧) ∈ ℂ)
42	21, 41	abs2difd 15400	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) − -(log‘𝑧))))
43	33	absnegd 15392	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘-(log‘𝑧)) = (abs‘(log‘𝑧)))
44	43	oveq2d 7421	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) = ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))))
45	21, 33	subnegd 11574	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) − -(log‘𝑧)) = ((log Γ‘𝑧) + (log‘𝑧)))
46	45	fveq2d 6892	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘((log Γ‘𝑧) − -(log‘𝑧))) = (abs‘((log Γ‘𝑧) + (log‘𝑧))))
47	42, 44, 46	3brtr3d 5178	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))))
48	22, 39, 35	lesubadd2d 11809	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log Γ‘𝑧)) ≤ ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
50	31, 32	absrpcld 15391	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘𝑧) ∈ ℝ⁺)
51	50	relogcld 26122	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ∈ ℝ)
52	51	recnd 11238	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ∈ ℂ)
53	52	abscld 15379	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘(abs‘𝑧))) ∈ ℝ)
54	53, 26	readdcld 11239	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ∈ ℝ)
55		abslogle 26117	. . . . . . . . . . . 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 11443	. . . . . . . . . . . . . . . 16 ⊢ -(log‘𝑅) = (0 − (log‘𝑅))
58		log1 26085	. . . . . . . . . . . . . . . . 17 ⊢ (log‘1) = 0
59	58	oveq1i 7415	. . . . . . . . . . . . . . . 16 ⊢ ((log‘1) − (log‘𝑅)) = (0 − (log‘𝑅))
60	57, 59	eqtr4i 2763	. . . . . . . . . . . . . . 15 ⊢ -(log‘𝑅) = ((log‘1) − (log‘𝑅))
61		1rp 12974	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
62		relogdiv 26092	. . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑅) = (log‘(1 / 𝑅)))
65		oveq2 7413	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝑧 + 𝑘) = (𝑧 + 0))
66	65	fveq2d 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (abs‘(𝑧 + 𝑘)) = (abs‘(𝑧 + 0)))
67	66	breq2d 5159	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → ((1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
68		fveq2 6888	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (abs‘𝑥) = (abs‘𝑧))
69	68	breq1d 5157	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑧) ≤ 𝑅))
70		fvoveq1 7428	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑧 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑧 + 𝑘)))
71	70	breq2d 5159	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
72	71	ralbidv 3177	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
73	69, 72	anbi12d 631	. . . . . . . . . . . . . . . . . . . . 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 3685	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑈 ↔ (𝑧 ∈ ℂ ∧ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
75	74	simprbi 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑈 → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
76	75	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
77	76	simprd 496	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))
78		0nn0 12483	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ₀
79	78	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → 0 ∈ ℕ₀)
80	67, 77, 79	rspcdva 3613	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0)))
81	31	addridd 11410	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (𝑧 + 0) = 𝑧)
82	81	fveq2d 6892	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(𝑧 + 0)) = (abs‘𝑧))
83	80, 82	breqtrd 5173	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ≤ (abs‘𝑧))
84	24	rpreccld 13022	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (1 / 𝑅) ∈ ℝ⁺)
85	84, 50	logled 26126	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((1 / 𝑅) ≤ (abs‘𝑧) ↔ (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧))))
86	83, 85	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧)))
87	64, 86	eqbrtrd 5169	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → -(log‘𝑅) ≤ (log‘(abs‘𝑧)))
88	76	simpld 495	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘𝑧) ≤ 𝑅)
89	50, 24	logled 26126	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘𝑧) ≤ 𝑅 ↔ (log‘(abs‘𝑧)) ≤ (log‘𝑅)))
90	88, 89	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (log‘(abs‘𝑧)) ≤ (log‘𝑅))
91	51, 25	absled 15373	. . . . . . . . . . . . 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 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅))
93	53, 25, 26, 92	leadd1dd 11824	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ≤ ((log‘𝑅) + π))
94	39, 54, 27, 56, 93	letrd 11367	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log‘𝑧)) ≤ ((log‘𝑅) + π))
95	39, 27, 35, 94	leadd1dd 11824	. . . . . . . . 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 11367	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
97	96	adantr 481	. . . . . . 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 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
99		simpllr 774	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → 𝑦 ∈ ℝ)
100	27	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → ((log‘𝑅) + π) ∈ ℝ)
101		simpr 485	. . . . . . . 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 11825	. . . . . . 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 11367	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
104	103	ex 413	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝑈) → ((abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
105	104	ralimdva 3167	. . . 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 455	. . 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 5207	. . 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 3161	1 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ∖ cdif 3944 ⊆ wss 3947 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 ≤ cle 11245 − cmin 11440 -cneg 11441 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℝ⁺crp 12970 seqcseq 13962 ↑cexp 14023 abscabs 15177 πcpi 16006 ⇝_𝑢culm 25879 logclog 26054 log Γclgam 26509

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-tan 16011 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-ulm 25880 df-log 26056 df-cxp 26057 df-lgam 26512

This theorem is referenced by: (None)

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