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Theorem lgambdd 27095
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 27094 . . . 4 (𝜑 → (∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘f + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))
54simprd 495 . . 3 (𝜑 → seq1( ∘f + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((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))))
71, 2, 3, 6lgamgulmlem6 27092 . . . 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 495 . . 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 13073 . . . . . . . 8 (𝜑𝑅 ∈ ℝ+)
1110adantr 480 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → 𝑅 ∈ ℝ+)
1211relogcld 26680 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (log‘𝑅) ∈ ℝ)
13 pire 26515 . . . . . . 7 π ∈ ℝ
1413a1i 11 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → π ∈ ℝ)
1512, 14readdcld 11288 . . . . 5 ((𝜑𝑦 ∈ ℝ) → ((log‘𝑅) + π) ∈ ℝ)
16 simpr 484 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
1715, 16readdcld 11288 . . . 4 ((𝜑𝑦 ∈ ℝ) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
1817adantrr 717 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
194simpld 494 . . . . . . . . . . 11 (𝜑 → ∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ)
2019adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → ∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ)
2120r19.21bi 3249 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log Γ‘𝑧) ∈ ℂ)
2221abscld 15472 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
2322adantr 480 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
2411adantr 480 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑅 ∈ ℝ+)
2524relogcld 26680 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑅) ∈ ℝ)
2613a1i 11 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → π ∈ ℝ)
2725, 26readdcld 11288 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log‘𝑅) + π) ∈ ℝ)
281, 2lgamgulmlem1 27087 . . . . . . . . . . . . . . 15 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2928adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ℝ) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
3029sselda 3995 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
3130eldifad 3975 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ ℂ)
3230dmgmn0 27084 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ≠ 0)
3331, 32logcld 26627 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑧) ∈ ℂ)
3421, 33addcld 11278 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) ∈ ℂ)
3534abscld 15472 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
3627, 35readdcld 11288 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
3736adantr 480 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
3817ad2antrr 726 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
3933abscld 15472 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ∈ ℝ)
4039, 35readdcld 11288 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
4133negcld 11605 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑧) ∈ ℂ)
4221, 41abs2difd 15493 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) − -(log‘𝑧))))
4333absnegd 15485 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘-(log‘𝑧)) = (abs‘(log‘𝑧)))
4443oveq2d 7447 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) = ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))))
4521, 33subnegd 11625 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) − -(log‘𝑧)) = ((log Γ‘𝑧) + (log‘𝑧)))
4645fveq2d 6911 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) − -(log‘𝑧))) = (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4742, 44, 463brtr3d 5179 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4822, 39, 35lesubadd2d 11860 . . . . . . . . . 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 232 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ≤ ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
5031, 32absrpcld 15484 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ∈ ℝ+)
5150relogcld 26680 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℝ)
5251recnd 11287 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℂ)
5352abscld 15472 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘(abs‘𝑧))) ∈ ℝ)
5453, 26readdcld 11288 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ∈ ℝ)
55 abslogle 26675 . . . . . . . . . . . 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 11493 . . . . . . . . . . . . . . . 16 -(log‘𝑅) = (0 − (log‘𝑅))
58 log1 26642 . . . . . . . . . . . . . . . . 17 (log‘1) = 0
5958oveq1i 7441 . . . . . . . . . . . . . . . 16 ((log‘1) − (log‘𝑅)) = (0 − (log‘𝑅))
6057, 59eqtr4i 2766 . . . . . . . . . . . . . . 15 -(log‘𝑅) = ((log‘1) − (log‘𝑅))
61 1rp 13036 . . . . . . . . . . . . . . . 16 1 ∈ ℝ+
62 relogdiv 26650 . . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) = (log‘(1 / 𝑅)))
65 oveq2 7439 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝑧 + 𝑘) = (𝑧 + 0))
6665fveq2d 6911 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (abs‘(𝑧 + 𝑘)) = (abs‘(𝑧 + 0)))
6766breq2d 5160 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → ((1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
68 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → (abs‘𝑥) = (abs‘𝑧))
6968breq1d 5158 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑧) ≤ 𝑅))
70 fvoveq1 7454 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑧 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑧 + 𝑘)))
7170breq2d 5160 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7271ralbidv 3176 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7369, 72anbi12d 632 . . . . . . . . . . . . . . . . . . . . 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 3698 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑈 ↔ (𝑧 ∈ ℂ ∧ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
7574simprbi 496 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑈 → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7675adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7776simprd 495 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))
78 0nn0 12539 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
7978a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 0 ∈ ℕ0)
8067, 77, 79rspcdva 3623 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0)))
8131addridd 11459 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (𝑧 + 0) = 𝑧)
8281fveq2d 6911 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(𝑧 + 0)) = (abs‘𝑧))
8380, 82breqtrd 5174 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘𝑧))
8424rpreccld 13085 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ∈ ℝ+)
8584, 50logled 26684 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((1 / 𝑅) ≤ (abs‘𝑧) ↔ (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧))))
8683, 85mpbid 232 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧)))
8764, 86eqbrtrd 5170 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) ≤ (log‘(abs‘𝑧)))
8876simpld 494 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ≤ 𝑅)
8950, 24logled 26684 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘𝑧) ≤ 𝑅 ↔ (log‘(abs‘𝑧)) ≤ (log‘𝑅)))
9088, 89mpbid 232 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ≤ (log‘𝑅))
9151, 25absled 15466 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅) ↔ (-(log‘𝑅) ≤ (log‘(abs‘𝑧)) ∧ (log‘(abs‘𝑧)) ≤ (log‘𝑅))))
9287, 90, 91mpbir2and 713 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅))
9353, 25, 26, 92leadd1dd 11875 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ≤ ((log‘𝑅) + π))
9439, 54, 27, 56, 93letrd 11416 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ≤ ((log‘𝑅) + π))
9539, 27, 35, 94leadd1dd 11875 . . . . . . . . 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 11416 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
9796adantr 480 . . . . . . 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 480 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
99 simpllr 776 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → 𝑦 ∈ ℝ)
10027adantr 480 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → ((log‘𝑅) + π) ∈ ℝ)
101 simpr 484 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)
10298, 99, 100, 101leadd2dd 11876 . . . . . . 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 11416 . . . . . 6 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
104103ex 412 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
105104ralimdva 3165 . . . 4 ((𝜑𝑦 ∈ ℝ) → (∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
106105impr 454 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
107 brralrspcev 5208 . . 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 3159 1 (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  cdif 3960  wss 3963  ifcif 4531   class class class wbr 5148  cmpt 5231  dom cdm 5689  cfv 6563  (class class class)co 7431  f cof 7695  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  cle 11294  cmin 11490  -cneg 11491   / cdiv 11918  cn 12264  2c2 12319  0cn0 12524  cz 12611  +crp 13032  seqcseq 14039  cexp 14099  abscabs 15270  πcpi 16099  𝑢culm 26434  logclog 26611  log Γclgam 27074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ioc 13389  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-shft 15103  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-limsup 15504  df-clim 15521  df-rlim 15522  df-sum 15720  df-ef 16100  df-sin 16102  df-cos 16103  df-tan 16104  df-pi 16105  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917  df-ulm 26435  df-log 26613  df-cxp 26614  df-lgam 27077
This theorem is referenced by: (None)
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