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Theorem lgambdd 24962
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 24961 . . . 4 (𝜑 → (∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘𝑓 + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))
54simprd 482 . . 3 (𝜑 → seq1( ∘𝑓 + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
6 eqid 2760 . . . . 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 24959 . . . 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 482 . . 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 12063 . . . . . . . 8 (𝜑𝑅 ∈ ℝ+)
1110adantr 472 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → 𝑅 ∈ ℝ+)
1211relogcld 24568 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (log‘𝑅) ∈ ℝ)
13 pire 24409 . . . . . . 7 π ∈ ℝ
1413a1i 11 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → π ∈ ℝ)
1512, 14readdcld 10261 . . . . 5 ((𝜑𝑦 ∈ ℝ) → ((log‘𝑅) + π) ∈ ℝ)
16 simpr 479 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
1715, 16readdcld 10261 . . . 4 ((𝜑𝑦 ∈ ℝ) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
1817adantrr 755 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
194simpld 477 . . . . . . . . . . 11 (𝜑 → ∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ)
2019adantr 472 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → ∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ)
2120r19.21bi 3070 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log Γ‘𝑧) ∈ ℂ)
2221abscld 14374 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
2322adantr 472 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ∈ ℝ)
2411adantr 472 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑅 ∈ ℝ+)
2524relogcld 24568 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑅) ∈ ℝ)
2613a1i 11 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → π ∈ ℝ)
2725, 26readdcld 10261 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log‘𝑅) + π) ∈ ℝ)
281, 2lgamgulmlem1 24954 . . . . . . . . . . . . . . 15 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2928adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ℝ) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
3029sselda 3744 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
3130eldifad 3727 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ ℂ)
3230dmgmn0 24951 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ≠ 0)
3331, 32logcld 24516 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑧) ∈ ℂ)
3421, 33addcld 10251 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) ∈ ℂ)
3534abscld 14374 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
3627, 35readdcld 10261 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
3736adantr 472 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
3817ad2antrr 764 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
3933abscld 14374 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ∈ ℝ)
4039, 35readdcld 10261 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
4133negcld 10571 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑧) ∈ ℂ)
4221, 41abs2difd 14395 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) − -(log‘𝑧))))
4333absnegd 14387 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘-(log‘𝑧)) = (abs‘(log‘𝑧)))
4443oveq2d 6829 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) = ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))))
4521, 33subnegd 10591 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) − -(log‘𝑧)) = ((log Γ‘𝑧) + (log‘𝑧)))
4645fveq2d 6356 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) − -(log‘𝑧))) = (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4742, 44, 463brtr3d 4835 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4822, 39, 35lesubadd2d 10818 . . . . . . . . . 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 222 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ≤ ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
5031, 32absrpcld 14386 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ∈ ℝ+)
5150relogcld 24568 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℝ)
5251recnd 10260 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℂ)
5352abscld 14374 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘(abs‘𝑧))) ∈ ℝ)
5453, 26readdcld 10261 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ∈ ℝ)
55 abslogle 24563 . . . . . . . . . . . 12 ((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
5631, 32, 55syl2anc 696 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
57 1rp 12029 . . . . . . . . . . . . . . . 16 1 ∈ ℝ+
58 relogdiv 24538 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ+𝑅 ∈ ℝ+) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
5957, 24, 58sylancr 698 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
60 df-neg 10461 . . . . . . . . . . . . . . . 16 -(log‘𝑅) = (0 − (log‘𝑅))
61 log1 24531 . . . . . . . . . . . . . . . . 17 (log‘1) = 0
6261oveq1i 6823 . . . . . . . . . . . . . . . 16 ((log‘1) − (log‘𝑅)) = (0 − (log‘𝑅))
6360, 62eqtr4i 2785 . . . . . . . . . . . . . . 15 -(log‘𝑅) = ((log‘1) − (log‘𝑅))
6459, 63syl6reqr 2813 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) = (log‘(1 / 𝑅)))
65 oveq2 6821 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝑧 + 𝑘) = (𝑧 + 0))
6665fveq2d 6356 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (abs‘(𝑧 + 𝑘)) = (abs‘(𝑧 + 0)))
6766breq2d 4816 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → ((1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
68 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → (abs‘𝑥) = (abs‘𝑧))
6968breq1d 4814 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑧) ≤ 𝑅))
70 oveq1 6820 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑧 → (𝑥 + 𝑘) = (𝑧 + 𝑘))
7170fveq2d 6356 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑧 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑧 + 𝑘)))
7271breq2d 4816 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7372ralbidv 3124 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7469, 73anbi12d 749 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑧 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
7574, 2elrab2 3507 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑈 ↔ (𝑧 ∈ ℂ ∧ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
7675simprbi 483 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑈 → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7776adantl 473 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7877simprd 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))
79 0nn0 11499 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
8079a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 0 ∈ ℕ0)
8167, 78, 80rspcdva 3455 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0)))
8231addid1d 10428 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (𝑧 + 0) = 𝑧)
8382fveq2d 6356 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(𝑧 + 0)) = (abs‘𝑧))
8481, 83breqtrd 4830 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘𝑧))
8524rpreccld 12075 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ∈ ℝ+)
8685, 50logled 24572 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((1 / 𝑅) ≤ (abs‘𝑧) ↔ (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧))))
8784, 86mpbid 222 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧)))
8864, 87eqbrtrd 4826 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) ≤ (log‘(abs‘𝑧)))
8977simpld 477 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ≤ 𝑅)
9050, 24logled 24572 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘𝑧) ≤ 𝑅 ↔ (log‘(abs‘𝑧)) ≤ (log‘𝑅)))
9189, 90mpbid 222 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ≤ (log‘𝑅))
9251, 25absled 14368 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅) ↔ (-(log‘𝑅) ≤ (log‘(abs‘𝑧)) ∧ (log‘(abs‘𝑧)) ≤ (log‘𝑅))))
9388, 91, 92mpbir2and 995 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅))
9453, 25, 26, 93leadd1dd 10833 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ≤ ((log‘𝑅) + π))
9539, 54, 27, 56, 94letrd 10386 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ≤ ((log‘𝑅) + π))
9639, 27, 35, 95leadd1dd 10833 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
9722, 40, 36, 49, 96letrd 10386 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
9897adantr 472 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
9935adantr 472 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
100 simpllr 817 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → 𝑦 ∈ ℝ)
10127adantr 472 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → ((log‘𝑅) + π) ∈ ℝ)
102 simpr 479 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)
10399, 100, 101, 102leadd2dd 10834 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ≤ (((log‘𝑅) + π) + 𝑦))
10423, 37, 38, 98, 103letrd 10386 . . . . . 6 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
105104ex 449 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
106105ralimdva 3100 . . . 4 ((𝜑𝑦 ∈ ℝ) → (∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
107106impr 650 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
108 breq2 4808 . . . . 5 (𝑟 = (((log‘𝑅) + π) + 𝑦) → ((abs‘(log Γ‘𝑧)) ≤ 𝑟 ↔ (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
109108ralbidv 3124 . . . 4 (𝑟 = (((log‘𝑅) + π) + 𝑦) → (∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟 ↔ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
110109rspcev 3449 . . 3 (((((log‘𝑅) + π) + 𝑦) ∈ ℝ ∧ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)) → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
11118, 107, 110syl2anc 696 . 2 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
1129, 111rexlimddv 3173 1 (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  {crab 3054  cdif 3712  wss 3715  ifcif 4230   class class class wbr 4804  cmpt 4881  dom cdm 5266  cfv 6049  (class class class)co 6813  𝑓 cof 7060  cc 10126  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   · cmul 10133  cle 10267  cmin 10458  -cneg 10459   / cdiv 10876  cn 11212  2c2 11262  0cn0 11484  cz 11569  +crp 12025  seqcseq 12995  cexp 13054  abscabs 14173  πcpi 14996  𝑢culm 24329  logclog 24500  log Γclgam 24941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-fi 8482  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ioo 12372  df-ioc 12373  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-fl 12787  df-mod 12863  df-seq 12996  df-exp 13055  df-fac 13255  df-bc 13284  df-hash 13312  df-shft 14006  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-limsup 14401  df-clim 14418  df-rlim 14419  df-sum 14616  df-ef 14997  df-sin 14999  df-cos 15000  df-tan 15001  df-pi 15002  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-starv 16158  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-unif 16167  df-hom 16168  df-cco 16169  df-rest 16285  df-topn 16286  df-0g 16304  df-gsum 16305  df-topgen 16306  df-pt 16307  df-prds 16310  df-xrs 16364  df-qtop 16369  df-imas 16370  df-xps 16372  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-fbas 19945  df-fg 19946  df-cnfld 19949  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027  df-nei 21104  df-lp 21142  df-perf 21143  df-cn 21233  df-cnp 21234  df-haus 21321  df-cmp 21392  df-tx 21567  df-hmeo 21760  df-fil 21851  df-fm 21943  df-flim 21944  df-flf 21945  df-xms 22326  df-ms 22327  df-tms 22328  df-cncf 22882  df-limc 23829  df-dv 23830  df-ulm 24330  df-log 24502  df-cxp 24503  df-lgam 24944
This theorem is referenced by: (None)
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