Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eulerpartlemgc Structured version   Visualization version   GIF version

Theorem eulerpartlemgc 29545
Description: Lemma for eulerpart 29565. (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 10940 . . . . 5 2 ∈ ℝ
21a1i 11 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 2 ∈ ℝ)
3 bitsss 14935 . . . . 5 (bits‘(𝐴𝑡)) ⊆ ℕ0
4 simprr 792 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ (bits‘(𝐴𝑡)))
53, 4sseldi 3566 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ ℕ0)
62, 5reexpcld 12845 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℝ)
7 simprl 790 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℕ)
87nnred 10885 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ)
96, 8remulcld 9927 . 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 29540 . . . . . . 7 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
1312simpld 474 . . . . . 6 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
1413ffvelrnda 6252 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℕ0)
1514adantrr 749 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℕ0)
1615nn0red 11202 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
1716, 8remulcld 9927 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ∈ ℝ)
1810, 11eulerpartlemsf 29542 . . . . 5 𝑆:((ℕ0𝑚 ℕ) ∩ 𝑅)⟶ℕ0
1918ffvelrni 6251 . . . 4 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (𝑆𝐴) ∈ ℕ0)
2019adantr 480 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℕ0)
2120nn0red 11202 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℝ)
2214nn0red 11202 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℝ)
2322adantrr 749 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
247nnrpd 11705 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ+)
2524rprege0d 11714 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡))
26 bitsfi 14946 . . . . . 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 3568 . . . . . 6 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 𝑖 ∈ ℕ0)
3128, 30reexpcld 12845 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → (2↑𝑖) ∈ ℝ)
32 0le2 10961 . . . . . . 7 0 ≤ 2
3332a1i 11 . . . . . 6 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ 2)
3428, 30, 33expge0d 12846 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ (2↑𝑖))
354snssd 4281 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → {𝑛} ⊆ (bits‘(𝐴𝑡)))
3627, 31, 34, 35fsumless 14318 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) ≤ Σ𝑖 ∈ (bits‘(𝐴𝑡))(2↑𝑖))
376recnd 9925 . . . . 5 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℂ)
38 oveq2 6535 . . . . . 6 (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛))
3938sumsn 14268 . . . . 5 ((𝑛 ∈ (bits‘(𝐴𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
404, 37, 39syl2anc 691 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41 bitsinv1 14951 . . . . 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 4609 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ≤ (𝐴𝑡))
44 lemul1a 10729 . . 3 ((((2↑𝑛) ∈ ℝ ∧ (𝐴𝑡) ∈ ℝ ∧ (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡)) ∧ (2↑𝑛) ≤ (𝐴𝑡)) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
456, 23, 25, 43, 44syl31anc 1321 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
46 fzfid 12592 . . . . . . 7 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → (1...(𝑆𝐴)) ∈ Fin)
47 elfznn 12199 . . . . . . . . . . 11 (𝑘 ∈ (1...(𝑆𝐴)) → 𝑘 ∈ ℕ)
48 ffvelrn 6250 . . . . . . . . . . 11 ((𝐴:ℕ⟶ℕ0𝑘 ∈ ℕ) → (𝐴𝑘) ∈ ℕ0)
4913, 47, 48syl2an 493 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℕ0)
5049nn0red 11202 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℝ)
5147adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℕ)
5251nnred 10885 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℝ)
5350, 52remulcld 9927 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5453adantlr 747 . . . . . . 7 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5549nn0ge0d 11204 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ (𝐴𝑘))
56 0red 9898 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ∈ ℝ)
5751nngt0d 10914 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 < 𝑘)
5856, 52, 57ltled 10037 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ 𝑘)
5950, 52, 55, 58mulge0d 10456 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
6059adantlr 747 . . . . . . 7 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
61 fveq2 6088 . . . . . . . 8 (𝑘 = 𝑡 → (𝐴𝑘) = (𝐴𝑡))
62 id 22 . . . . . . . 8 (𝑘 = 𝑡𝑘 = 𝑡)
6361, 62oveq12d 6545 . . . . . . 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 14319 . . . . . 6 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
6665adantlr 747 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
67 eldif 3550 . . . . . . 7 (𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))))
68 nndiffz1 28730 . . . . . . . . . . . . . 14 ((𝑆𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆𝐴))) = (ℤ‘((𝑆𝐴) + 1)))
6968eleq2d 2673 . . . . . . . . . . . . 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 667 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) ↔ (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))))
7210, 11eulerpartlems 29543 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))) → (𝐴𝑡) = 0)
7371, 72sylbi 206 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → (𝐴𝑡) = 0)
7473oveq1d 6542 . . . . . . . . 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 3552 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℕ)
7776nncnd 10886 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℂ)
7877mul02d 10086 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → (0 · 𝑡) = 0)
7974, 78eqtrd 2644 . . . . . . . 8 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) = 0)
80 fzfid 12592 . . . . . . . . . 10 (𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) → (1...(𝑆𝐴)) ∈ Fin)
8180, 53, 59fsumge0 14317 . . . . . . . . 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 4600 . . . . . . 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 678 . . . . 5 (((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8666, 85pm2.61dan 828 . . . 4 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8710, 11eulerpartlemsv3 29544 . . . . 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 4606 . . 3 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
9089adantrr 749 . 2 ((𝐴 ∈ ((ℕ0𝑚 ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
919, 17, 21, 45, 90letrd 10046 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 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  cdif 3537  cin 3539  wss 3540  {csn 4125   class class class wbr 4578  cmpt 4638  ccnv 5027  cima 5031  wf 5786  cfv 5790  (class class class)co 6527  𝑚 cmap 7722  Fincfn 7819  cc 9791  cr 9792  0cc0 9793  1c1 9794   + caddc 9796   · cmul 9798  cle 9932  cn 10870  2c2 10920  0cn0 11142  cuz 11522  ...cfz 12155  cexp 12680  Σcsu 14213  bitscbits 14928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-int 4406  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-supp 7161  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-sup 8209  df-inf 8210  df-oi 8276  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-n0 11143  df-z 11214  df-uz 11523  df-rp 11668  df-ico 12011  df-fz 12156  df-fzo 12293  df-fl 12413  df-mod 12489  df-seq 12622  df-exp 12681  df-hash 12938  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-clim 14016  df-rlim 14017  df-sum 14214  df-dvds 14771  df-bits 14931
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator