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Theorem lgamgulmlem5 24504
Description: Lemma for lgamgulm 24506. (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 4581 . . 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 4581 . . 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 479 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℕ)
54adantr 479 . . . 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 6088 . . . . . . . 8 (𝑥 = 𝑡 → (abs‘𝑥) = (abs‘𝑡))
87breq1d 4587 . . . . . . 7 (𝑥 = 𝑡 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑡) ≤ 𝑅))
9 oveq1 6534 . . . . . . . . . 10 (𝑥 = 𝑡 → (𝑥 + 𝑘) = (𝑡 + 𝑘))
109fveq2d 6092 . . . . . . . . 9 (𝑥 = 𝑡 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑡 + 𝑘)))
1110breq2d 4589 . . . . . . . 8 (𝑥 = 𝑡 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
1211ralbidv 2968 . . . . . . 7 (𝑥 = 𝑡 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
138, 12anbi12d 742 . . . . . 6 (𝑥 = 𝑡 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))))
1413cbvrabv 3171 . . . . 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 2631 . . . 4 𝑈 = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))}
16 simplrl 795 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑛 ∈ ℕ)
17 simprr 791 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦𝑈)
1817adantr 479 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑦𝑈)
19 simpr 475 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → (2 · 𝑅) ≤ 𝑛)
205, 15, 16, 18, 19lgamgulmlem3 24502 . . 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 24500 . . . . . . . . . . 11 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2221adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2322, 17sseldd 3568 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
2423eldifad 3551 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ ℂ)
25 simprl 789 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℕ)
2625peano2nnd 10887 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℕ)
2726nnrpd 11705 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ+)
2825nnrpd 11705 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ+)
2927, 28rpdivcld 11724 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ+)
3029relogcld 24118 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
3130recnd 9925 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
3224, 31mulcld 9917 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
3325nncnd 10886 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℂ)
3425nnne0d 10915 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≠ 0)
3524, 33, 34divcld 10653 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 / 𝑛) ∈ ℂ)
36 1cnd 9913 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℂ)
3735, 36addcld 9916 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ∈ ℂ)
3823, 25dmgmdivn0 24499 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ≠ 0)
3937, 38logcld 24066 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑦 / 𝑛) + 1)) ∈ ℂ)
4032, 39subcld 10244 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
4140abscld 13972 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
4232abscld 13972 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ∈ ℝ)
4339abscld 13972 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
4442, 43readdcld 9926 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
454nnred 10885 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ)
4645, 30remulcld 9927 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
474peano2nnd 10887 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℕ)
4847nnrpd 11705 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ+)
4948, 28rpmulcld 11723 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ+)
5049relogcld 24118 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
51 pire 23959 . . . . . . . 8 π ∈ ℝ
5251a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → π ∈ ℝ)
5350, 52readdcld 9926 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
5446, 53readdcld 9926 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
5532, 39abs2dif2d 13994 . . . . 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 13990 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))))
5729rpred 11707 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ)
5833mulid2d 9915 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) = 𝑛)
5925nnred 10885 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ)
6059lep1d 10807 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≤ (𝑛 + 1))
6158, 60eqbrtrd 4599 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) ≤ (𝑛 + 1))
62 1red 9912 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ)
6359, 62readdcld 9926 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ)
6462, 63, 28lemuldivd 11756 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 · 𝑛) ≤ (𝑛 + 1) ↔ 1 ≤ ((𝑛 + 1) / 𝑛)))
6561, 64mpbid 220 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ ((𝑛 + 1) / 𝑛))
6657, 65logge0d 24125 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (log‘((𝑛 + 1) / 𝑛)))
6730, 66absidd 13958 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑛 + 1) / 𝑛))) = (log‘((𝑛 + 1) / 𝑛)))
6867oveq2d 6543 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
6956, 68eqtrd 2643 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
7024abscld 13972 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑦) ∈ ℝ)
71 fveq2 6088 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (abs‘𝑥) = (abs‘𝑦))
7271breq1d 4587 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
73 oveq1 6534 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑥 + 𝑘) = (𝑦 + 𝑘))
7473fveq2d 6092 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑦 + 𝑘)))
7574breq2d 4589 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7675ralbidv 2968 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7772, 76anbi12d 742 . . . . . . . . . . . 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 3332 . . . . . . . . . . 11 (𝑦𝑈 ↔ (𝑦 ∈ ℂ ∧ ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))))
7978simprbi 478 . . . . . . . . . 10 (𝑦𝑈 → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
8079ad2antll 760 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
8180simpld 473 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑦) ≤ 𝑅)
8270, 45, 30, 66, 81lemul1ad 10815 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8369, 82eqbrtrd 4599 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8437, 38absrpcld 13984 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ+)
8584relogcld 24118 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
8685recnd 9925 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
8786abscld 13972 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
8887, 52readdcld 9926 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ∈ ℝ)
89 abslogle 24113 . . . . . . . 8 ((((𝑦 / 𝑛) + 1) ∈ ℂ ∧ ((𝑦 / 𝑛) + 1) ≠ 0) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π))
9037, 38, 89syl2anc 690 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π))
91 1rp 11671 . . . . . . . . . . . 12 1 ∈ ℝ+
92 relogdiv 24088 . . . . . . . . . . . 12 ((1 ∈ ℝ+ ∧ ((𝑅 + 1) · 𝑛) ∈ ℝ+) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
9391, 49, 92sylancr 693 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
94 log1 24081 . . . . . . . . . . . . 13 (log‘1) = 0
9594oveq1i 6537 . . . . . . . . . . . 12 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = (0 − (log‘((𝑅 + 1) · 𝑛)))
96 df-neg 10121 . . . . . . . . . . . 12 -(log‘((𝑅 + 1) · 𝑛)) = (0 − (log‘((𝑅 + 1) · 𝑛)))
9795, 96eqtr4i 2634 . . . . . . . . . . 11 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = -(log‘((𝑅 + 1) · 𝑛))
9893, 97syl6req 2660 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) = (log‘(1 / ((𝑅 + 1) · 𝑛))))
9947nnrecred 10916 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ∈ ℝ)
10024, 33addcld 9916 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 + 𝑛) ∈ ℂ)
101100abscld 13972 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 + 𝑛)) ∈ ℝ)
1024nnrecred 10916 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / 𝑅) ∈ ℝ)
1034nnrpd 11705 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ+)
104 0le1 10403 . . . . . . . . . . . . . . . 16 0 ≤ 1
105104a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 1)
10645lep1d 10807 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ≤ (𝑅 + 1))
107103, 48, 62, 105, 106lediv2ad 11729 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (1 / 𝑅))
10825nnnn0d 11201 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℕ0)
10980simprd 477 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))
110 oveq2 6535 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (𝑦 + 𝑘) = (𝑦 + 𝑛))
111110fveq2d 6092 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (abs‘(𝑦 + 𝑘)) = (abs‘(𝑦 + 𝑛)))
112111breq2d 4589 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛))))
113112rspcv 3277 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)) → (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛))))
114108, 109, 113sylc 62 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛)))
11599, 102, 101, 107, 114letrd 10046 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (abs‘(𝑦 + 𝑛)))
11699, 101, 28, 115lediv1dd 11765 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) ≤ ((abs‘(𝑦 + 𝑛)) / 𝑛))
11747nncnd 10886 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℂ)
11847nnne0d 10915 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≠ 0)
119117, 33, 118, 34recdiv2d 10671 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) = (1 / ((𝑅 + 1) · 𝑛)))
12024, 33, 33, 34divdird 10691 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 + 𝑛) / 𝑛) = ((𝑦 / 𝑛) + (𝑛 / 𝑛)))
12133, 34dividd 10651 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 / 𝑛) = 1)
122121oveq2d 6543 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + (𝑛 / 𝑛)) = ((𝑦 / 𝑛) + 1))
123120, 122eqtr2d 2644 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) = ((𝑦 + 𝑛) / 𝑛))
124123fveq2d 6092 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) = (abs‘((𝑦 + 𝑛) / 𝑛)))
125100, 33, 34absdivd 13991 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 + 𝑛) / 𝑛)) = ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)))
12628rpge0d 11711 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 𝑛)
12759, 126absidd 13958 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑛) = 𝑛)
128127oveq2d 6543 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)) = ((abs‘(𝑦 + 𝑛)) / 𝑛))
129124, 125, 1283eqtrrd 2648 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / 𝑛) = (abs‘((𝑦 / 𝑛) + 1)))
130116, 119, 1293brtr3d 4608 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)))
13149rpreccld 11717 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ∈ ℝ+)
132131, 84logled 24122 . . . . . . . . . . 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 220 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13498, 133eqbrtrd 4599 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13537abscld 13972 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ)
13645, 62readdcld 9926 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ)
13749rpred 11707 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ)
13835abscld 13972 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ∈ ℝ)
139138, 62readdcld 9926 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ∈ ℝ)
14035, 36abstrid 13992 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + (abs‘1)))
141 abs1 13834 . . . . . . . . . . . . . 14 (abs‘1) = 1
142141oveq2i 6538 . . . . . . . . . . . . 13 ((abs‘(𝑦 / 𝑛)) + (abs‘1)) = ((abs‘(𝑦 / 𝑛)) + 1)
143140, 142syl6breq 4618 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + 1))
14491a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ+)
14524absge0d 13980 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (abs‘𝑦))
14625nnge1d 10913 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ 𝑛)
14770, 45, 144, 59, 145, 81, 146lediv12ad 11766 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) ≤ (𝑅 / 1))
14824, 33, 34absdivd 13991 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) = ((abs‘𝑦) / (abs‘𝑛)))
149127oveq2d 6543 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / (abs‘𝑛)) = ((abs‘𝑦) / 𝑛))
150148, 149eqtr2d 2644 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) = (abs‘(𝑦 / 𝑛)))
1514nncnd 10886 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℂ)
152151div1d 10645 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 / 1) = 𝑅)
153147, 150, 1523brtr3d 4608 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ≤ 𝑅)
154138, 45, 62, 153leadd1dd 10493 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ≤ (𝑅 + 1))
155135, 139, 136, 143, 154letrd 10046 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ (𝑅 + 1))
15648rpge0d 11711 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (𝑅 + 1))
157136, 59, 156, 146lemulge11d 10813 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≤ ((𝑅 + 1) · 𝑛))
158135, 136, 137, 155, 157letrd 10046 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛))
15984, 49logled 24122 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛) ↔ (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛))))
160158, 159mpbid 220 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛)))
16185, 50absled 13966 . . . . . . . . 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 958 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ≤ (log‘((𝑅 + 1) · 𝑛)))
16387, 50, 52, 162leadd1dd 10493 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16443, 88, 53, 90, 163letrd 10046 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16542, 43, 46, 53, 83, 164le2addd 10498 . . . . 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 10046 . . . 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 479 . . 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 4072 . 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 6534 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
170 id 22 . . . . . . . . . . . 12 (𝑚 = 𝑛𝑚 = 𝑛)
171169, 170oveq12d 6545 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
172171fveq2d 6092 . . . . . . . . . 10 (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
173172oveq2d 6543 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
174 oveq2 6535 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
175174oveq1d 6542 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝑧 / 𝑚) + 1) = ((𝑧 / 𝑛) + 1))
176175fveq2d 6092 . . . . . . . . 9 (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
177173, 176oveq12d 6545 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
178177mpteq2dv 4667 . . . . . . 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 9874 . . . . . . . . 9 ℂ ∈ V
1816, 180rabex2 4737 . . . . . . . 8 𝑈 ∈ V
182181mptex 6368 . . . . . . 7 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) ∈ V
183178, 179, 182fvmpt 6176 . . . . . 6 (𝑛 ∈ ℕ → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
184183ad2antrl 759 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
185184fveq1d 6090 . . . 4 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦))
186 oveq1 6534 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) = (𝑦 · (log‘((𝑛 + 1) / 𝑛))))
187 oveq1 6534 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧 / 𝑛) = (𝑦 / 𝑛))
188187oveq1d 6542 . . . . . . . 8 (𝑧 = 𝑦 → ((𝑧 / 𝑛) + 1) = ((𝑦 / 𝑛) + 1))
189188fveq2d 6092 . . . . . . 7 (𝑧 = 𝑦 → (log‘((𝑧 / 𝑛) + 1)) = (log‘((𝑦 / 𝑛) + 1)))
190186, 189oveq12d 6545 . . . . . 6 (𝑧 = 𝑦 → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
191 eqid 2609 . . . . . 6 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
192 ovex 6555 . . . . . 6 ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ V
193190, 191, 192fvmpt 6176 . . . . 5 (𝑦𝑈 → ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
194193ad2antll 760 . . . 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 2643 . . 3 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
196195fveq2d 6092 . 2 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) = (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))))
197 breq2 4581 . . . . 5 (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
198 oveq1 6534 . . . . . . 7 (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
199198oveq2d 6543 . . . . . 6 (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
200199oveq2d 6543 . . . . 5 (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
201172oveq2d 6543 . . . . . 6 (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
202 oveq2 6535 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
203202fveq2d 6092 . . . . . . 7 (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
204203oveq1d 6542 . . . . . 6 (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
205201, 204oveq12d 6545 . . . . 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 4062 . . . 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 6555 . . . . 5 (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
209 ovex 6555 . . . . 5 ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
210208, 209ifex 4105 . . . 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 6176 . . 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 759 . 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 4609 1 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) ≤ (𝑇𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  {crab 2899  cdif 3536  wss 3539  ifcif 4035   class class class wbr 4577  cmpt 4637  cfv 5790  (class class class)co 6527  cc 9791  cr 9792  0cc0 9793  1c1 9794   + caddc 9796   · cmul 9798  cle 9932  cmin 10118  -cneg 10119   / cdiv 10536  cn 10870  2c2 10920  0cn0 11142  cz 11213  +crp 11667  cexp 12680  abscabs 13771  πcpi 14585  logclog 24050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871  ax-addf 9872  ax-mulf 9873
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6773  df-om 6936  df-1st 7037  df-2nd 7038  df-supp 7161  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-2o 7426  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-ixp 7773  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-fsupp 8137  df-fi 8178  df-sup 8209  df-inf 8210  df-oi 8276  df-card 8626  df-cda 8851  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-dec 11329  df-uz 11523  df-q 11624  df-rp 11668  df-xneg 11781  df-xadd 11782  df-xmul 11783  df-ioo 12009  df-ioc 12010  df-ico 12011  df-icc 12012  df-fz 12156  df-fzo 12293  df-fl 12413  df-mod 12489  df-seq 12622  df-exp 12681  df-fac 12881  df-bc 12910  df-hash 12938  df-shft 13604  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-limsup 13999  df-clim 14016  df-rlim 14017  df-sum 14214  df-ef 14586  df-sin 14588  df-cos 14589  df-tan 14590  df-pi 14591  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-plusg 15730  df-mulr 15731  df-starv 15732  df-sca 15733  df-vsca 15734  df-ip 15735  df-tset 15736  df-ple 15737  df-ds 15740  df-unif 15741  df-hom 15742  df-cco 15743  df-rest 15855  df-topn 15856  df-0g 15874  df-gsum 15875  df-topgen 15876  df-pt 15877  df-prds 15880  df-xrs 15934  df-qtop 15939  df-imas 15940  df-xps 15942  df-mre 16018  df-mrc 16019  df-acs 16021  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-submnd 17108  df-mulg 17313  df-cntz 17522  df-cmn 17967  df-psmet 19508  df-xmet 19509  df-met 19510  df-bl 19511  df-mopn 19512  df-fbas 19513  df-fg 19514  df-cnfld 19517  df-top 20469  df-bases 20470  df-topon 20471  df-topsp 20472  df-cld 20581  df-ntr 20582  df-cls 20583  df-nei 20660  df-lp 20698  df-perf 20699  df-cn 20789  df-cnp 20790  df-haus 20877  df-cmp 20948  df-tx 21123  df-hmeo 21316  df-fil 21408  df-fm 21500  df-flim 21501  df-flf 21502  df-xms 21883  df-ms 21884  df-tms 21885  df-cncf 22437  df-limc 23381  df-dv 23382  df-log 24052
This theorem is referenced by:  lgamgulmlem6  24505
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