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Theorem lgamgulmlem5 27077
Description: Lemma for lgamgulm 27079. (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 5146 . . 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 5146 . . 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 6905 . . . . . . . 8 (𝑥 = 𝑡 → (abs‘𝑥) = (abs‘𝑡))
87breq1d 5152 . . . . . . 7 (𝑥 = 𝑡 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑡) ≤ 𝑅))
9 fvoveq1 7455 . . . . . . . . 9 (𝑥 = 𝑡 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑡 + 𝑘)))
109breq2d 5154 . . . . . . . 8 (𝑥 = 𝑡 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
1110ralbidv 3177 . . . . . . 7 (𝑥 = 𝑡 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘))))
128, 11anbi12d 632 . . . . . 6 (𝑥 = 𝑡 → (((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))))
1312cbvrabv 3446 . . . . 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 2764 . . . 4 𝑈 = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑡 + 𝑘)))}
15 simplrl 776 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑛 ∈ ℕ)
16 simprr 772 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦𝑈)
1716adantr 480 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → 𝑦𝑈)
18 simpr 484 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) ∧ (2 · 𝑅) ≤ 𝑛) → (2 · 𝑅) ≤ 𝑛)
195, 14, 15, 17, 18lgamgulmlem3 27075 . . 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 27073 . . . . . . . . . . 11 (𝜑𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2120adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
2221, 16sseldd 3983 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
2322eldifad 3962 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑦 ∈ ℂ)
24 simprl 770 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℕ)
2524peano2nnd 12284 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℕ)
2625nnrpd 13076 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ+)
2724nnrpd 13076 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ+)
2826, 27rpdivcld 13095 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ+)
2928relogcld 26666 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
3029recnd 11290 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
3123, 30mulcld 11282 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
3224nncnd 12283 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℂ)
3324nnne0d 12317 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≠ 0)
3423, 32, 33divcld 12044 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 / 𝑛) ∈ ℂ)
35 1cnd 11257 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℂ)
3634, 35addcld 11281 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ∈ ℂ)
3722, 24dmgmdivn0 27072 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) ≠ 0)
3836, 37logcld 26613 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑦 / 𝑛) + 1)) ∈ ℂ)
3931, 38subcld 11621 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
4039abscld 15476 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
4131abscld 15476 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ∈ ℝ)
4238abscld 15476 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
4341, 42readdcld 11291 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) + (abs‘(log‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
444nnred 12282 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ)
4544, 29remulcld 11292 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
464peano2nnd 12284 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℕ)
4746nnrpd 13076 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ+)
4847, 27rpmulcld 13094 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ+)
4948relogcld 26666 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
50 pire 26501 . . . . . . . 8 π ∈ ℝ
5150a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → π ∈ ℝ)
5249, 51readdcld 11291 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
5345, 52readdcld 11291 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
5431, 38abs2dif2d 15498 . . . . 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 15494 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))))
5628rpred 13078 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑛 + 1) / 𝑛) ∈ ℝ)
5732mullidd 11280 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) = 𝑛)
5824nnred 12282 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ∈ ℝ)
5958lep1d 12200 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑛 ≤ (𝑛 + 1))
6057, 59eqbrtrd 5164 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 · 𝑛) ≤ (𝑛 + 1))
61 1red 11263 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ)
6258, 61readdcld 11291 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 + 1) ∈ ℝ)
6361, 62, 27lemuldivd 13127 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 · 𝑛) ≤ (𝑛 + 1) ↔ 1 ≤ ((𝑛 + 1) / 𝑛)))
6460, 63mpbid 232 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ ((𝑛 + 1) / 𝑛))
6556, 64logge0d 26673 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (log‘((𝑛 + 1) / 𝑛)))
6629, 65absidd 15462 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑛 + 1) / 𝑛))) = (log‘((𝑛 + 1) / 𝑛)))
6766oveq2d 7448 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (abs‘(log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
6855, 67eqtrd 2776 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) = ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))))
6923abscld 15476 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑦) ∈ ℝ)
70 fveq2 6905 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (abs‘𝑥) = (abs‘𝑦))
7170breq1d 5152 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((abs‘𝑥) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
72 fvoveq1 7455 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑦 + 𝑘)))
7372breq2d 5154 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7473ralbidv 3177 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7571, 74anbi12d 632 . . . . . . . . . . . 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 3694 . . . . . . . . . . 11 (𝑦𝑈 ↔ (𝑦 ∈ ℂ ∧ ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))))
7776simprbi 496 . . . . . . . . . 10 (𝑦𝑈 → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7877ad2antll 729 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘))))
7978simpld 494 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑦) ≤ 𝑅)
8069, 44, 29, 65, 79lemul1ad 12208 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) · (log‘((𝑛 + 1) / 𝑛))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8168, 80eqbrtrd 5164 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 · (log‘((𝑛 + 1) / 𝑛)))) ≤ (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
8236, 37absrpcld 15488 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ+)
8382relogcld 26666 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℝ)
8483recnd 11290 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ∈ ℂ)
8584abscld 15476 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ∈ ℝ)
8685, 51readdcld 11291 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ∈ ℝ)
87 abslogle 26661 . . . . . . . 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 13039 . . . . . . . . . . . 12 1 ∈ ℝ+
90 relogdiv 26636 . . . . . . . . . . . 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 26628 . . . . . . . . . . . . 13 (log‘1) = 0
9392oveq1i 7442 . . . . . . . . . . . 12 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = (0 − (log‘((𝑅 + 1) · 𝑛)))
94 df-neg 11496 . . . . . . . . . . . 12 -(log‘((𝑅 + 1) · 𝑛)) = (0 − (log‘((𝑅 + 1) · 𝑛)))
9593, 94eqtr4i 2767 . . . . . . . . . . 11 ((log‘1) − (log‘((𝑅 + 1) · 𝑛))) = -(log‘((𝑅 + 1) · 𝑛))
9691, 95eqtr2di 2793 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) = (log‘(1 / ((𝑅 + 1) · 𝑛))))
9746nnrecred 12318 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ∈ ℝ)
9823, 32addcld 11281 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑦 + 𝑛) ∈ ℂ)
9998abscld 15476 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 + 𝑛)) ∈ ℝ)
1004nnrecred 12318 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / 𝑅) ∈ ℝ)
1014nnrpd 13076 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℝ+)
102 0le1 11787 . . . . . . . . . . . . . . . 16 0 ≤ 1
103102a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 1)
10444lep1d 12200 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ≤ (𝑅 + 1))
105101, 47, 61, 103, 104lediv2ad 13100 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (1 / 𝑅))
106 oveq2 7440 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑦 + 𝑘) = (𝑦 + 𝑛))
107106fveq2d 6909 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (abs‘(𝑦 + 𝑘)) = (abs‘(𝑦 + 𝑛)))
108107breq2d 5154 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)) ↔ (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑛))))
10978simprd 495 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑦 + 𝑘)))
11024nnnn0d 12589 . . . . . . . . . . . . . . 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 11419 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / (𝑅 + 1)) ≤ (abs‘(𝑦 + 𝑛)))
11397, 99, 27, 112lediv1dd 13136 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) ≤ ((abs‘(𝑦 + 𝑛)) / 𝑛))
11446nncnd 12283 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℂ)
11546nnne0d 12317 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≠ 0)
116114, 32, 115, 33recdiv2d 12062 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((1 / (𝑅 + 1)) / 𝑛) = (1 / ((𝑅 + 1) · 𝑛)))
11723, 32, 32, 33divdird 12082 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 + 𝑛) / 𝑛) = ((𝑦 / 𝑛) + (𝑛 / 𝑛)))
11832, 33dividd 12042 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑛 / 𝑛) = 1)
119118oveq2d 7448 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + (𝑛 / 𝑛)) = ((𝑦 / 𝑛) + 1))
120117, 119eqtr2d 2777 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑦 / 𝑛) + 1) = ((𝑦 + 𝑛) / 𝑛))
121120fveq2d 6909 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) = (abs‘((𝑦 + 𝑛) / 𝑛)))
12298, 32, 33absdivd 15495 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 + 𝑛) / 𝑛)) = ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)))
12327rpge0d 13082 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ 𝑛)
12458, 123absidd 15462 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘𝑛) = 𝑛)
125124oveq2d 7448 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / (abs‘𝑛)) = ((abs‘(𝑦 + 𝑛)) / 𝑛))
126121, 122, 1253eqtrrd 2781 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 + 𝑛)) / 𝑛) = (abs‘((𝑦 / 𝑛) + 1)))
127113, 116, 1263brtr3d 5173 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ≤ (abs‘((𝑦 / 𝑛) + 1)))
12848rpreccld 13088 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (1 / ((𝑅 + 1) · 𝑛)) ∈ ℝ+)
129128, 82logled 26670 . . . . . . . . . . 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 232 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(1 / ((𝑅 + 1) · 𝑛))) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13196, 130eqbrtrd 5164 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → -(log‘((𝑅 + 1) · 𝑛)) ≤ (log‘(abs‘((𝑦 / 𝑛) + 1))))
13236abscld 15476 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ∈ ℝ)
13344, 61readdcld 11291 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ∈ ℝ)
13448rpred 13078 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝑅 + 1) · 𝑛) ∈ ℝ)
13534abscld 15476 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ∈ ℝ)
136135, 61readdcld 11291 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ∈ ℝ)
13734, 35abstrid 15496 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + (abs‘1)))
138 abs1 15337 . . . . . . . . . . . . . 14 (abs‘1) = 1
139138oveq2i 7443 . . . . . . . . . . . . 13 ((abs‘(𝑦 / 𝑛)) + (abs‘1)) = ((abs‘(𝑦 / 𝑛)) + 1)
140137, 139breqtrdi 5183 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((abs‘(𝑦 / 𝑛)) + 1))
14189a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ∈ ℝ+)
14223absge0d 15484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (abs‘𝑦))
14324nnge1d 12315 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 1 ≤ 𝑛)
14469, 44, 141, 58, 142, 79, 143lediv12ad 13137 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) ≤ (𝑅 / 1))
14523, 32, 33absdivd 15495 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) = ((abs‘𝑦) / (abs‘𝑛)))
146124oveq2d 7448 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / (abs‘𝑛)) = ((abs‘𝑦) / 𝑛))
147145, 146eqtr2d 2777 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘𝑦) / 𝑛) = (abs‘(𝑦 / 𝑛)))
1484nncnd 12283 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 𝑅 ∈ ℂ)
149148div1d 12036 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 / 1) = 𝑅)
150144, 147, 1493brtr3d 5173 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(𝑦 / 𝑛)) ≤ 𝑅)
151135, 44, 61, 150leadd1dd 11878 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(𝑦 / 𝑛)) + 1) ≤ (𝑅 + 1))
152132, 136, 133, 140, 151letrd 11419 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ (𝑅 + 1))
15347rpge0d 13082 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → 0 ≤ (𝑅 + 1))
154133, 58, 153, 143lemulge11d 12206 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝑅 + 1) ≤ ((𝑅 + 1) · 𝑛))
155132, 133, 134, 152, 154letrd 11419 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛))
15682, 48logled 26670 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘((𝑦 / 𝑛) + 1)) ≤ ((𝑅 + 1) · 𝑛) ↔ (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛))))
157155, 156mpbid 232 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (log‘(abs‘((𝑦 / 𝑛) + 1))) ≤ (log‘((𝑅 + 1) · 𝑛)))
15883, 49absled 15470 . . . . . . . . 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 713 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) ≤ (log‘((𝑅 + 1) · 𝑛)))
16085, 49, 51, 159leadd1dd 11878 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((abs‘(log‘(abs‘((𝑦 / 𝑛) + 1)))) + π) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16142, 86, 52, 88, 160letrd 11419 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘(log‘((𝑦 / 𝑛) + 1))) ≤ ((log‘((𝑅 + 1) · 𝑛)) + π))
16241, 42, 45, 52, 81, 161le2addd 11883 . . . . 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 11419 . . . 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 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)))
1651, 2, 19, 164ifbothda 4563 . 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 7439 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
167 id 22 . . . . . . . . . . . 12 (𝑚 = 𝑛𝑚 = 𝑛)
168166, 167oveq12d 7450 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
169168fveq2d 6909 . . . . . . . . . 10 (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
170169oveq2d 7448 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
171 oveq2 7440 . . . . . . . . . 10 (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
172171fvoveq1d 7454 . . . . . . . . 9 (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
173170, 172oveq12d 7450 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
174173mpteq2dv 5243 . . . . . . 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 11237 . . . . . . . . 9 ℂ ∈ V
1776, 176rabex2 5340 . . . . . . . 8 𝑈 ∈ V
178177mptex 7244 . . . . . . 7 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) ∈ V
179174, 175, 178fvmpt 7015 . . . . . 6 (𝑛 ∈ ℕ → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
180179ad2antrl 728 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (𝐺𝑛) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))))
181180fveq1d 6907 . . . 4 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦))
182 oveq1 7439 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) = (𝑦 · (log‘((𝑛 + 1) / 𝑛))))
183 oveq1 7439 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧 / 𝑛) = (𝑦 / 𝑛))
184183fvoveq1d 7454 . . . . . . 7 (𝑧 = 𝑦 → (log‘((𝑧 / 𝑛) + 1)) = (log‘((𝑦 / 𝑛) + 1)))
185182, 184oveq12d 7450 . . . . . 6 (𝑧 = 𝑦 → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
186 eqid 2736 . . . . . 6 (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1)))) = (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
187 ovex 7465 . . . . . 6 ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))) ∈ V
188185, 186, 187fvmpt 7015 . . . . 5 (𝑦𝑈 → ((𝑧𝑈 ↦ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
189188ad2antll 729 . . . 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 2776 . . 3 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → ((𝐺𝑛)‘𝑦) = ((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1))))
191190fveq2d 6909 . 2 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) = (abs‘((𝑦 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑦 / 𝑛) + 1)))))
192 breq2 5146 . . . . 5 (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
193 oveq1 7439 . . . . . . 7 (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
194193oveq2d 7448 . . . . . 6 (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
195194oveq2d 7448 . . . . 5 (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
196169oveq2d 7448 . . . . . 6 (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
197 oveq2 7440 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
198197fveq2d 6909 . . . . . . 7 (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
199198oveq1d 7447 . . . . . 6 (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
200196, 199oveq12d 7450 . . . . 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 4553 . . . 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 7465 . . . . 5 (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
204 ovex 7465 . . . . 5 ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
205203, 204ifex 4575 . . . 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 7015 . . 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 728 . 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 5174 1 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) ≤ (𝑇𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wcel 2107  wne 2939  wral 3060  {crab 3435  cdif 3947  wss 3950  ifcif 4524   class class class wbr 5142  cmpt 5224  cfv 6560  (class class class)co 7432  cc 11154  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161  cle 11297  cmin 11493  -cneg 11494   / cdiv 11921  cn 12267  2c2 12322  0cn0 12528  cz 12615  +crp 13035  cexp 14103  abscabs 15274  πcpi 16103  logclog 26597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234  ax-addf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-fi 9452  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ioo 13392  df-ioc 13393  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-fl 13833  df-mod 13911  df-seq 14044  df-exp 14104  df-fac 14314  df-bc 14343  df-hash 14371  df-shft 15107  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-limsup 15508  df-clim 15525  df-rlim 15526  df-sum 15724  df-ef 16104  df-sin 16106  df-cos 16107  df-tan 16108  df-pi 16109  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-submnd 18798  df-mulg 19087  df-cntz 19336  df-cmn 19801  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-fbas 21362  df-fg 21363  df-cnfld 21366  df-top 22901  df-topon 22918  df-topsp 22940  df-bases 22954  df-cld 23028  df-ntr 23029  df-cls 23030  df-nei 23107  df-lp 23145  df-perf 23146  df-cn 23236  df-cnp 23237  df-haus 23324  df-cmp 23396  df-tx 23571  df-hmeo 23764  df-fil 23855  df-fm 23947  df-flim 23948  df-flf 23949  df-xms 24331  df-ms 24332  df-tms 24333  df-cncf 24905  df-limc 25902  df-dv 25903  df-log 26599
This theorem is referenced by:  lgamgulmlem6  27078
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