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Theorem eulerpartlemgc 30552
 Description: Lemma for eulerpart 30572. (Contributed by Thierry Arnoux, 9-Aug-2018.)
Hypotheses
Ref Expression
eulerpartlems.r 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
eulerpartlems.s 𝑆 = (𝑓 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))
Assertion
Ref Expression
eulerpartlemgc ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆𝐴))
Distinct variable groups:   𝑓,𝑘,𝐴   𝑅,𝑓,𝑘   𝑡,𝑘,𝐴   𝑡,𝑅   𝑡,𝑆,𝑘
Allowed substitution hints:   𝐴(𝑛)   𝑅(𝑛)   𝑆(𝑓,𝑛)

Proof of Theorem eulerpartlemgc
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 2re 11128 . . . . 5 2 ∈ ℝ
21a1i 11 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 2 ∈ ℝ)
3 bitsss 15195 . . . . 5 (bits‘(𝐴𝑡)) ⊆ ℕ0
4 simprr 811 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ (bits‘(𝐴𝑡)))
53, 4sseldi 3634 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ ℕ0)
62, 5reexpcld 13065 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℝ)
7 simprl 809 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℕ)
87nnred 11073 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ)
96, 8remulcld 10108 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ∈ ℝ)
10 eulerpartlems.r . . . . . . . 8 𝑅 = {𝑓 ∣ (𝑓 “ ℕ) ∈ Fin}
11 eulerpartlems.s . . . . . . . 8 𝑆 = (𝑓 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘))
1210, 11eulerpartlemelr 30547 . . . . . . 7 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
1312simpld 474 . . . . . 6 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
1413ffvelrnda 6399 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℕ0)
1514adantrr 753 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℕ0)
1615nn0red 11390 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
1716, 8remulcld 10108 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ∈ ℝ)
1810, 11eulerpartlemsf 30549 . . . . 5 𝑆:((ℕ0𝑚 ℕ) ∩ 𝑅)⟶ℕ0
1918ffvelrni 6398 . . . 4 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝑆𝐴) ∈ ℕ0)
2019adantr 480 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℕ0)
2120nn0red 11390 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℝ)
2214nn0red 11390 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℝ)
2322adantrr 753 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
247nnrpd 11908 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ+)
2524rprege0d 11917 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡))
26 bitsfi 15206 . . . . . 6 ((𝐴𝑡) ∈ ℕ0 → (bits‘(𝐴𝑡)) ∈ Fin)
2715, 26syl 17 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (bits‘(𝐴𝑡)) ∈ Fin)
281a1i 11 . . . . . 6 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 2 ∈ ℝ)
293a1i 11 . . . . . . 7 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (bits‘(𝐴𝑡)) ⊆ ℕ0)
3029sselda 3636 . . . . . 6 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 𝑖 ∈ ℕ0)
3128, 30reexpcld 13065 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → (2↑𝑖) ∈ ℝ)
32 0le2 11149 . . . . . . 7 0 ≤ 2
3332a1i 11 . . . . . 6 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ 2)
3428, 30, 33expge0d 13066 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ (2↑𝑖))
354snssd 4372 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → {𝑛} ⊆ (bits‘(𝐴𝑡)))
3627, 31, 34, 35fsumless 14572 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) ≤ Σ𝑖 ∈ (bits‘(𝐴𝑡))(2↑𝑖))
376recnd 10106 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℂ)
38 oveq2 6698 . . . . . 6 (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛))
3938sumsn 14519 . . . . 5 ((𝑛 ∈ (bits‘(𝐴𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
404, 37, 39syl2anc 694 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41 bitsinv1 15211 . . . . 5 ((𝐴𝑡) ∈ ℕ0 → Σ𝑖 ∈ (bits‘(𝐴𝑡))(2↑𝑖) = (𝐴𝑡))
4215, 41syl 17 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ (bits‘(𝐴𝑡))(2↑𝑖) = (𝐴𝑡))
4336, 40, 423brtr3d 4716 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ≤ (𝐴𝑡))
44 lemul1a 10915 . . 3 ((((2↑𝑛) ∈ ℝ ∧ (𝐴𝑡) ∈ ℝ ∧ (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡)) ∧ (2↑𝑛) ≤ (𝐴𝑡)) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
456, 23, 25, 43, 44syl31anc 1369 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
46 fzfid 12812 . . . . . . 7 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → (1...(𝑆𝐴)) ∈ Fin)
47 elfznn 12408 . . . . . . . . . . 11 (𝑘 ∈ (1...(𝑆𝐴)) → 𝑘 ∈ ℕ)
48 ffvelrn 6397 . . . . . . . . . . 11 ((𝐴:ℕ⟶ℕ0𝑘 ∈ ℕ) → (𝐴𝑘) ∈ ℕ0)
4913, 47, 48syl2an 493 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℕ0)
5049nn0red 11390 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℝ)
5147adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℕ)
5251nnred 11073 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℝ)
5350, 52remulcld 10108 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5453adantlr 751 . . . . . . 7 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5549nn0ge0d 11392 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ (𝐴𝑘))
56 0red 10079 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ∈ ℝ)
5751nngt0d 11102 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 < 𝑘)
5856, 52, 57ltled 10223 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ 𝑘)
5950, 52, 55, 58mulge0d 10642 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
6059adantlr 751 . . . . . . 7 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
61 fveq2 6229 . . . . . . . 8 (𝑘 = 𝑡 → (𝐴𝑘) = (𝐴𝑡))
62 id 22 . . . . . . . 8 (𝑘 = 𝑡𝑘 = 𝑡)
6361, 62oveq12d 6708 . . . . . . 7 (𝑘 = 𝑡 → ((𝐴𝑘) · 𝑘) = ((𝐴𝑡) · 𝑡))
64 simpr 476 . . . . . . 7 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → 𝑡 ∈ (1...(𝑆𝐴)))
6546, 54, 60, 63, 64fsumge1 14573 . . . . . 6 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
6665adantlr 751 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
67 eldif 3617 . . . . . . 7 (𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))))
68 nndiffz1 29676 . . . . . . . . . . . . . 14 ((𝑆𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆𝐴))) = (ℤ‘((𝑆𝐴) + 1)))
6968eleq2d 2716 . . . . . . . . . . . . 13 ((𝑆𝐴) ∈ ℕ0 → (𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))) ↔ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))))
7019, 69syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))) ↔ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))))
7170pm5.32i 670 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) ↔ (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))))
7210, 11eulerpartlems 30550 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))) → (𝐴𝑡) = 0)
7371, 72sylbi 207 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → (𝐴𝑡) = 0)
7473oveq1d 6705 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) = (0 · 𝑡))
75 simpr 476 . . . . . . . . . . . 12 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))))
7675eldifad 3619 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℕ)
7776nncnd 11074 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℂ)
7877mul02d 10272 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → (0 · 𝑡) = 0)
7974, 78eqtrd 2685 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) = 0)
80 fzfid 12812 . . . . . . . . . 10 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (1...(𝑆𝐴)) ∈ Fin)
8180, 53, 59fsumge0 14571 . . . . . . . . 9 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 0 ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8281adantr 480 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 0 ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8379, 82eqbrtrd 4707 . . . . . . 7 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8467, 83sylan2br 492 . . . . . 6 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8584anassrs 681 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8666, 85pm2.61dan 849 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8710, 11eulerpartlemsv3 30551 . . . . 5 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8887adantr 480 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝑆𝐴) = Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8986, 88breqtrrd 4713 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
9089adantrr 753 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
919, 17, 21, 45, 90letrd 10232 1 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637   ∖ cdif 3604   ∩ cin 3606   ⊆ wss 3607  {csn 4210   class class class wbr 4685   ↦ cmpt 4762  ◡ccnv 5142   “ cima 5146  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  Fincfn 7997  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   ≤ cle 10113  ℕcn 11058  2c2 11108  ℕ0cn0 11330  ℤ≥cuz 11725  ...cfz 12364  ↑cexp 12900  Σcsu 14460  bitscbits 15188 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-dvds 15028  df-bits 15191 This theorem is referenced by: (None)
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