MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgamgulmlem5 Structured version   Visualization version   GIF version

Theorem lgamgulmlem5 26182
Description: Lemma for lgamgulm 26184. (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 5078 . . 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 5078 . . 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 481 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℕ)
54adantr 481 . . . 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 6774 . . . . . . . 8 (𝑥 = 𝑡 → (abs‘𝑥) = (abs‘𝑡))
87breq1d 5084 . . . . . . 7 (𝑥 = 𝑡 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑡) ≤ 𝑅))
9 fvoveq1 7298 . . . . . . . . 9 (𝑥 = 𝑡 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑡 + 𝑘)))
109breq2d 5086 . . . . . . . 8 (𝑥 = 𝑡 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
1110ralbidv 3112 . . . . . . 7 (𝑥 = 𝑡 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
128, 11anbi12d 631 . . . . . 6 (𝑥 = 𝑡 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))))
1312cbvrabv 3426 . . . . 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)))}
146, 13eqtri 2766 . . . 4 𝑈 = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))}
15 simplrl 774 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑛 ∈ ℕ)
16 simprr 770 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦𝑈)
1716adantr 481 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑦𝑈)
18 simpr 485 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → (2 · 𝑅) ≤ 𝑛)
195, 14, 15, 17, 18lgamgulmlem3 26180 . . 3 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
203, 6lgamgulmlem1 26178 . . . . . . . . . . 11 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2120adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2221, 16sseldd 3922 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
2322eldifad 3899 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ ℂ)
24 simprl 768 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℕ)
2524peano2nnd 11990 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℕ)
2625nnrpd 12770 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ+)
2724nnrpd 12770 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ+)
2826, 27rpdivcld 12789 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ+)
2928relogcld 25778 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
3029recnd 11003 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
3123, 30mulcld 10995 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
3224nncnd 11989 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℂ)
3324nnne0d 12023 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≠ 0)
3423, 32, 33divcld 11751 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 / 𝑛) ∈ ℂ)
35 1cnd 10970 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℂ)
3634, 35addcld 10994 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ∈ ℂ)
3722, 24dmgmdivn0 26177 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ≠ 0)
3836, 37logcld 25726 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑦 / 𝑛) + 1)) ∈ ℂ)
3931, 38subcld 11332 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
4039abscld 15148 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
4131abscld 15148 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ∈ ℝ)
4238abscld 15148 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
4341, 42readdcld 11004 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
444nnred 11988 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ)
4544, 29remulcld 11005 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
464peano2nnd 11990 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℕ)
4746nnrpd 12770 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ+)
4847, 27rpmulcld 12788 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ+)
4948relogcld 25778 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
50 pire 25615 . . . . . . . 8 π ∈ ℝ
5150a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → π ∈ ℝ)
5249, 51readdcld 11004 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
5345, 52readdcld 11004 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
5431, 38abs2dif2d 15170 . . . . 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)))))
5523, 30absmuld 15166 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))))
5628rpred 12772 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ)
5732mulid2d 10993 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) = 𝑛)
5824nnred 11988 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ)
5958lep1d 11906 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≤ (𝑛 + 1))
6057, 59eqbrtrd 5096 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) ≤ (𝑛 + 1))
61 1red 10976 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ)
6258, 61readdcld 11004 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ)
6361, 62, 27lemuldivd 12821 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 · 𝑛) ≤ (𝑛 + 1) ↔ 1 ≤ ((𝑛 + 1) / 𝑛)))
6460, 63mpbid 231 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ ((𝑛 + 1) / 𝑛))
6556, 64logge0d 25785 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (log‘((𝑛 + 1) / 𝑛)))
6629, 65absidd 15134 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑛 + 1) / 𝑛))) = (log‘((𝑛 + 1) / 𝑛)))
6766oveq2d 7291 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
6855, 67eqtrd 2778 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
6923abscld 15148 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑦) ∈ ℝ)
70 fveq2 6774 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (abs‘𝑥) = (abs‘𝑦))
7170breq1d 5084 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
72 fvoveq1 7298 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑦 + 𝑘)))
7372breq2d 5086 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7473ralbidv 3112 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7571, 74anbi12d 631 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))))
7675, 6elrab2 3627 . . . . . . . . . . 11 (𝑦𝑈 ↔ (𝑦 ∈ ℂ ∧ ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))))
7776simprbi 497 . . . . . . . . . 10 (𝑦𝑈 → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7877ad2antll 726 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7978simpld 495 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑦) ≤ 𝑅)
8069, 44, 29, 65, 79lemul1ad 11914 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8168, 80eqbrtrd 5096 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8236, 37absrpcld 15160 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ+)
8382relogcld 25778 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
8483recnd 11003 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
8584abscld 15148 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
8685, 51readdcld 11004 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ∈ ℝ)
87 abslogle 25773 . . . . . . . 8 ((((𝑦 / 𝑛) + 1) ∈ ℂ ∧ ((𝑦 / 𝑛) + 1) ≠ 0) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π))
8836, 37, 87syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π))
89 1rp 12734 . . . . . . . . . . . 12 1 ∈ ℝ+
90 relogdiv 25748 . . . . . . . . . . . 12 ((1 ∈ ℝ+ ∧ ((𝑅 + 1) · 𝑛) ∈ ℝ+) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
9189, 48, 90sylancr 587 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) = ((log‘1) − (log‘((𝑅 + 1) · 𝑛))))
92 log1 25741 . . . . . . . . . . . . 13 (log‘1) = 0
9392oveq1i 7285 . . . . . . . . . . . 12 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = (0 − (log‘((𝑅 + 1) · 𝑛)))
94 df-neg 11208 . . . . . . . . . . . 12 -(log‘((𝑅 + 1) · 𝑛)) = (0 − (log‘((𝑅 + 1) · 𝑛)))
9593, 94eqtr4i 2769 . . . . . . . . . . 11 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = -(log‘((𝑅 + 1) · 𝑛))
9691, 95eqtr2di 2795 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) = (log‘(1 / ((𝑅 + 1) · 𝑛))))
9746nnrecred 12024 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ∈ ℝ)
9823, 32addcld 10994 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 + 𝑛) ∈ ℂ)
9998abscld 15148 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 + 𝑛)) ∈ ℝ)
1004nnrecred 12024 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / 𝑅) ∈ ℝ)
1014nnrpd 12770 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ+)
102 0le1 11498 . . . . . . . . . . . . . . . 16 0 ≤ 1
103102a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 1)
10444lep1d 11906 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ≤ (𝑅 + 1))
105101, 47, 61, 103, 104lediv2ad 12794 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (1 / 𝑅))
106 oveq2 7283 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑦 + 𝑘) = (𝑦 + 𝑛))
107106fveq2d 6778 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (abs‘(𝑦 + 𝑘)) = (abs‘(𝑦 + 𝑛)))
108107breq2d 5086 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛))))
10978simprd 496 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))
11024nnnn0d 12293 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℕ0)
111108, 109, 110rspcdva 3562 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛)))
11297, 100, 99, 105, 111letrd 11132 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (abs‘(𝑦 + 𝑛)))
11397, 99, 27, 112lediv1dd 12830 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) ≤ ((abs‘(𝑦 + 𝑛)) / 𝑛))
11446nncnd 11989 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℂ)
11546nnne0d 12023 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≠ 0)
116114, 32, 115, 33recdiv2d 11769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) = (1 / ((𝑅 + 1) · 𝑛)))
11723, 32, 32, 33divdird 11789 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 + 𝑛) / 𝑛) = ((𝑦 / 𝑛) + (𝑛 / 𝑛)))
11832, 33dividd 11749 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 / 𝑛) = 1)
119118oveq2d 7291 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + (𝑛 / 𝑛)) = ((𝑦 / 𝑛) + 1))
120117, 119eqtr2d 2779 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) = ((𝑦 + 𝑛) / 𝑛))
121120fveq2d 6778 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) = (abs‘((𝑦 + 𝑛) / 𝑛)))
12298, 32, 33absdivd 15167 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 + 𝑛) / 𝑛)) = ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)))
12327rpge0d 12776 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 𝑛)
12458, 123absidd 15134 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑛) = 𝑛)
125124oveq2d 7291 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)) = ((abs‘(𝑦 + 𝑛)) / 𝑛))
126121, 122, 1253eqtrrd 2783 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / 𝑛) = (abs‘((𝑦 / 𝑛) + 1)))
127113, 116, 1263brtr3d 5105 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)))
12848rpreccld 12782 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ∈ ℝ+)
129128, 82logled 25782 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)) ↔ (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1)))))
130127, 129mpbid 231 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13196, 130eqbrtrd 5096 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13236abscld 15148 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ)
13344, 61readdcld 11004 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ)
13448rpred 12772 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ)
13534abscld 15148 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ∈ ℝ)
136135, 61readdcld 11004 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ∈ ℝ)
13734, 35abstrid 15168 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + (abs‘1)))
138 abs1 15009 . . . . . . . . . . . . . 14 (abs‘1) = 1
139138oveq2i 7286 . . . . . . . . . . . . 13 ((abs‘(𝑦 / 𝑛)) + (abs‘1)) = ((abs‘(𝑦 / 𝑛)) + 1)
140137, 139breqtrdi 5115 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + 1))
14189a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ+)
14223absge0d 15156 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (abs‘𝑦))
14324nnge1d 12021 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ 𝑛)
14469, 44, 141, 58, 142, 79, 143lediv12ad 12831 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) ≤ (𝑅 / 1))
14523, 32, 33absdivd 15167 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) = ((abs‘𝑦) / (abs‘𝑛)))
146124oveq2d 7291 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / (abs‘𝑛)) = ((abs‘𝑦) / 𝑛))
147145, 146eqtr2d 2779 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) = (abs‘(𝑦 / 𝑛)))
1484nncnd 11989 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℂ)
149148div1d 11743 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 / 1) = 𝑅)
150144, 147, 1493brtr3d 5105 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ≤ 𝑅)
151135, 44, 61, 150leadd1dd 11589 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ≤ (𝑅 + 1))
152132, 136, 133, 140, 151letrd 11132 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ (𝑅 + 1))
15347rpge0d 12776 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (𝑅 + 1))
154133, 58, 153, 143lemulge11d 11912 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≤ ((𝑅 + 1) · 𝑛))
155132, 133, 134, 152, 154letrd 11132 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛))
15682, 48logled 25782 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛) ↔ (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛))))
157155, 156mpbid 231 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛)))
15883, 49absled 15142 . . . . . . . . 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)))))
159131, 157, 158mpbir2and 710 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ≤ (log‘((𝑅 + 1) · 𝑛)))
16085, 49, 51, 159leadd1dd 11589 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16142, 86, 52, 88, 160letrd 11132 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16241, 42, 45, 52, 81, 161le2addd 11594 . . . . 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)))
16340, 43, 53, 54, 162letrd 11132 . . . 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)))
164163adantr 481 . . 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)))
1651, 2, 19, 164ifbothda 4497 . 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))))
166 oveq1 7282 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
167 id 22 . . . . . . . . . . . 12 (𝑚 = 𝑛𝑚 = 𝑛)
168166, 167oveq12d 7293 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
169168fveq2d 6778 . . . . . . . . . 10 (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
170169oveq2d 7291 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
171 oveq2 7283 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
172171fvoveq1d 7297 . . . . . . . . 9 (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
173170, 172oveq12d 7293 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
174173mpteq2dv 5176 . . . . . . 7 (𝑚 = 𝑛 → (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
175 lgamgulm.g . . . . . . 7 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
176 cnex 10952 . . . . . . . . 9 ℂ ∈ V
1776, 176rabex2 5258 . . . . . . . 8 𝑈 ∈ V
178177mptex 7099 . . . . . . 7 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) ∈ V
179174, 175, 178fvmpt 6875 . . . . . 6 (𝑛 ∈ ℕ → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
180179ad2antrl 725 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
181180fveq1d 6776 . . . 4 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦))
182 oveq1 7282 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) = (𝑦 · (log‘((𝑛 + 1) / 𝑛))))
183 oveq1 7282 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧 / 𝑛) = (𝑦 / 𝑛))
184183fvoveq1d 7297 . . . . . . 7 (𝑧 = 𝑦 → (log‘((𝑧 / 𝑛) + 1)) = (log‘((𝑦 / 𝑛) + 1)))
185182, 184oveq12d 7293 . . . . . 6 (𝑧 = 𝑦 → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
186 eqid 2738 . . . . . 6 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
187 ovex 7308 . . . . . 6 ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ V
188185, 186, 187fvmpt 6875 . . . . 5 (𝑦𝑈 → ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
189188ad2antll 726 . . . 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))))
190181, 189eqtrd 2778 . . 3 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
191190fveq2d 6778 . 2 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) = (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))))
192 breq2 5078 . . . . 5 (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
193 oveq1 7282 . . . . . . 7 (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
194193oveq2d 7291 . . . . . 6 (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
195194oveq2d 7291 . . . . 5 (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
196169oveq2d 7291 . . . . . 6 (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
197 oveq2 7283 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
198197fveq2d 6778 . . . . . . 7 (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
199198oveq1d 7290 . . . . . 6 (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
200196, 199oveq12d 7293 . . . . 5 (𝑚 = 𝑛 → ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π)) = ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
201192, 195, 200ifbieq12d 4487 . . . 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))))
202 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))))
203 ovex 7308 . . . . 5 (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
204 ovex 7308 . . . . 5 ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
205203, 204ifex 4509 . . . 4 if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) ∈ V
206201, 202, 205fvmpt 6875 . . 3 (𝑛 ∈ ℕ → (𝑇𝑛) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
207206ad2antrl 725 . 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))))
208165, 191, 2073brtr4d 5106 1 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) ≤ (𝑇𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  {crab 3068  cdif 3884  wss 3887  ifcif 4459   class class class wbr 5074  cmpt 5157  cfv 6433  (class class class)co 7275  cc 10869  cr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876  cle 11010  cmin 11205  -cneg 11206   / cdiv 11632  cn 11973  2c2 12028  0cn0 12233  cz 12319  +crp 12730  cexp 13782  abscabs 14945  πcpi 15776  logclog 25710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ioc 13084  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-fac 13988  df-bc 14017  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-sum 15398  df-ef 15777  df-sin 15779  df-cos 15780  df-tan 15781  df-pi 15782  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lp 22287  df-perf 22288  df-cn 22378  df-cnp 22379  df-haus 22466  df-cmp 22538  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041  df-limc 25030  df-dv 25031  df-log 25712
This theorem is referenced by:  lgamgulmlem6  26183
  Copyright terms: Public domain W3C validator