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Theorem eulerpartlemgc 31520
Description: Lemma for eulerpart 31540. (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 11700 . . . . 5 2 ∈ ℝ
21a1i 11 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 2 ∈ ℝ)
3 bitsss 15765 . . . . 5 (bits‘(𝐴𝑡)) ⊆ ℕ0
4 simprr 769 . . . . 5 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ (bits‘(𝐴𝑡)))
53, 4sseldi 3964 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ ℕ0)
62, 5reexpcld 13517 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℝ)
7 simprl 767 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℕ)
87nnred 11642 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ)
96, 8remulcld 10660 . 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 31515 . . . . . . 7 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
1312simpld 495 . . . . . 6 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
1413ffvelrnda 6844 . . . . 5 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℕ0)
1514adantrr 713 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℕ0)
1615nn0red 11945 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
1716, 8remulcld 10660 . 2 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ∈ ℝ)
1810, 11eulerpartlemsf 31517 . . . . 5 𝑆:((ℕ0m ℕ) ∩ 𝑅)⟶ℕ0
1918ffvelrni 6843 . . . 4 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝐴) ∈ ℕ0)
2019adantr 481 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℕ0)
2120nn0red 11945 . 2 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℝ)
2214nn0red 11945 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℝ)
2322adantrr 713 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
247nnrpd 12419 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ+)
2524rprege0d 12428 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡))
26 bitsfi 15776 . . . . . 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 3966 . . . . . 6 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 𝑖 ∈ ℕ0)
3128, 30reexpcld 13517 . . . . 5 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → (2↑𝑖) ∈ ℝ)
32 0le2 11728 . . . . . . 7 0 ≤ 2
3332a1i 11 . . . . . 6 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ 2)
3428, 30, 33expge0d 13518 . . . . 5 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ (2↑𝑖))
354snssd 4736 . . . . 5 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → {𝑛} ⊆ (bits‘(𝐴𝑡)))
3627, 31, 34, 35fsumless 15141 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) ≤ Σ𝑖 ∈ (bits‘(𝐴𝑡))(2↑𝑖))
376recnd 10658 . . . . 5 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℂ)
38 oveq2 7153 . . . . . 6 (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛))
3938sumsn 15091 . . . . 5 ((𝑛 ∈ (bits‘(𝐴𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
404, 37, 39syl2anc 584 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41 bitsinv1 15781 . . . . 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 5089 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ≤ (𝐴𝑡))
44 lemul1a 11483 . . 3 ((((2↑𝑛) ∈ ℝ ∧ (𝐴𝑡) ∈ ℝ ∧ (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡)) ∧ (2↑𝑛) ≤ (𝐴𝑡)) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
456, 23, 25, 43, 44syl31anc 1365 . 2 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
46 fzfid 13331 . . . . . . 7 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → (1...(𝑆𝐴)) ∈ Fin)
47 elfznn 12926 . . . . . . . . . . 11 (𝑘 ∈ (1...(𝑆𝐴)) → 𝑘 ∈ ℕ)
48 ffvelrn 6842 . . . . . . . . . . 11 ((𝐴:ℕ⟶ℕ0𝑘 ∈ ℕ) → (𝐴𝑘) ∈ ℕ0)
4913, 47, 48syl2an 595 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℕ0)
5049nn0red 11945 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℝ)
5147adantl 482 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℕ)
5251nnred 11642 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℝ)
5350, 52remulcld 10660 . . . . . . . 8 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5453adantlr 711 . . . . . . 7 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5549nn0ge0d 11947 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ (𝐴𝑘))
56 0red 10633 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ∈ ℝ)
5751nngt0d 11675 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 < 𝑘)
5856, 52, 57ltled 10777 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ 𝑘)
5950, 52, 55, 58mulge0d 11206 . . . . . . . 8 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
6059adantlr 711 . . . . . . 7 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
61 fveq2 6664 . . . . . . . 8 (𝑘 = 𝑡 → (𝐴𝑘) = (𝐴𝑡))
62 id 22 . . . . . . . 8 (𝑘 = 𝑡𝑘 = 𝑡)
6361, 62oveq12d 7163 . . . . . . 7 (𝑘 = 𝑡 → ((𝐴𝑘) · 𝑘) = ((𝐴𝑡) · 𝑡))
64 simpr 485 . . . . . . 7 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → 𝑡 ∈ (1...(𝑆𝐴)))
6546, 54, 60, 63, 64fsumge1 15142 . . . . . 6 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
6665adantlr 711 . . . . 5 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
67 eldif 3945 . . . . . . 7 (𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))))
68 nndiffz1 30436 . . . . . . . . . . . . . 14 ((𝑆𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆𝐴))) = (ℤ‘((𝑆𝐴) + 1)))
6968eleq2d 2898 . . . . . . . . . . . . 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 575 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) ↔ (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))))
7210, 11eulerpartlems 31518 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))) → (𝐴𝑡) = 0)
7371, 72sylbi 218 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → (𝐴𝑡) = 0)
7473oveq1d 7160 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) = (0 · 𝑡))
75 simpr 485 . . . . . . . . . . . 12 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))))
7675eldifad 3947 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℕ)
7776nncnd 11643 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℂ)
7877mul02d 10827 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → (0 · 𝑡) = 0)
7974, 78eqtrd 2856 . . . . . . . 8 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) = 0)
80 fzfid 13331 . . . . . . . . . 10 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (1...(𝑆𝐴)) ∈ Fin)
8180, 53, 59fsumge0 15140 . . . . . . . . 9 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 0 ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8281adantr 481 . . . . . . . 8 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 0 ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8379, 82eqbrtrd 5080 . . . . . . 7 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8467, 83sylan2br 594 . . . . . 6 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8584anassrs 468 . . . . 5 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8666, 85pm2.61dan 809 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8710, 11eulerpartlemsv3 31519 . . . . 5 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8887adantr 481 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝑆𝐴) = Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8986, 88breqtrrd 5086 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
9089adantrr 713 . 2 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
919, 17, 21, 45, 90letrd 10786 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 207  wa 396   = wceq 1528  wcel 2105  {cab 2799  cdif 3932  cin 3934  wss 3935  {csn 4559   class class class wbr 5058  cmpt 5138  ccnv 5548  cima 5552  wf 6345  cfv 6349  (class class class)co 7145  m cmap 8396  Fincfn 8498  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  cle 10665  cn 11627  2c2 11681  0cn0 11886  cuz 12232  ...cfz 12882  cexp 13419  Σcsu 15032  bitscbits 15758
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 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-supp 7822  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-ico 12734  df-fz 12883  df-fzo 13024  df-fl 13152  df-mod 13228  df-seq 13360  df-exp 13420  df-hash 13681  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-clim 14835  df-rlim 14836  df-sum 15033  df-dvds 15598  df-bits 15761
This theorem is referenced by: (None)
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