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Theorem lgambdd 24563
 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 24562 . . . 4 (𝜑 → (∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘𝑓 + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))
54simprd 478 . . 3 (𝜑 → seq1( ∘𝑓 + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
6 eqid 2610 . . . . 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 24560 . . . 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 478 . . 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 11746 . . . . . . . 8 (𝜑𝑅 ∈ ℝ+)
1110adantr 480 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → 𝑅 ∈ ℝ+)
1211relogcld 24173 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (log‘𝑅) ∈ ℝ)
13 pire 24014 . . . . . . 7 π ∈ ℝ
1413a1i 11 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → π ∈ ℝ)
1512, 14readdcld 9948 . . . . 5 ((𝜑𝑦 ∈ ℝ) → ((log‘𝑅) + π) ∈ ℝ)
16 simpr 476 . . . . 5 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
1715, 16readdcld 9948 . . . 4 ((𝜑𝑦 ∈ ℝ) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
1817adantrr 749 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
194simpld 474 . . . . . . . . . . 11 (𝜑 → ∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ)
2019adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → ∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ)
2120r19.21bi 2916 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log Γ‘𝑧) ∈ ℂ)
2221abscld 14023 . . . . . . . 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 24173 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑅) ∈ ℝ)
2613a1i 11 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → π ∈ ℝ)
2725, 26readdcld 9948 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log‘𝑅) + π) ∈ ℝ)
281, 2lgamgulmlem1 24555 . . . . . . . . . . . . . . 15 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2928adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ℝ) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
3029sselda 3568 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
3130eldifad 3552 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ∈ ℂ)
3230dmgmn0 24552 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → 𝑧 ≠ 0)
3331, 32logcld 24121 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘𝑧) ∈ ℂ)
3421, 33addcld 9938 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) ∈ ℂ)
3534abscld 14023 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
3627, 35readdcld 9948 . . . . . . . 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 758 . . . . . . 7 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (((log‘𝑅) + π) + 𝑦) ∈ ℝ)
3933abscld 14023 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ∈ ℝ)
4039, 35readdcld 9948 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))) ∈ ℝ)
4133negcld 10258 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑧) ∈ ℂ)
4221, 41abs2difd 14044 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) − -(log‘𝑧))))
4333absnegd 14036 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘-(log‘𝑧)) = (abs‘(log‘𝑧)))
4443oveq2d 6565 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘-(log‘𝑧))) = ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))))
4521, 33subnegd 10278 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((log Γ‘𝑧) − -(log‘𝑧)) = ((log Γ‘𝑧) + (log‘𝑧)))
4645fveq2d 6107 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘((log Γ‘𝑧) − -(log‘𝑧))) = (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4742, 44, 463brtr3d 4614 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log Γ‘𝑧)) − (abs‘(log‘𝑧))) ≤ (abs‘((log Γ‘𝑧) + (log‘𝑧))))
4822, 39, 35lesubadd2d 10505 . . . . . . . . . 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 221 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ≤ ((abs‘(log‘𝑧)) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
5031, 32absrpcld 14035 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ∈ ℝ+)
5150relogcld 24173 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℝ)
5251recnd 9947 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ∈ ℂ)
5352abscld 14023 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘(abs‘𝑧))) ∈ ℝ)
5453, 26readdcld 9948 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ∈ ℝ)
55 abslogle 24168 . . . . . . . . . . . 12 ((𝑧 ∈ ℂ ∧ 𝑧 ≠ 0) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
5631, 32, 55syl2anc 691 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ≤ ((abs‘(log‘(abs‘𝑧))) + π))
57 1rp 11712 . . . . . . . . . . . . . . . 16 1 ∈ ℝ+
58 relogdiv 24143 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ+𝑅 ∈ ℝ+) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
5957, 24, 58sylancr 694 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(1 / 𝑅)) = ((log‘1) − (log‘𝑅)))
60 df-neg 10148 . . . . . . . . . . . . . . . 16 -(log‘𝑅) = (0 − (log‘𝑅))
61 log1 24136 . . . . . . . . . . . . . . . . 17 (log‘1) = 0
6261oveq1i 6559 . . . . . . . . . . . . . . . 16 ((log‘1) − (log‘𝑅)) = (0 − (log‘𝑅))
6360, 62eqtr4i 2635 . . . . . . . . . . . . . . 15 -(log‘𝑅) = ((log‘1) − (log‘𝑅))
6459, 63syl6reqr 2663 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) = (log‘(1 / 𝑅)))
65 0nn0 11184 . . . . . . . . . . . . . . . . 17 0 ∈ ℕ0
66 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → (abs‘𝑥) = (abs‘𝑧))
6766breq1d 4593 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑧) ≤ 𝑅))
68 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑧 → (𝑥 + 𝑘) = (𝑧 + 𝑘))
6968fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑧 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑧 + 𝑘)))
7069breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑧 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7170ralbidv 2969 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑧 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7267, 71anbi12d 743 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑧 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
7372, 2elrab2 3333 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑈 ↔ (𝑧 ∈ ℂ ∧ ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))))
7473simprbi 479 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑈 → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7574adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘𝑧) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘))))
7675simprd 478 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)))
77 oveq2 6557 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → (𝑧 + 𝑘) = (𝑧 + 0))
7877fveq2d 6107 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (abs‘(𝑧 + 𝑘)) = (abs‘(𝑧 + 0)))
7978breq2d 4595 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
8079rspcv 3278 . . . . . . . . . . . . . . . . 17 (0 ∈ ℕ0 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑧 + 𝑘)) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0))))
8165, 76, 80mpsyl 66 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘(𝑧 + 0)))
8231addid1d 10115 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (𝑧 + 0) = 𝑧)
8382fveq2d 6107 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(𝑧 + 0)) = (abs‘𝑧))
8481, 83breqtrd 4609 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ≤ (abs‘𝑧))
8524rpreccld 11758 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (1 / 𝑅) ∈ ℝ+)
8685, 50logled 24177 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((1 / 𝑅) ≤ (abs‘𝑧) ↔ (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧))))
8784, 86mpbid 221 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(1 / 𝑅)) ≤ (log‘(abs‘𝑧)))
8864, 87eqbrtrd 4605 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → -(log‘𝑅) ≤ (log‘(abs‘𝑧)))
8975simpld 474 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘𝑧) ≤ 𝑅)
9050, 24logled 24177 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘𝑧) ≤ 𝑅 ↔ (log‘(abs‘𝑧)) ≤ (log‘𝑅)))
9189, 90mpbid 221 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (log‘(abs‘𝑧)) ≤ (log‘𝑅))
9251, 25absled 14017 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅) ↔ (-(log‘𝑅) ≤ (log‘(abs‘𝑧)) ∧ (log‘(abs‘𝑧)) ≤ (log‘𝑅))))
9388, 91, 92mpbir2and 959 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘(abs‘𝑧))) ≤ (log‘𝑅))
9453, 25, 26, 93leadd1dd 10520 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → ((abs‘(log‘(abs‘𝑧))) + π) ≤ ((log‘𝑅) + π))
9539, 54, 27, 56, 94letrd 10073 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log‘𝑧)) ≤ ((log‘𝑅) + π))
9639, 27, 35, 95leadd1dd 10520 . . . . . . . . 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 10073 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) → (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + (abs‘((log Γ‘𝑧) + (log‘𝑧)))))
9897adantr 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)))))
9935adantr 480 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ∈ ℝ)
100 simpllr 795 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → 𝑦 ∈ ℝ)
10127adantr 480 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → ((log‘𝑅) + π) ∈ ℝ)
102 simpr 476 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑧𝑈) ∧ (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦) → (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)
10399, 100, 101, 102leadd2dd 10521 . . . . . . 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 10073 . . . . . 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 2945 . . . 4 ((𝜑𝑦 ∈ ℝ) → (∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦 → ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
107106impr 647 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦))
108 breq2 4587 . . . . 5 (𝑟 = (((log‘𝑅) + π) + 𝑦) → ((abs‘(log Γ‘𝑧)) ≤ 𝑟 ↔ (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
109108ralbidv 2969 . . . 4 (𝑟 = (((log‘𝑅) + π) + 𝑦) → (∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟 ↔ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ (((log‘𝑅) + π) + 𝑦)))
110109rspcev 3282 . . 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 691 . 2 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑧𝑈 (abs‘((log Γ‘𝑧) + (log‘𝑧))) ≤ 𝑦)) → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
1129, 111rexlimddv 3017 1 (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954   − cmin 10145  -cneg 10146   / cdiv 10563  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ℝ+crp 11708  seqcseq 12663  ↑cexp 12722  abscabs 13822  πcpi 14636  ⇝𝑢culm 23934  logclog 24105  log Γclgam 24542 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-tan 14641  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-ulm 23935  df-log 24107  df-cxp 24108  df-lgam 24545 This theorem is referenced by: (None)
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