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Theorem lgambdd 27081
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 27080 . . . 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 2736 . . . . 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 27078 . . . 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 13076 . . . . . . . 8 (𝜑𝑅 ∈ ℝ+)
1110adantr 480 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → 𝑅 ∈ ℝ+)
1211relogcld 26666 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (log‘𝑅) ∈ ℝ)
13 pire 26501 . . . . . . 7 π ∈ ℝ
1413a1i 11 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → π ∈ ℝ)
1512, 14readdcld 11291 . . . . 5 ((𝜑𝑦 ∈ ℝ) → ((log‘𝑅) + π) ∈ ℝ)
16 simpr 484 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
1715, 16readdcld 11291 . . . 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 3250 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log Γ‘𝑧) ∈ ℂ)
2221abscld 15476 . . . . . . . 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 26666 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑅) ∈ ℝ)
2613a1i 11 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → π ∈ ℝ)
2725, 26readdcld 11291 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log‘𝑅) + π) ∈ ℝ)
281, 2lgamgulmlem1 27073 . . . . . . . . . . . . . . 15 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2928adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ℝ) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
3029sselda 3982 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
3130eldifad 3962 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ ℂ)
3230dmgmn0 27070 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ≠ 0)
3331, 32logcld 26613 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑧) ∈ ℂ)
3421, 33addcld 11281 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) ∈ ℂ)
3534abscld 15476 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
3627, 35readdcld 11291 . . . . . . . 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 15476 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ∈ ℝ)
4039, 35readdcld 11291 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
4133negcld 11608 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑧) ∈ ℂ)
4221, 41abs2difd 15497 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) − -(log‘𝑧))))
4333absnegd 15489 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘-(log‘𝑧)) = (abs‘(log‘𝑧)))
4443oveq2d 7448 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) = ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))))
4521, 33subnegd 11628 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) − -(log‘𝑧)) = ((log Γ‘𝑧) + (log‘𝑧)))
4645fveq2d 6909 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) − -(log‘𝑧))) = (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4742, 44, 463brtr3d 5173 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4822, 39, 35lesubadd2d 11863 . . . . . . . . . 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 15488 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ∈ ℝ+)
5150relogcld 26666 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℝ)
5251recnd 11290 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℂ)
5352abscld 15476 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘(abs‘𝑧))) ∈ ℝ)
5453, 26readdcld 11291 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ∈ ℝ)
55 abslogle 26661 . . . . . . . . . . . 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 11496 . . . . . . . . . . . . . . . 16 -(log‘𝑅) = (0 − (log‘𝑅))
58 log1 26628 . . . . . . . . . . . . . . . . 17 (log‘1) = 0
5958oveq1i 7442 . . . . . . . . . . . . . . . 16 ((log‘1) − (log‘𝑅)) = (0 − (log‘𝑅))
6057, 59eqtr4i 2767 . . . . . . . . . . . . . . 15 -(log‘𝑅) = ((log‘1) − (log‘𝑅))
61 1rp 13039 . . . . . . . . . . . . . . . 16 1 ∈ ℝ+
62 relogdiv 26636 . . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) = (log‘(1 / 𝑅)))
65 oveq2 7440 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝑧 + 𝑘) = (𝑧 + 0))
6665fveq2d 6909 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (abs‘(𝑧 + 𝑘)) = (abs‘(𝑧 + 0)))
6766breq2d 5154 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → ((1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
68 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → (abs‘𝑥) = (abs‘𝑧))
6968breq1d 5152 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑧) ≤ 𝑅))
70 fvoveq1 7455 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑧 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑧 + 𝑘)))
7170breq2d 5154 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7271ralbidv 3177 . . . . . . . . . . . . . . . . . . . . . 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 3694 . . . . . . . . . . . . . . . . . . . 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 12543 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
7978a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 0 ∈ ℕ0)
8067, 77, 79rspcdva 3622 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0)))
8131addridd 11462 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (𝑧 + 0) = 𝑧)
8281fveq2d 6909 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(𝑧 + 0)) = (abs‘𝑧))
8380, 82breqtrd 5168 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘𝑧))
8424rpreccld 13088 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ∈ ℝ+)
8584, 50logled 26670 . . . . . . . . . . . . . . 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 5164 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) ≤ (log‘(abs‘𝑧)))
8876simpld 494 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ≤ 𝑅)
8950, 24logled 26670 . . . . . . . . . . . . . 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 15470 . . . . . . . . . . . . 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 11878 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ≤ ((log‘𝑅) + π))
9439, 54, 27, 56, 93letrd 11419 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ≤ ((log‘𝑅) + π))
9539, 27, 35, 94leadd1dd 11878 . . . . . . . . 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 11419 . . . . . . . 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 775 . . . . . . . 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 11879 . . . . . . 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 11419 . . . . . 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 3166 . . . 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 5202 . . 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 3160 1 (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  cdif 3947  wss 3950  ifcif 4524   class class class wbr 5142  cmpt 5224  dom cdm 5684  cfv 6560  (class class class)co 7432  f cof 7696  cc 11154  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161  cle 11297  cmin 11493  -cneg 11494   / cdiv 11921  cn 12267  2c2 12322  0cn0 12528  cz 12615  +crp 13035  seqcseq 14043  cexp 14103  abscabs 15274  πcpi 16103  𝑢culm 26420  logclog 26597  log Γclgam 27060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234  ax-addf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-fi 9452  df-sup 9483  df-inf 9484  df-oi 9551  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ioo 13392  df-ioc 13393  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-fl 13833  df-mod 13911  df-seq 14044  df-exp 14104  df-fac 14314  df-bc 14343  df-hash 14371  df-shft 15107  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-limsup 15508  df-clim 15525  df-rlim 15526  df-sum 15724  df-ef 16104  df-sin 16106  df-cos 16107  df-tan 16108  df-pi 16109  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  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 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-submnd 18798  df-mulg 19087  df-cntz 19336  df-cmn 19801  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-fbas 21362  df-fg 21363  df-cnfld 21366  df-top 22901  df-topon 22918  df-topsp 22940  df-bases 22954  df-cld 23028  df-ntr 23029  df-cls 23030  df-nei 23107  df-lp 23145  df-perf 23146  df-cn 23236  df-cnp 23237  df-haus 23324  df-cmp 23396  df-tx 23571  df-hmeo 23764  df-fil 23855  df-fm 23947  df-flim 23948  df-flf 23949  df-xms 24331  df-ms 24332  df-tms 24333  df-cncf 24905  df-limc 25902  df-dv 25903  df-ulm 26421  df-log 26599  df-cxp 26600  df-lgam 27063
This theorem is referenced by: (None)
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