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Theorem eulerpartlemgc 33002
Description: Lemma for eulerpart 33022. (Contributed by Thierry Arnoux, 9-Aug-2018.)
Hypotheses
Ref Expression
eulerpartlems.r 𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}
eulerpartlems.s 𝑆 = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))
Assertion
Ref Expression
eulerpartlemgc ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ ((2↑𝑛) Β· 𝑑) ≀ (π‘†β€˜π΄))
Distinct variable groups:   𝑓,π‘˜,𝐴   𝑅,𝑓,π‘˜   𝑑,π‘˜,𝐴   𝑑,𝑅   𝑑,𝑆,π‘˜
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 12234 . . . . 5 2 ∈ ℝ
21a1i 11 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 2 ∈ ℝ)
3 bitsss 16313 . . . . 5 (bitsβ€˜(π΄β€˜π‘‘)) βŠ† β„•0
4 simprr 772 . . . . 5 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))
53, 4sselid 3947 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 𝑛 ∈ β„•0)
62, 5reexpcld 14075 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (2↑𝑛) ∈ ℝ)
7 simprl 770 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 𝑑 ∈ β„•)
87nnred 12175 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 𝑑 ∈ ℝ)
96, 8remulcld 11192 . 2 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ ((2↑𝑛) Β· 𝑑) ∈ ℝ)
10 eulerpartlems.r . . . . . . . 8 𝑅 = {𝑓 ∣ (◑𝑓 β€œ β„•) ∈ Fin}
11 eulerpartlems.s . . . . . . . 8 𝑆 = (𝑓 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ↦ Ξ£π‘˜ ∈ β„• ((π‘“β€˜π‘˜) Β· π‘˜))
1210, 11eulerpartlemelr 32997 . . . . . . 7 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (𝐴:β„•βŸΆβ„•0 ∧ (◑𝐴 β€œ β„•) ∈ Fin))
1312simpld 496 . . . . . 6 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 𝐴:β„•βŸΆβ„•0)
1413ffvelcdmda 7040 . . . . 5 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ β„•) β†’ (π΄β€˜π‘‘) ∈ β„•0)
1514adantrr 716 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (π΄β€˜π‘‘) ∈ β„•0)
1615nn0red 12481 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (π΄β€˜π‘‘) ∈ ℝ)
1716, 8remulcld 11192 . 2 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ∈ ℝ)
1810, 11eulerpartlemsf 32999 . . . . 5 𝑆:((β„•0 ↑m β„•) ∩ 𝑅)βŸΆβ„•0
1918ffvelcdmi 7039 . . . 4 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (π‘†β€˜π΄) ∈ β„•0)
2019adantr 482 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (π‘†β€˜π΄) ∈ β„•0)
2120nn0red 12481 . 2 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (π‘†β€˜π΄) ∈ ℝ)
2214nn0red 12481 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ β„•) β†’ (π΄β€˜π‘‘) ∈ ℝ)
2322adantrr 716 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (π΄β€˜π‘‘) ∈ ℝ)
247nnrpd 12962 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 𝑑 ∈ ℝ+)
2524rprege0d 12971 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (𝑑 ∈ ℝ ∧ 0 ≀ 𝑑))
26 bitsfi 16324 . . . . . 6 ((π΄β€˜π‘‘) ∈ β„•0 β†’ (bitsβ€˜(π΄β€˜π‘‘)) ∈ Fin)
2715, 26syl 17 . . . . 5 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (bitsβ€˜(π΄β€˜π‘‘)) ∈ Fin)
281a1i 11 . . . . . 6 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) ∧ 𝑖 ∈ (bitsβ€˜(π΄β€˜π‘‘))) β†’ 2 ∈ ℝ)
293a1i 11 . . . . . . 7 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (bitsβ€˜(π΄β€˜π‘‘)) βŠ† β„•0)
3029sselda 3949 . . . . . 6 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) ∧ 𝑖 ∈ (bitsβ€˜(π΄β€˜π‘‘))) β†’ 𝑖 ∈ β„•0)
3128, 30reexpcld 14075 . . . . 5 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) ∧ 𝑖 ∈ (bitsβ€˜(π΄β€˜π‘‘))) β†’ (2↑𝑖) ∈ ℝ)
32 0le2 12262 . . . . . . 7 0 ≀ 2
3332a1i 11 . . . . . 6 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) ∧ 𝑖 ∈ (bitsβ€˜(π΄β€˜π‘‘))) β†’ 0 ≀ 2)
3428, 30, 33expge0d 14076 . . . . 5 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) ∧ 𝑖 ∈ (bitsβ€˜(π΄β€˜π‘‘))) β†’ 0 ≀ (2↑𝑖))
354snssd 4774 . . . . 5 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ {𝑛} βŠ† (bitsβ€˜(π΄β€˜π‘‘)))
3627, 31, 34, 35fsumless 15688 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ Σ𝑖 ∈ {𝑛} (2↑𝑖) ≀ Σ𝑖 ∈ (bitsβ€˜(π΄β€˜π‘‘))(2↑𝑖))
376recnd 11190 . . . . 5 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (2↑𝑛) ∈ β„‚)
38 oveq2 7370 . . . . . 6 (𝑖 = 𝑛 β†’ (2↑𝑖) = (2↑𝑛))
3938sumsn 15638 . . . . 5 ((𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)) ∧ (2↑𝑛) ∈ β„‚) β†’ Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
404, 37, 39syl2anc 585 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41 bitsinv1 16329 . . . . 5 ((π΄β€˜π‘‘) ∈ β„•0 β†’ Σ𝑖 ∈ (bitsβ€˜(π΄β€˜π‘‘))(2↑𝑖) = (π΄β€˜π‘‘))
4215, 41syl 17 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ Σ𝑖 ∈ (bitsβ€˜(π΄β€˜π‘‘))(2↑𝑖) = (π΄β€˜π‘‘))
4336, 40, 423brtr3d 5141 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (2↑𝑛) ≀ (π΄β€˜π‘‘))
44 lemul1a 12016 . . 3 ((((2↑𝑛) ∈ ℝ ∧ (π΄β€˜π‘‘) ∈ ℝ ∧ (𝑑 ∈ ℝ ∧ 0 ≀ 𝑑)) ∧ (2↑𝑛) ≀ (π΄β€˜π‘‘)) β†’ ((2↑𝑛) Β· 𝑑) ≀ ((π΄β€˜π‘‘) Β· 𝑑))
456, 23, 25, 43, 44syl31anc 1374 . 2 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ ((2↑𝑛) Β· 𝑑) ≀ ((π΄β€˜π‘‘) Β· 𝑑))
46 fzfid 13885 . . . . . . 7 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (1...(π‘†β€˜π΄))) β†’ (1...(π‘†β€˜π΄)) ∈ Fin)
47 elfznn 13477 . . . . . . . . . . 11 (π‘˜ ∈ (1...(π‘†β€˜π΄)) β†’ π‘˜ ∈ β„•)
48 ffvelcdm 7037 . . . . . . . . . . 11 ((𝐴:β„•βŸΆβ„•0 ∧ π‘˜ ∈ β„•) β†’ (π΄β€˜π‘˜) ∈ β„•0)
4913, 47, 48syl2an 597 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ (π΄β€˜π‘˜) ∈ β„•0)
5049nn0red 12481 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ (π΄β€˜π‘˜) ∈ ℝ)
5147adantl 483 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ π‘˜ ∈ β„•)
5251nnred 12175 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ π‘˜ ∈ ℝ)
5350, 52remulcld 11192 . . . . . . . 8 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ ((π΄β€˜π‘˜) Β· π‘˜) ∈ ℝ)
5453adantlr 714 . . . . . . 7 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (1...(π‘†β€˜π΄))) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ ((π΄β€˜π‘˜) Β· π‘˜) ∈ ℝ)
5549nn0ge0d 12483 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 ≀ (π΄β€˜π‘˜))
56 0red 11165 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 ∈ ℝ)
5751nngt0d 12209 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 < π‘˜)
5856, 52, 57ltled 11310 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 ≀ π‘˜)
5950, 52, 55, 58mulge0d 11739 . . . . . . . 8 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 ≀ ((π΄β€˜π‘˜) Β· π‘˜))
6059adantlr 714 . . . . . . 7 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (1...(π‘†β€˜π΄))) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 ≀ ((π΄β€˜π‘˜) Β· π‘˜))
61 fveq2 6847 . . . . . . . 8 (π‘˜ = 𝑑 β†’ (π΄β€˜π‘˜) = (π΄β€˜π‘‘))
62 id 22 . . . . . . . 8 (π‘˜ = 𝑑 β†’ π‘˜ = 𝑑)
6361, 62oveq12d 7380 . . . . . . 7 (π‘˜ = 𝑑 β†’ ((π΄β€˜π‘˜) Β· π‘˜) = ((π΄β€˜π‘‘) Β· 𝑑))
64 simpr 486 . . . . . . 7 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (1...(π‘†β€˜π΄))) β†’ 𝑑 ∈ (1...(π‘†β€˜π΄)))
6546, 54, 60, 63, 64fsumge1 15689 . . . . . 6 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (1...(π‘†β€˜π΄))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ≀ Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
6665adantlr 714 . . . . 5 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ β„•) ∧ 𝑑 ∈ (1...(π‘†β€˜π΄))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ≀ Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
67 eldif 3925 . . . . . . 7 (𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄))) ↔ (𝑑 ∈ β„• ∧ Β¬ 𝑑 ∈ (1...(π‘†β€˜π΄))))
68 nndiffz1 31731 . . . . . . . . . . . . . 14 ((π‘†β€˜π΄) ∈ β„•0 β†’ (β„• βˆ– (1...(π‘†β€˜π΄))) = (β„€β‰₯β€˜((π‘†β€˜π΄) + 1)))
6968eleq2d 2824 . . . . . . . . . . . . 13 ((π‘†β€˜π΄) ∈ β„•0 β†’ (𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄))) ↔ 𝑑 ∈ (β„€β‰₯β€˜((π‘†β€˜π΄) + 1))))
7019, 69syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄))) ↔ 𝑑 ∈ (β„€β‰₯β€˜((π‘†β€˜π΄) + 1))))
7170pm5.32i 576 . . . . . . . . . . 11 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) ↔ (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„€β‰₯β€˜((π‘†β€˜π΄) + 1))))
7210, 11eulerpartlems 33000 . . . . . . . . . . 11 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„€β‰₯β€˜((π‘†β€˜π΄) + 1))) β†’ (π΄β€˜π‘‘) = 0)
7371, 72sylbi 216 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ (π΄β€˜π‘‘) = 0)
7473oveq1d 7377 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) = (0 Β· 𝑑))
75 simpr 486 . . . . . . . . . . . 12 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄))))
7675eldifad 3927 . . . . . . . . . . 11 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ 𝑑 ∈ β„•)
7776nncnd 12176 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ 𝑑 ∈ β„‚)
7877mul02d 11360 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ (0 Β· 𝑑) = 0)
7974, 78eqtrd 2777 . . . . . . . 8 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) = 0)
80 fzfid 13885 . . . . . . . . . 10 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (1...(π‘†β€˜π΄)) ∈ Fin)
8180, 53, 59fsumge0 15687 . . . . . . . . 9 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 0 ≀ Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8281adantr 482 . . . . . . . 8 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ 0 ≀ Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8379, 82eqbrtrd 5132 . . . . . . 7 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ≀ Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8467, 83sylan2br 596 . . . . . 6 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ Β¬ 𝑑 ∈ (1...(π‘†β€˜π΄)))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ≀ Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8584anassrs 469 . . . . 5 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ β„•) ∧ Β¬ 𝑑 ∈ (1...(π‘†β€˜π΄))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ≀ Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8666, 85pm2.61dan 812 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ β„•) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ≀ Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8710, 11eulerpartlemsv3 33001 . . . . 5 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (π‘†β€˜π΄) = Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8887adantr 482 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ β„•) β†’ (π‘†β€˜π΄) = Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8986, 88breqtrrd 5138 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ β„•) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ≀ (π‘†β€˜π΄))
9089adantrr 716 . 2 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ≀ (π‘†β€˜π΄))
919, 17, 21, 45, 90letrd 11319 1 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ ((2↑𝑛) Β· 𝑑) ≀ (π‘†β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2714   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915  {csn 4591   class class class wbr 5110   ↦ cmpt 5193  β—‘ccnv 5637   β€œ cima 5641  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ↑m cmap 8772  Fincfn 8890  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   Β· cmul 11063   ≀ cle 11197  β„•cn 12160  2c2 12215  β„•0cn0 12420  β„€β‰₯cuz 12770  ...cfz 13431  β†‘cexp 13974  Ξ£csu 15577  bitscbits 16306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-inf 9386  df-oi 9453  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-z 12507  df-uz 12771  df-rp 12923  df-ico 13277  df-fz 13432  df-fzo 13575  df-fl 13704  df-mod 13782  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-rlim 15378  df-sum 15578  df-dvds 16144  df-bits 16309
This theorem is referenced by: (None)
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