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Theorem eulerpartlemgc 31728
 Description: Lemma for eulerpart 31748. (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 11703 . . . . 5 2 ∈ ℝ
21a1i 11 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 2 ∈ ℝ)
3 bitsss 15768 . . . . 5 (bits‘(𝐴𝑡)) ⊆ ℕ0
4 simprr 772 . . . . 5 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ (bits‘(𝐴𝑡)))
53, 4sseldi 3916 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑛 ∈ ℕ0)
62, 5reexpcld 13527 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℝ)
7 simprl 770 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℕ)
87nnred 11644 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ)
96, 8remulcld 10664 . 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 31723 . . . . . . 7 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (𝐴 “ ℕ) ∈ Fin))
1312simpld 498 . . . . . 6 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
1413ffvelrnda 6832 . . . . 5 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℕ0)
1514adantrr 716 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℕ0)
1615nn0red 11948 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
1716, 8remulcld 10664 . 2 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ∈ ℝ)
1810, 11eulerpartlemsf 31725 . . . . 5 𝑆:((ℕ0m ℕ) ∩ 𝑅)⟶ℕ0
1918ffvelrni 6831 . . . 4 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝐴) ∈ ℕ0)
2019adantr 484 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℕ0)
2120nn0red 11948 . 2 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑆𝐴) ∈ ℝ)
2214nn0red 11948 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴𝑡) ∈ ℝ)
2322adantrr 716 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝐴𝑡) ∈ ℝ)
247nnrpd 12421 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → 𝑡 ∈ ℝ+)
2524rprege0d 12430 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡))
26 bitsfi 15779 . . . . . 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 3918 . . . . . 6 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 𝑖 ∈ ℕ0)
3128, 30reexpcld 13527 . . . . 5 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → (2↑𝑖) ∈ ℝ)
32 0le2 11731 . . . . . . 7 0 ≤ 2
3332a1i 11 . . . . . 6 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ 2)
3428, 30, 33expge0d 13528 . . . . 5 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴𝑡))) → 0 ≤ (2↑𝑖))
354snssd 4705 . . . . 5 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → {𝑛} ⊆ (bits‘(𝐴𝑡)))
3627, 31, 34, 35fsumless 15146 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) ≤ Σ𝑖 ∈ (bits‘(𝐴𝑡))(2↑𝑖))
376recnd 10662 . . . . 5 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ∈ ℂ)
38 oveq2 7147 . . . . . 6 (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛))
3938sumsn 15096 . . . . 5 ((𝑛 ∈ (bits‘(𝐴𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
404, 37, 39syl2anc 587 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41 bitsinv1 15784 . . . . 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 5064 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → (2↑𝑛) ≤ (𝐴𝑡))
44 lemul1a 11487 . . 3 ((((2↑𝑛) ∈ ℝ ∧ (𝐴𝑡) ∈ ℝ ∧ (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡)) ∧ (2↑𝑛) ≤ (𝐴𝑡)) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
456, 23, 25, 43, 44syl31anc 1370 . 2 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴𝑡) · 𝑡))
46 fzfid 13340 . . . . . . 7 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → (1...(𝑆𝐴)) ∈ Fin)
47 elfznn 12935 . . . . . . . . . . 11 (𝑘 ∈ (1...(𝑆𝐴)) → 𝑘 ∈ ℕ)
48 ffvelrn 6830 . . . . . . . . . . 11 ((𝐴:ℕ⟶ℕ0𝑘 ∈ ℕ) → (𝐴𝑘) ∈ ℕ0)
4913, 47, 48syl2an 598 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℕ0)
5049nn0red 11948 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → (𝐴𝑘) ∈ ℝ)
5147adantl 485 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℕ)
5251nnred 11644 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 𝑘 ∈ ℝ)
5350, 52remulcld 10664 . . . . . . . 8 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5453adantlr 714 . . . . . . 7 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → ((𝐴𝑘) · 𝑘) ∈ ℝ)
5549nn0ge0d 11950 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ (𝐴𝑘))
56 0red 10637 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ∈ ℝ)
5751nngt0d 11678 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 < 𝑘)
5856, 52, 57ltled 10781 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ 𝑘)
5950, 52, 55, 58mulge0d 11210 . . . . . . . 8 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
6059adantlr 714 . . . . . . 7 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) ∧ 𝑘 ∈ (1...(𝑆𝐴))) → 0 ≤ ((𝐴𝑘) · 𝑘))
61 fveq2 6649 . . . . . . . 8 (𝑘 = 𝑡 → (𝐴𝑘) = (𝐴𝑡))
62 id 22 . . . . . . . 8 (𝑘 = 𝑡𝑘 = 𝑡)
6361, 62oveq12d 7157 . . . . . . 7 (𝑘 = 𝑡 → ((𝐴𝑘) · 𝑘) = ((𝐴𝑡) · 𝑡))
64 simpr 488 . . . . . . 7 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → 𝑡 ∈ (1...(𝑆𝐴)))
6546, 54, 60, 63, 64fsumge1 15147 . . . . . 6 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
6665adantlr 714 . . . . 5 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
67 eldif 3894 . . . . . . 7 (𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))))
68 nndiffz1 30538 . . . . . . . . . . . . . 14 ((𝑆𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆𝐴))) = (ℤ‘((𝑆𝐴) + 1)))
6968eleq2d 2878 . . . . . . . . . . . . 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 578 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) ↔ (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))))
7210, 11eulerpartlems 31726 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ‘((𝑆𝐴) + 1))) → (𝐴𝑡) = 0)
7371, 72sylbi 220 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → (𝐴𝑡) = 0)
7473oveq1d 7154 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) = (0 · 𝑡))
75 simpr 488 . . . . . . . . . . . 12 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴))))
7675eldifad 3896 . . . . . . . . . . 11 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℕ)
7776nncnd 11645 . . . . . . . . . 10 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 𝑡 ∈ ℂ)
7877mul02d 10831 . . . . . . . . 9 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → (0 · 𝑡) = 0)
7974, 78eqtrd 2836 . . . . . . . 8 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) = 0)
80 fzfid 13340 . . . . . . . . . 10 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (1...(𝑆𝐴)) ∈ Fin)
8180, 53, 59fsumge0 15145 . . . . . . . . 9 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → 0 ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8281adantr 484 . . . . . . . 8 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → 0 ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8379, 82eqbrtrd 5055 . . . . . . 7 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8467, 83sylan2br 597 . . . . . 6 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴)))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8584anassrs 471 . . . . 5 (((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (1...(𝑆𝐴))) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8666, 85pm2.61dan 812 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8710, 11eulerpartlemsv3 31727 . . . . 5 (𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) → (𝑆𝐴) = Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8887adantr 484 . . . 4 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝑆𝐴) = Σ𝑘 ∈ (1...(𝑆𝐴))((𝐴𝑘) · 𝑘))
8986, 88breqtrrd 5061 . . 3 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
9089adantrr 716 . 2 ((𝐴 ∈ ((ℕ0m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴𝑡)))) → ((𝐴𝑡) · 𝑡) ≤ (𝑆𝐴))
919, 17, 21, 45, 90letrd 10790 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  {cab 2779   ∖ cdif 3881   ∩ cin 3883   ⊆ wss 3884  {csn 4528   class class class wbr 5033   ↦ cmpt 5113  ◡ccnv 5522   “ cima 5526  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393  Fincfn 8496  ℂcc 10528  ℝcr 10529  0cc0 10530  1c1 10531   + caddc 10533   · cmul 10535   ≤ cle 10669  ℕcn 11629  2c2 11684  ℕ0cn0 11889  ℤ≥cuz 12235  ...cfz 12889  ↑cexp 13429  Σcsu 15037  bitscbits 15761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-ico 12736  df-fz 12890  df-fzo 13033  df-fl 13161  df-mod 13237  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-rlim 14841  df-sum 15038  df-dvds 15603  df-bits 15764 This theorem is referenced by: (None)
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