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Theorem lgamgulmlem5 24804
Description: Lemma for lgamgulm 24806. (Contributed by Mario Carneiro, 3-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)))))
lgamgulm.t 𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))
Assertion
Ref Expression
lgamgulmlem5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) ≤ (𝑇𝑛))
Distinct variable groups:   𝑦,𝑛,𝐺   𝑥,𝑦   𝑘,𝑚,𝑛,𝑥,𝑦,𝑧,𝑅   𝑈,𝑚,𝑛,𝑦,𝑧   𝜑,𝑚,𝑛,𝑥,𝑦,𝑧   𝑇,𝑛,𝑦
Allowed substitution hints:   𝜑(𝑘)   𝑇(𝑥,𝑧,𝑘,𝑚)   𝑈(𝑥,𝑘)   𝐺(𝑥,𝑧,𝑘,𝑚)

Proof of Theorem lgamgulmlem5
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 breq2 4689 . . 3 ((𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) → ((abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ↔ (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))))
2 breq2 4689 . . 3 (((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) → ((abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ↔ (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))))
3 lgamgulm.r . . . . . 6 (𝜑𝑅 ∈ ℕ)
43adantr 480 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℕ)
54adantr 480 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑅 ∈ ℕ)
6 lgamgulm.u . . . . 5 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}
7 fveq2 6229 . . . . . . . 8 (𝑥 = 𝑡 → (abs‘𝑥) = (abs‘𝑡))
87breq1d 4695 . . . . . . 7 (𝑥 = 𝑡 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑡) ≤ 𝑅))
9 oveq1 6697 . . . . . . . . . 10 (𝑥 = 𝑡 → (𝑥 + 𝑘) = (𝑡 + 𝑘))
109fveq2d 6233 . . . . . . . . 9 (𝑥 = 𝑡 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑡 + 𝑘)))
1110breq2d 4697 . . . . . . . 8 (𝑥 = 𝑡 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
1211ralbidv 3015 . . . . . . 7 (𝑥 = 𝑡 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
138, 12anbi12d 747 . . . . . 6 (𝑥 = 𝑡 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))))
1413cbvrabv 3230 . . . . 5 {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))}
156, 14eqtri 2673 . . . 4 𝑈 = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))}
16 simplrl 817 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑛 ∈ ℕ)
17 simprr 811 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦𝑈)
1817adantr 480 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑦𝑈)
19 simpr 476 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → (2 · 𝑅) ≤ 𝑛)
205, 15, 16, 18, 19lgamgulmlem3 24802 . . 3 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
213, 6lgamgulmlem1 24800 . . . . . . . . . . 11 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2221adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2322, 17sseldd 3637 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
2423eldifad 3619 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ ℂ)
25 simprl 809 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℕ)
2625peano2nnd 11075 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℕ)
2726nnrpd 11908 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ+)
2825nnrpd 11908 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ+)
2927, 28rpdivcld 11927 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ+)
3029relogcld 24414 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
3130recnd 10106 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
3224, 31mulcld 10098 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
3325nncnd 11074 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℂ)
3425nnne0d 11103 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≠ 0)
3524, 33, 34divcld 10839 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 / 𝑛) ∈ ℂ)
36 1cnd 10094 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℂ)
3735, 36addcld 10097 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ∈ ℂ)
3823, 25dmgmdivn0 24799 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ≠ 0)
3937, 38logcld 24362 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑦 / 𝑛) + 1)) ∈ ℂ)
4032, 39subcld 10430 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
4140abscld 14219 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
4232abscld 14219 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ∈ ℝ)
4339abscld 14219 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
4442, 43readdcld 10107 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
454nnred 11073 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ)
4645, 30remulcld 10108 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
474peano2nnd 11075 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℕ)
4847nnrpd 11908 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ+)
4948, 28rpmulcld 11926 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ+)
5049relogcld 24414 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
51 pire 24255 . . . . . . . 8 π ∈ ℝ
5251a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → π ∈ ℝ)
5350, 52readdcld 10107 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
5446, 53readdcld 10107 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
5532, 39abs2dif2d 14241 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))))
5624, 31absmuld 14237 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))))
5729rpred 11910 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ)
5833mulid2d 10096 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) = 𝑛)
5925nnred 11073 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ)
6059lep1d 10993 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≤ (𝑛 + 1))
6158, 60eqbrtrd 4707 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) ≤ (𝑛 + 1))
62 1red 10093 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ)
6359, 62readdcld 10107 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ)
6462, 63, 28lemuldivd 11959 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 · 𝑛) ≤ (𝑛 + 1) ↔ 1 ≤ ((𝑛 + 1) / 𝑛)))
6561, 64mpbid 222 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ ((𝑛 + 1) / 𝑛))
6657, 65logge0d 24421 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (log‘((𝑛 + 1) / 𝑛)))
6730, 66absidd 14205 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑛 + 1) / 𝑛))) = (log‘((𝑛 + 1) / 𝑛)))
6867oveq2d 6706 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
6956, 68eqtrd 2685 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
7024abscld 14219 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑦) ∈ ℝ)
71 fveq2 6229 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (abs‘𝑥) = (abs‘𝑦))
7271breq1d 4695 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
73 oveq1 6697 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑥 + 𝑘) = (𝑦 + 𝑘))
7473fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑦 + 𝑘)))
7574breq2d 4697 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7675ralbidv 3015 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7772, 76anbi12d 747 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))))
7877, 6elrab2 3399 . . . . . . . . . . 11 (𝑦𝑈 ↔ (𝑦 ∈ ℂ ∧ ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))))
7978simprbi 479 . . . . . . . . . 10 (𝑦𝑈 → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
8079ad2antll 765 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
8180simpld 474 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑦) ≤ 𝑅)
8270, 45, 30, 66, 81lemul1ad 11001 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8369, 82eqbrtrd 4707 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8437, 38absrpcld 14231 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ+)
8584relogcld 24414 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
8685recnd 10106 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
8786abscld 14219 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
8887, 52readdcld 10107 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ∈ ℝ)
89 abslogle 24409 . . . . . . . 8 ((((𝑦 / 𝑛) + 1) ∈ ℂ ∧ ((𝑦 / 𝑛) + 1) ≠ 0) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π))
9037, 38, 89syl2anc 694 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π))
91 1rp 11874 . . . . . . . . . . . 12 1 ∈ ℝ+
92 relogdiv 24384 . . . . . . . . . . . 12 ((1 ∈ ℝ+ ∧ ((𝑅 + 1) · 𝑛) ∈ ℝ+) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
9391, 49, 92sylancr 696 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
94 log1 24377 . . . . . . . . . . . . 13 (log‘1) = 0
9594oveq1i 6700 . . . . . . . . . . . 12 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = (0 − (log‘((𝑅 + 1) · 𝑛)))
96 df-neg 10307 . . . . . . . . . . . 12 -(log‘((𝑅 + 1) · 𝑛)) = (0 − (log‘((𝑅 + 1) · 𝑛)))
9795, 96eqtr4i 2676 . . . . . . . . . . 11 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = -(log‘((𝑅 + 1) · 𝑛))
9893, 97syl6req 2702 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) = (log‘(1 / ((𝑅 + 1) · 𝑛))))
9947nnrecred 11104 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ∈ ℝ)
10024, 33addcld 10097 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 + 𝑛) ∈ ℂ)
101100abscld 14219 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 + 𝑛)) ∈ ℝ)
1024nnrecred 11104 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / 𝑅) ∈ ℝ)
1034nnrpd 11908 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ+)
104 0le1 10589 . . . . . . . . . . . . . . . 16 0 ≤ 1
105104a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 1)
10645lep1d 10993 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ≤ (𝑅 + 1))
107103, 48, 62, 105, 106lediv2ad 11932 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (1 / 𝑅))
10825nnnn0d 11389 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℕ0)
10980simprd 478 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))
110 oveq2 6698 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (𝑦 + 𝑘) = (𝑦 + 𝑛))
111110fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (abs‘(𝑦 + 𝑘)) = (abs‘(𝑦 + 𝑛)))
112111breq2d 4697 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛))))
113112rspcv 3336 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)) → (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛))))
114108, 109, 113sylc 65 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛)))
11599, 102, 101, 107, 114letrd 10232 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (abs‘(𝑦 + 𝑛)))
11699, 101, 28, 115lediv1dd 11968 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) ≤ ((abs‘(𝑦 + 𝑛)) / 𝑛))
11747nncnd 11074 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℂ)
11847nnne0d 11103 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≠ 0)
119117, 33, 118, 34recdiv2d 10857 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) = (1 / ((𝑅 + 1) · 𝑛)))
12024, 33, 33, 34divdird 10877 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 + 𝑛) / 𝑛) = ((𝑦 / 𝑛) + (𝑛 / 𝑛)))
12133, 34dividd 10837 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 / 𝑛) = 1)
122121oveq2d 6706 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + (𝑛 / 𝑛)) = ((𝑦 / 𝑛) + 1))
123120, 122eqtr2d 2686 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) = ((𝑦 + 𝑛) / 𝑛))
124123fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) = (abs‘((𝑦 + 𝑛) / 𝑛)))
125100, 33, 34absdivd 14238 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 + 𝑛) / 𝑛)) = ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)))
12628rpge0d 11914 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 𝑛)
12759, 126absidd 14205 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑛) = 𝑛)
128127oveq2d 6706 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)) = ((abs‘(𝑦 + 𝑛)) / 𝑛))
129124, 125, 1283eqtrrd 2690 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / 𝑛) = (abs‘((𝑦 / 𝑛) + 1)))
130116, 119, 1293brtr3d 4716 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)))
13149rpreccld 11920 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ∈ ℝ+)
132131, 84logled 24418 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)) ↔ (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1)))))
133130, 132mpbid 222 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13498, 133eqbrtrd 4707 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13537abscld 14219 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ)
13645, 62readdcld 10107 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ)
13749rpred 11910 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ)
13835abscld 14219 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ∈ ℝ)
139138, 62readdcld 10107 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ∈ ℝ)
14035, 36abstrid 14239 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + (abs‘1)))
141 abs1 14081 . . . . . . . . . . . . . 14 (abs‘1) = 1
142141oveq2i 6701 . . . . . . . . . . . . 13 ((abs‘(𝑦 / 𝑛)) + (abs‘1)) = ((abs‘(𝑦 / 𝑛)) + 1)
143140, 142syl6breq 4726 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + 1))
14491a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ+)
14524absge0d 14227 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (abs‘𝑦))
14625nnge1d 11101 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ 𝑛)
14770, 45, 144, 59, 145, 81, 146lediv12ad 11969 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) ≤ (𝑅 / 1))
14824, 33, 34absdivd 14238 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) = ((abs‘𝑦) / (abs‘𝑛)))
149127oveq2d 6706 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / (abs‘𝑛)) = ((abs‘𝑦) / 𝑛))
150148, 149eqtr2d 2686 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) = (abs‘(𝑦 / 𝑛)))
1514nncnd 11074 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℂ)
152151div1d 10831 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 / 1) = 𝑅)
153147, 150, 1523brtr3d 4716 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ≤ 𝑅)
154138, 45, 62, 153leadd1dd 10679 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ≤ (𝑅 + 1))
155135, 139, 136, 143, 154letrd 10232 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ (𝑅 + 1))
15648rpge0d 11914 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (𝑅 + 1))
157136, 59, 156, 146lemulge11d 10999 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≤ ((𝑅 + 1) · 𝑛))
158135, 136, 137, 155, 157letrd 10232 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛))
15984, 49logled 24418 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛) ↔ (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛))))
160158, 159mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛)))
16185, 50absled 14213 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ≤ (log‘((𝑅 + 1) · 𝑛)) ↔ (-(log‘((𝑅 + 1) · 𝑛)) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))) ∧ (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛)))))
162134, 160, 161mpbir2and 977 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ≤ (log‘((𝑅 + 1) · 𝑛)))
16387, 50, 52, 162leadd1dd 10679 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16443, 88, 53, 90, 163letrd 10232 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16542, 43, 46, 53, 83, 164le2addd 10684 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))) ≤ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
16641, 44, 54, 55, 165letrd 10232 . . . 4 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
167166adantr 480 . . 3 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ ¬ (2 · 𝑅) ≤ 𝑛) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
1681, 2, 20, 167ifbothda 4156 . 2 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
169 oveq1 6697 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
170 id 22 . . . . . . . . . . . 12 (𝑚 = 𝑛𝑚 = 𝑛)
171169, 170oveq12d 6708 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
172171fveq2d 6233 . . . . . . . . . 10 (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
173172oveq2d 6706 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
174 oveq2 6698 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
175174oveq1d 6705 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝑧 / 𝑚) + 1) = ((𝑧 / 𝑛) + 1))
176175fveq2d 6233 . . . . . . . . 9 (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
177173, 176oveq12d 6708 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
178177mpteq2dv 4778 . . . . . . 7 (𝑚 = 𝑛 → (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
179 lgamgulm.g . . . . . . 7 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
180 cnex 10055 . . . . . . . . 9 ℂ ∈ V
1816, 180rabex2 4847 . . . . . . . 8 𝑈 ∈ V
182181mptex 6527 . . . . . . 7 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) ∈ V
183178, 179, 182fvmpt 6321 . . . . . 6 (𝑛 ∈ ℕ → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
184183ad2antrl 764 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
185184fveq1d 6231 . . . 4 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦))
186 oveq1 6697 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) = (𝑦 · (log‘((𝑛 + 1) / 𝑛))))
187 oveq1 6697 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧 / 𝑛) = (𝑦 / 𝑛))
188187oveq1d 6705 . . . . . . . 8 (𝑧 = 𝑦 → ((𝑧 / 𝑛) + 1) = ((𝑦 / 𝑛) + 1))
189188fveq2d 6233 . . . . . . 7 (𝑧 = 𝑦 → (log‘((𝑧 / 𝑛) + 1)) = (log‘((𝑦 / 𝑛) + 1)))
190186, 189oveq12d 6708 . . . . . 6 (𝑧 = 𝑦 → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
191 eqid 2651 . . . . . 6 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
192 ovex 6718 . . . . . 6 ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ V
193190, 191, 192fvmpt 6321 . . . . 5 (𝑦𝑈 → ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
194193ad2antll 765 . . . 4 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
195185, 194eqtrd 2685 . . 3 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
196195fveq2d 6233 . 2 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) = (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))))
197 breq2 4689 . . . . 5 (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
198 oveq1 6697 . . . . . . 7 (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
199198oveq2d 6706 . . . . . 6 (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
200199oveq2d 6706 . . . . 5 (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
201172oveq2d 6706 . . . . . 6 (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
202 oveq2 6698 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
203202fveq2d 6233 . . . . . . 7 (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
204203oveq1d 6705 . . . . . 6 (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
205201, 204oveq12d 6708 . . . . 5 (𝑚 = 𝑛 → ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π)) = ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
206197, 200, 205ifbieq12d 4146 . . . 4 (𝑚 = 𝑛 → if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
207 lgamgulm.t . . . 4 𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))
208 ovex 6718 . . . . 5 (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
209 ovex 6718 . . . . 5 ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
210208, 209ifex 4189 . . . 4 if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) ∈ V
211206, 207, 210fvmpt 6321 . . 3 (𝑛 ∈ ℕ → (𝑇𝑛) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
212211ad2antrl 764 . 2 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑇𝑛) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
213168, 196, 2123brtr4d 4717 1 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) ≤ (𝑇𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  cdif 3604  wss 3607  ifcif 4119   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  cle 10113  cmin 10304  -cneg 10305   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cz 11415  +crp 11870  cexp 12900  abscabs 14018  πcpi 14841  logclog 24346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-tan 14846  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348
This theorem is referenced by:  lgamgulmlem6  24805
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