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Theorem lgamgulmlem5 25537
Description: Lemma for lgamgulm 25539. (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 5061 . . 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 5061 . . 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 6663 . . . . . . . 8 (𝑥 = 𝑡 → (abs‘𝑥) = (abs‘𝑡))
87breq1d 5067 . . . . . . 7 (𝑥 = 𝑡 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑡) ≤ 𝑅))
9 fvoveq1 7168 . . . . . . . . 9 (𝑥 = 𝑡 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑡 + 𝑘)))
109breq2d 5069 . . . . . . . 8 (𝑥 = 𝑡 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
1110ralbidv 3194 . . . . . . 7 (𝑥 = 𝑡 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
128, 11anbi12d 630 . . . . . 6 (𝑥 = 𝑡 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))))
1312cbvrabv 3489 . . . . 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 2841 . . . 4 𝑈 = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))}
15 simplrl 773 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑛 ∈ ℕ)
16 simprr 769 . . . . 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 25535 . . 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 25533 . . . . . . . . . . 11 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2120adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2221, 16sseldd 3965 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
2322eldifad 3945 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ ℂ)
24 simprl 767 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℕ)
2524peano2nnd 11643 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℕ)
2625nnrpd 12417 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ+)
2724nnrpd 12417 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ+)
2826, 27rpdivcld 12436 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ+)
2928relogcld 25133 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
3029recnd 10657 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
3123, 30mulcld 10649 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
3224nncnd 11642 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℂ)
3324nnne0d 11675 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≠ 0)
3423, 32, 33divcld 11404 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 / 𝑛) ∈ ℂ)
35 1cnd 10624 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℂ)
3634, 35addcld 10648 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ∈ ℂ)
3722, 24dmgmdivn0 25532 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ≠ 0)
3836, 37logcld 25081 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑦 / 𝑛) + 1)) ∈ ℂ)
3931, 38subcld 10985 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
4039abscld 14784 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
4131abscld 14784 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ∈ ℝ)
4238abscld 14784 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
4341, 42readdcld 10658 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
444nnred 11641 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ)
4544, 29remulcld 10659 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
464peano2nnd 11643 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℕ)
4746nnrpd 12417 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ+)
4847, 27rpmulcld 12435 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ+)
4948relogcld 25133 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
50 pire 24971 . . . . . . . 8 π ∈ ℝ
5150a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → π ∈ ℝ)
5249, 51readdcld 10658 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
5345, 52readdcld 10658 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
5431, 38abs2dif2d 14806 . . . . 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 14802 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))))
5628rpred 12419 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ)
5732mulid2d 10647 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) = 𝑛)
5824nnred 11641 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ)
5958lep1d 11559 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≤ (𝑛 + 1))
6057, 59eqbrtrd 5079 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) ≤ (𝑛 + 1))
61 1red 10630 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ)
6258, 61readdcld 10658 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ)
6361, 62, 27lemuldivd 12468 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 · 𝑛) ≤ (𝑛 + 1) ↔ 1 ≤ ((𝑛 + 1) / 𝑛)))
6460, 63mpbid 233 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ ((𝑛 + 1) / 𝑛))
6556, 64logge0d 25140 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (log‘((𝑛 + 1) / 𝑛)))
6629, 65absidd 14770 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑛 + 1) / 𝑛))) = (log‘((𝑛 + 1) / 𝑛)))
6766oveq2d 7161 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
6855, 67eqtrd 2853 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
6923abscld 14784 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑦) ∈ ℝ)
70 fveq2 6663 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (abs‘𝑥) = (abs‘𝑦))
7170breq1d 5067 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
72 fvoveq1 7168 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑦 + 𝑘)))
7372breq2d 5069 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7473ralbidv 3194 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7571, 74anbi12d 630 . . . . . . . . . . . 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 3680 . . . . . . . . . . 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 725 . . . . . . . . 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 11567 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8168, 80eqbrtrd 5079 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8236, 37absrpcld 14796 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ+)
8382relogcld 25133 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
8483recnd 10657 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
8584abscld 14784 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
8685, 51readdcld 10658 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ∈ ℝ)
87 abslogle 25128 . . . . . . . 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 12381 . . . . . . . . . . . 12 1 ∈ ℝ+
90 relogdiv 25103 . . . . . . . . . . . 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 25096 . . . . . . . . . . . . 13 (log‘1) = 0
9392oveq1i 7155 . . . . . . . . . . . 12 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = (0 − (log‘((𝑅 + 1) · 𝑛)))
94 df-neg 10861 . . . . . . . . . . . 12 -(log‘((𝑅 + 1) · 𝑛)) = (0 − (log‘((𝑅 + 1) · 𝑛)))
9593, 94eqtr4i 2844 . . . . . . . . . . 11 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = -(log‘((𝑅 + 1) · 𝑛))
9691, 95syl6req 2870 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) = (log‘(1 / ((𝑅 + 1) · 𝑛))))
9746nnrecred 11676 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ∈ ℝ)
9823, 32addcld 10648 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 + 𝑛) ∈ ℂ)
9998abscld 14784 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 + 𝑛)) ∈ ℝ)
1004nnrecred 11676 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / 𝑅) ∈ ℝ)
1014nnrpd 12417 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ+)
102 0le1 11151 . . . . . . . . . . . . . . . 16 0 ≤ 1
103102a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 1)
10444lep1d 11559 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ≤ (𝑅 + 1))
105101, 47, 61, 103, 104lediv2ad 12441 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (1 / 𝑅))
106 oveq2 7153 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑦 + 𝑘) = (𝑦 + 𝑛))
107106fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (abs‘(𝑦 + 𝑘)) = (abs‘(𝑦 + 𝑛)))
108107breq2d 5069 . . . . . . . . . . . . . . 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 11943 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℕ0)
111108, 109, 110rspcdva 3622 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛)))
11297, 100, 99, 105, 111letrd 10785 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (abs‘(𝑦 + 𝑛)))
11397, 99, 27, 112lediv1dd 12477 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) ≤ ((abs‘(𝑦 + 𝑛)) / 𝑛))
11446nncnd 11642 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℂ)
11546nnne0d 11675 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≠ 0)
116114, 32, 115, 33recdiv2d 11422 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) = (1 / ((𝑅 + 1) · 𝑛)))
11723, 32, 32, 33divdird 11442 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 + 𝑛) / 𝑛) = ((𝑦 / 𝑛) + (𝑛 / 𝑛)))
11832, 33dividd 11402 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 / 𝑛) = 1)
119118oveq2d 7161 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + (𝑛 / 𝑛)) = ((𝑦 / 𝑛) + 1))
120117, 119eqtr2d 2854 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) = ((𝑦 + 𝑛) / 𝑛))
121120fveq2d 6667 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) = (abs‘((𝑦 + 𝑛) / 𝑛)))
12298, 32, 33absdivd 14803 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 + 𝑛) / 𝑛)) = ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)))
12327rpge0d 12423 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 𝑛)
12458, 123absidd 14770 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑛) = 𝑛)
125124oveq2d 7161 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)) = ((abs‘(𝑦 + 𝑛)) / 𝑛))
126121, 122, 1253eqtrrd 2858 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / 𝑛) = (abs‘((𝑦 / 𝑛) + 1)))
127113, 116, 1263brtr3d 5088 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)))
12848rpreccld 12429 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ∈ ℝ+)
129128, 82logled 25137 . . . . . . . . . . 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 233 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13196, 130eqbrtrd 5079 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13236abscld 14784 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ)
13344, 61readdcld 10658 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ)
13448rpred 12419 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ)
13534abscld 14784 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ∈ ℝ)
136135, 61readdcld 10658 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ∈ ℝ)
13734, 35abstrid 14804 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + (abs‘1)))
138 abs1 14645 . . . . . . . . . . . . . 14 (abs‘1) = 1
139138oveq2i 7156 . . . . . . . . . . . . 13 ((abs‘(𝑦 / 𝑛)) + (abs‘1)) = ((abs‘(𝑦 / 𝑛)) + 1)
140137, 139breqtrdi 5098 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + 1))
14189a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ+)
14223absge0d 14792 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (abs‘𝑦))
14324nnge1d 11673 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ 𝑛)
14469, 44, 141, 58, 142, 79, 143lediv12ad 12478 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) ≤ (𝑅 / 1))
14523, 32, 33absdivd 14803 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) = ((abs‘𝑦) / (abs‘𝑛)))
146124oveq2d 7161 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / (abs‘𝑛)) = ((abs‘𝑦) / 𝑛))
147145, 146eqtr2d 2854 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) = (abs‘(𝑦 / 𝑛)))
1484nncnd 11642 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℂ)
149148div1d 11396 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 / 1) = 𝑅)
150144, 147, 1493brtr3d 5088 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ≤ 𝑅)
151135, 44, 61, 150leadd1dd 11242 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ≤ (𝑅 + 1))
152132, 136, 133, 140, 151letrd 10785 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ (𝑅 + 1))
15347rpge0d 12423 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (𝑅 + 1))
154133, 58, 153, 143lemulge11d 11565 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≤ ((𝑅 + 1) · 𝑛))
155132, 133, 134, 152, 154letrd 10785 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛))
15682, 48logled 25137 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛) ↔ (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛))))
157155, 156mpbid 233 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛)))
15883, 49absled 14778 . . . . . . . . 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 709 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ≤ (log‘((𝑅 + 1) · 𝑛)))
16085, 49, 51, 159leadd1dd 11242 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16142, 86, 52, 88, 160letrd 10785 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16241, 42, 45, 52, 81, 161le2addd 11247 . . . . 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 10785 . . . 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 4500 . 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 7152 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
167 id 22 . . . . . . . . . . . 12 (𝑚 = 𝑛𝑚 = 𝑛)
168166, 167oveq12d 7163 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
169168fveq2d 6667 . . . . . . . . . 10 (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
170169oveq2d 7161 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
171 oveq2 7153 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
172171fvoveq1d 7167 . . . . . . . . 9 (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
173170, 172oveq12d 7163 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
174173mpteq2dv 5153 . . . . . . 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 10606 . . . . . . . . 9 ℂ ∈ V
1776, 176rabex2 5228 . . . . . . . 8 𝑈 ∈ V
178177mptex 6977 . . . . . . 7 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) ∈ V
179174, 175, 178fvmpt 6761 . . . . . 6 (𝑛 ∈ ℕ → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
180179ad2antrl 724 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
181180fveq1d 6665 . . . 4 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦))
182 oveq1 7152 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) = (𝑦 · (log‘((𝑛 + 1) / 𝑛))))
183 oveq1 7152 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧 / 𝑛) = (𝑦 / 𝑛))
184183fvoveq1d 7167 . . . . . . 7 (𝑧 = 𝑦 → (log‘((𝑧 / 𝑛) + 1)) = (log‘((𝑦 / 𝑛) + 1)))
185182, 184oveq12d 7163 . . . . . 6 (𝑧 = 𝑦 → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
186 eqid 2818 . . . . . 6 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
187 ovex 7178 . . . . . 6 ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ V
188185, 186, 187fvmpt 6761 . . . . 5 (𝑦𝑈 → ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
189188ad2antll 725 . . . 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 2853 . . 3 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
191190fveq2d 6667 . 2 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) = (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))))
192 breq2 5061 . . . . 5 (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
193 oveq1 7152 . . . . . . 7 (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
194193oveq2d 7161 . . . . . 6 (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
195194oveq2d 7161 . . . . 5 (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
196169oveq2d 7161 . . . . . 6 (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
197 oveq2 7153 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
198197fveq2d 6667 . . . . . . 7 (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
199198oveq1d 7160 . . . . . 6 (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
200196, 199oveq12d 7163 . . . . 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 4490 . . . 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 7178 . . . . 5 (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
204 ovex 7178 . . . . 5 ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
205203, 204ifex 4511 . . . 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 6761 . . 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 724 . 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 5089 1 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) ≤ (𝑇𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  {crab 3139  cdif 3930  wss 3933  ifcif 4463   class class class wbr 5057  cmpt 5137  cfv 6348  (class class class)co 7145  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  cle 10664  cmin 10858  -cneg 10859   / cdiv 11285  cn 11626  2c2 11680  0cn0 11885  cz 11969  +crp 12377  cexp 13417  abscabs 14581  πcpi 15408  logclog 25065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-bc 13651  df-hash 13679  df-shft 14414  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-limsup 14816  df-clim 14833  df-rlim 14834  df-sum 15031  df-ef 15409  df-sin 15411  df-cos 15412  df-tan 15413  df-pi 15414  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-xrs 16763  df-qtop 16768  df-imas 16769  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-mulg 18163  df-cntz 18385  df-cmn 18837  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-lp 21672  df-perf 21673  df-cn 21763  df-cnp 21764  df-haus 21851  df-cmp 21923  df-tx 22098  df-hmeo 22291  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-xms 22857  df-ms 22858  df-tms 22859  df-cncf 23413  df-limc 24391  df-dv 24392  df-log 25067
This theorem is referenced by:  lgamgulmlem6  25538
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