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Theorem lgambdd 26184
Description: The log-Gamma function is bounded on the region 𝑈. (Contributed by Mario Carneiro, 9-Jul-2017.)
Hypotheses
Ref Expression
lgamgulm.r (𝜑𝑅 ∈ ℕ)
lgamgulm.u 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}
lgamgulm.g 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
Assertion
Ref Expression
lgambdd (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (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.
StepHypRef Expression
1 lgamgulm.r . . . . 5 (𝜑𝑅 ∈ 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 ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
41, 2, 3lgamgulm2 26183 . . . 4 (𝜑 → (∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘f + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))
54simprd 496 . . 3 (𝜑 → seq1( ∘f + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
6 eqid 2740 . . . . 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))))
71, 2, 3, 6lgamgulmlem6 26181 . . . 4 (𝜑 → (seq1( ∘f + , 𝐺) ∈ dom (⇝𝑢𝑈) ∧ (seq1( ∘f + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))) → ∃𝑦 ∈ ℝ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)))
87simprd 496 . . 3 (𝜑 → (seq1( ∘f + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))) → ∃𝑦 ∈ ℝ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦))
95, 8mpd 15 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)
101nnrpd 12769 . . . . . . . 8 (𝜑𝑅 ∈ ℝ+)
1110adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → 𝑅 ∈ ℝ+)
1211relogcld 25776 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (log‘𝑅) ∈ ℝ)
13 pire 25613 . . . . . . 7 π ∈ ℝ
1413a1i 11 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → π ∈ ℝ)
1512, 14readdcld 11005 . . . . 5 ((𝜑𝑦 ∈ ℝ) → ((log‘𝑅) + π) ∈ ℝ)
16 simpr 485 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
1715, 16readdcld 11005 . . . 4 ((𝜑𝑦 ∈ ℝ) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
1817adantrr 714 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
194simpld 495 . . . . . . . . . . 11 (𝜑 → ∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ)
2019adantr 481 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → ∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ)
2120r19.21bi 3135 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log Γ‘𝑧) ∈ ℂ)
2221abscld 15146 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
2322adantr 481 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
2411adantr 481 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑅 ∈ ℝ+)
2524relogcld 25776 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑅) ∈ ℝ)
2613a1i 11 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → π ∈ ℝ)
2725, 26readdcld 11005 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log‘𝑅) + π) ∈ ℝ)
281, 2lgamgulmlem1 26176 . . . . . . . . . . . . . . 15 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2928adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ℝ) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
3029sselda 3926 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
3130eldifad 3904 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ ℂ)
3230dmgmn0 26173 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ≠ 0)
3331, 32logcld 25724 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑧) ∈ ℂ)
3421, 33addcld 10995 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) ∈ ℂ)
3534abscld 15146 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
3627, 35readdcld 11005 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
3736adantr 481 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
3817ad2antrr 723 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
3933abscld 15146 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ∈ ℝ)
4039, 35readdcld 11005 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
4133negcld 11319 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑧) ∈ ℂ)
4221, 41abs2difd 15167 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) − -(log‘𝑧))))
4333absnegd 15159 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘-(log‘𝑧)) = (abs‘(log‘𝑧)))
4443oveq2d 7287 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) = ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))))
4521, 33subnegd 11339 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) − -(log‘𝑧)) = ((log Γ‘𝑧) + (log‘𝑧)))
4645fveq2d 6775 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) − -(log‘𝑧))) = (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4742, 44, 463brtr3d 5110 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4822, 39, 35lesubadd2d 11574 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (((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))))))
4947, 48mpbid 231 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ≤ ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
5031, 32absrpcld 15158 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ∈ ℝ+)
5150relogcld 25776 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℝ)
5251recnd 11004 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℂ)
5352abscld 15146 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘(abs‘𝑧))) ∈ ℝ)
5453, 26readdcld 11005 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ∈ ℝ)
55 abslogle 25771 . . . . . . . . . . . 12 ((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
5631, 32, 55syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
57 df-neg 11208 . . . . . . . . . . . . . . . 16 -(log‘𝑅) = (0 − (log‘𝑅))
58 log1 25739 . . . . . . . . . . . . . . . . 17 (log‘1) = 0
5958oveq1i 7281 . . . . . . . . . . . . . . . 16 ((log‘1) − (log‘𝑅)) = (0 − (log‘𝑅))
6057, 59eqtr4i 2771 . . . . . . . . . . . . . . 15 -(log‘𝑅) = ((log‘1) − (log‘𝑅))
61 1rp 12733 . . . . . . . . . . . . . . . 16 1 ∈ ℝ+
62 relogdiv 25746 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ+𝑅 ∈ ℝ+) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
6361, 24, 62sylancr 587 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
6460, 63eqtr4id 2799 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) = (log‘(1 / 𝑅)))
65 oveq2 7279 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝑧 + 𝑘) = (𝑧 + 0))
6665fveq2d 6775 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (abs‘(𝑧 + 𝑘)) = (abs‘(𝑧 + 0)))
6766breq2d 5091 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → ((1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
68 fveq2 6771 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → (abs‘𝑥) = (abs‘𝑧))
6968breq1d 5089 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑧) ≤ 𝑅))
70 fvoveq1 7294 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑧 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑧 + 𝑘)))
7170breq2d 5091 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7271ralbidv 3123 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7369, 72anbi12d 631 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑧 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
7473, 2elrab2 3629 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑈 ↔ (𝑧 ∈ ℂ ∧ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
7574simprbi 497 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑈 → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7675adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7776simprd 496 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))
78 0nn0 12248 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
7978a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 0 ∈ ℕ0)
8067, 77, 79rspcdva 3563 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0)))
8131addid1d 11175 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (𝑧 + 0) = 𝑧)
8281fveq2d 6775 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(𝑧 + 0)) = (abs‘𝑧))
8380, 82breqtrd 5105 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘𝑧))
8424rpreccld 12781 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ∈ ℝ+)
8584, 50logled 25780 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((1 / 𝑅) ≤ (abs‘𝑧) ↔ (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧))))
8683, 85mpbid 231 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧)))
8764, 86eqbrtrd 5101 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) ≤ (log‘(abs‘𝑧)))
8876simpld 495 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ≤ 𝑅)
8950, 24logled 25780 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘𝑧) ≤ 𝑅 ↔ (log‘(abs‘𝑧)) ≤ (log‘𝑅)))
9088, 89mpbid 231 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ≤ (log‘𝑅))
9151, 25absled 15140 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅) ↔ (-(log‘𝑅) ≤ (log‘(abs‘𝑧)) ∧ (log‘(abs‘𝑧)) ≤ (log‘𝑅))))
9287, 90, 91mpbir2and 710 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅))
9353, 25, 26, 92leadd1dd 11589 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ≤ ((log‘𝑅) + π))
9439, 54, 27, 56, 93letrd 11132 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ≤ ((log‘𝑅) + π))
9539, 27, 35, 94leadd1dd 11589 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
9622, 40, 36, 49, 95letrd 11132 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
9796adantr 481 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
9835adantr 481 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
99 simpllr 773 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → 𝑦 ∈ ℝ)
10027adantr 481 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → ((log‘𝑅) + π) ∈ ℝ)
101 simpr 485 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)
10298, 99, 100, 101leadd2dd 11590 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ≤ (((log‘𝑅) + π) + 𝑦))
10323, 37, 38, 97, 102letrd 11132 . . . . . 6 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
104103ex 413 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
105104ralimdva 3105 . . . 4 ((𝜑𝑦 ∈ ℝ) → (∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
106105impr 455 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
107 brralrspcev 5139 . . 3 (((((log‘𝑅) + π) + 𝑦) ∈ ℝ ∧ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)) → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
10818, 106, 107syl2anc 584 . 2 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
1099, 108rexlimddv 3222 1 (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1542  wcel 2110  wne 2945  wral 3066  wrex 3067  {crab 3070  cdif 3889  wss 3892  ifcif 4465   class class class wbr 5079  cmpt 5162  dom cdm 5590  cfv 6432  (class class class)co 7271  f cof 7525  cc 10870  cr 10871  0cc0 10872  1c1 10873   + caddc 10875   · cmul 10877  cle 11011  cmin 11205  -cneg 11206   / cdiv 11632  cn 11973  2c2 12028  0cn0 12233  cz 12319  +crp 12729  seqcseq 13719  cexp 13780  abscabs 14943  πcpi 15774  𝑢culm 25533  logclog 25708  log Γclgam 26163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-inf2 9377  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949  ax-pre-sup 10950  ax-addf 10951  ax-mulf 10952
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-of 7527  df-om 7707  df-1st 7824  df-2nd 7825  df-supp 7969  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-2o 8289  df-oadd 8292  df-er 8481  df-map 8600  df-pm 8601  df-ixp 8669  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-fsupp 9107  df-fi 9148  df-sup 9179  df-inf 9180  df-oi 9247  df-dju 9660  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12437  df-uz 12582  df-q 12688  df-rp 12730  df-xneg 12847  df-xadd 12848  df-xmul 12849  df-ioo 13082  df-ioc 13083  df-ico 13084  df-icc 13085  df-fz 13239  df-fzo 13382  df-fl 13510  df-mod 13588  df-seq 13720  df-exp 13781  df-fac 13986  df-bc 14015  df-hash 14043  df-shft 14776  df-cj 14808  df-re 14809  df-im 14810  df-sqrt 14944  df-abs 14945  df-limsup 15178  df-clim 15195  df-rlim 15196  df-sum 15396  df-ef 15775  df-sin 15777  df-cos 15778  df-tan 15779  df-pi 15780  df-struct 16846  df-sets 16863  df-slot 16881  df-ndx 16893  df-base 16911  df-ress 16940  df-plusg 16973  df-mulr 16974  df-starv 16975  df-sca 16976  df-vsca 16977  df-ip 16978  df-tset 16979  df-ple 16980  df-ds 16982  df-unif 16983  df-hom 16984  df-cco 16985  df-rest 17131  df-topn 17132  df-0g 17150  df-gsum 17151  df-topgen 17152  df-pt 17153  df-prds 17156  df-xrs 17211  df-qtop 17216  df-imas 17217  df-xps 17219  df-mre 17293  df-mrc 17294  df-acs 17296  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-submnd 18429  df-mulg 18699  df-cntz 18921  df-cmn 19386  df-psmet 20587  df-xmet 20588  df-met 20589  df-bl 20590  df-mopn 20591  df-fbas 20592  df-fg 20593  df-cnfld 20596  df-top 22041  df-topon 22058  df-topsp 22080  df-bases 22094  df-cld 22168  df-ntr 22169  df-cls 22170  df-nei 22247  df-lp 22285  df-perf 22286  df-cn 22376  df-cnp 22377  df-haus 22464  df-cmp 22536  df-tx 22711  df-hmeo 22904  df-fil 22995  df-fm 23087  df-flim 23088  df-flf 23089  df-xms 23471  df-ms 23472  df-tms 23473  df-cncf 24039  df-limc 25028  df-dv 25029  df-ulm 25534  df-log 25710  df-cxp 25711  df-lgam 26166
This theorem is referenced by: (None)
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