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Theorem eulerpartlemgc 30654
Description: Lemma for eulerpart 30674. (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 11203 . . . . 5 2 ∈ ℝ
21a1i 11 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 2 ∈ ℝ)
3 bitsss 15271 . . . . 5 (bits‘(𝐴𝑡)) ⊆ ℕ0
4 simprr 813 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ (bits‘(𝐴𝑡)))
53, 4sseldi 3707 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ ℕ0)
62, 5reexpcld 13140 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℝ)
7 simprl 811 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℕ)
87nnred 11148 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ)
96, 8remulcld 10183 . 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 30649 . . . . . . 7 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
1312simpld 477 . . . . . 6 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
1413ffvelrnda 6474 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℕ0)
1514adantrr 755 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℕ0)
1615nn0red 11465 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
1716, 8remulcld 10183 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ∈ ℝ)
1810, 11eulerpartlemsf 30651 . . . . 5 𝑆:((ℕ0𝑚 ℕ) ∩ 𝑅)⟶ℕ0
1918ffvelrni 6473 . . . 4 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝑆𝐴) ∈ ℕ0)
2019adantr 472 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℕ0)
2120nn0red 11465 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℝ)
2214nn0red 11465 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℝ)
2322adantrr 755 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
247nnrpd 11984 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ+)
2524rprege0d 11993 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡))
26 bitsfi 15282 . . . . . 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 3709 . . . . . 6 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 𝑖 ∈ ℕ0)
3128, 30reexpcld 13140 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → (2↑𝑖) ∈ ℝ)
32 0le2 11224 . . . . . . 7 0 ≤ 2
3332a1i 11 . . . . . 6 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ 2)
3428, 30, 33expge0d 13141 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ (2↑𝑖))
354snssd 4448 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → {𝑛} ⊆ (bits‘(𝐴𝑡)))
3627, 31, 34, 35fsumless 14648 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) ≤ Σ𝑖 ∈ (bits‘(𝐴𝑡))(2↑𝑖))
376recnd 10181 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℂ)
38 oveq2 6773 . . . . . 6 (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛))
3938sumsn 14595 . . . . 5 ((𝑛 ∈ (bits‘(𝐴𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
404, 37, 39syl2anc 696 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41 bitsinv1 15287 . . . . 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 4791 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ≤ (𝐴𝑡))
44 lemul1a 10990 . . 3 ((((2↑𝑛) ∈ ℝ ∧ (𝐴𝑡) ∈ ℝ ∧ (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡)) ∧ (2↑𝑛) ≤ (𝐴𝑡)) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
456, 23, 25, 43, 44syl31anc 1442 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
46 fzfid 12887 . . . . . . 7 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → (1...(𝑆𝐴)) ∈ Fin)
47 elfznn 12484 . . . . . . . . . . 11 (𝑘 ∈ (1...(𝑆𝐴)) → 𝑘 ∈ ℕ)
48 ffvelrn 6472 . . . . . . . . . . 11 ((𝐴:ℕ⟶ℕ0𝑘 ∈ ℕ) → (𝐴𝑘) ∈ ℕ0)
4913, 47, 48syl2an 495 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℕ0)
5049nn0red 11465 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℝ)
5147adantl 473 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℕ)
5251nnred 11148 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℝ)
5350, 52remulcld 10183 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5453adantlr 753 . . . . . . 7 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5549nn0ge0d 11467 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ (𝐴𝑘))
56 0red 10154 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ∈ ℝ)
5751nngt0d 11177 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 < 𝑘)
5856, 52, 57ltled 10298 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ 𝑘)
5950, 52, 55, 58mulge0d 10717 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
6059adantlr 753 . . . . . . 7 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
61 fveq2 6304 . . . . . . . 8 (𝑘 = 𝑡 → (𝐴𝑘) = (𝐴𝑡))
62 id 22 . . . . . . . 8 (𝑘 = 𝑡𝑘 = 𝑡)
6361, 62oveq12d 6783 . . . . . . 7 (𝑘 = 𝑡 → ((𝐴𝑘) · 𝑘) = ((𝐴𝑡) · 𝑡))
64 simpr 479 . . . . . . 7 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → 𝑡 ∈ (1...(𝑆𝐴)))
6546, 54, 60, 63, 64fsumge1 14649 . . . . . 6 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
6665adantlr 753 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
67 eldif 3690 . . . . . . 7 (𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))))
68 nndiffz1 29778 . . . . . . . . . . . . . 14 ((𝑆𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆𝐴))) = (ℤ‘((𝑆𝐴) + 1)))
6968eleq2d 2789 . . . . . . . . . . . . 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 672 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) ↔ (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))))
7210, 11eulerpartlems 30652 . . . . . . . . . . 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 6780 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) = (0 · 𝑡))
75 simpr 479 . . . . . . . . . . . 12 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))))
7675eldifad 3692 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℕ)
7776nncnd 11149 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℂ)
7877mul02d 10347 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → (0 · 𝑡) = 0)
7974, 78eqtrd 2758 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) = 0)
80 fzfid 12887 . . . . . . . . . 10 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (1...(𝑆𝐴)) ∈ Fin)
8180, 53, 59fsumge0 14647 . . . . . . . . 9 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 0 ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8281adantr 472 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 0 ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8379, 82eqbrtrd 4782 . . . . . . 7 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8467, 83sylan2br 494 . . . . . 6 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8584anassrs 683 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8666, 85pm2.61dan 867 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8710, 11eulerpartlemsv3 30653 . . . . 5 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8887adantr 472 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝑆𝐴) = Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8986, 88breqtrrd 4788 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
9089adantrr 755 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
919, 17, 21, 45, 90letrd 10307 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 1596  wcel 2103  {cab 2710  cdif 3677  cin 3679  wss 3680  {csn 4285   class class class wbr 4760  cmpt 4837  ccnv 5217  cima 5221  wf 5997  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  Fincfn 8072  cc 10047  cr 10048  0cc0 10049  1c1 10050   + caddc 10052   · cmul 10054  cle 10188  cn 11133  2c2 11183  0cn0 11405  cuz 11800  ...cfz 12440  cexp 12975  Σcsu 14536  bitscbits 15264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-supp 7416  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-sup 8464  df-inf 8465  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-n0 11406  df-z 11491  df-uz 11801  df-rp 11947  df-ico 12295  df-fz 12441  df-fzo 12581  df-fl 12708  df-mod 12784  df-seq 12917  df-exp 12976  df-hash 13233  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-clim 14339  df-rlim 14340  df-sum 14537  df-dvds 15104  df-bits 15267
This theorem is referenced by: (None)
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