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Theorem eulerpartlemgc 33918
Description: Lemma for eulerpart 33938. (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 12308 . . . . 5 2 ∈ ℝ
21a1i 11 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 2 ∈ ℝ)
3 bitsss 16392 . . . . 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 3976 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 𝑛 ∈ β„•0)
62, 5reexpcld 14151 . . 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 12249 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 𝑑 ∈ ℝ)
96, 8remulcld 11266 . 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 33913 . . . . . . 7 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (𝐴:β„•βŸΆβ„•0 ∧ (◑𝐴 β€œ β„•) ∈ Fin))
1312simpld 494 . . . . . 6 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 𝐴:β„•βŸΆβ„•0)
1413ffvelcdmda 7088 . . . . 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 12555 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (π΄β€˜π‘‘) ∈ ℝ)
1716, 8remulcld 11266 . 2 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) ∈ ℝ)
1810, 11eulerpartlemsf 33915 . . . . 5 𝑆:((β„•0 ↑m β„•) ∩ 𝑅)βŸΆβ„•0
1918ffvelcdmi 7087 . . . 4 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (π‘†β€˜π΄) ∈ β„•0)
2019adantr 480 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (π‘†β€˜π΄) ∈ β„•0)
2120nn0red 12555 . 2 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (π‘†β€˜π΄) ∈ ℝ)
2214nn0red 12555 . . . 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 13038 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ 𝑑 ∈ ℝ+)
2524rprege0d 13047 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (𝑑 ∈ ℝ ∧ 0 ≀ 𝑑))
26 bitsfi 16403 . . . . . 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 3978 . . . . . 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 14151 . . . . 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 12336 . . . . . . 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 14152 . . . . 5 (((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) ∧ 𝑖 ∈ (bitsβ€˜(π΄β€˜π‘‘))) β†’ 0 ≀ (2↑𝑖))
354snssd 4808 . . . . 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 15766 . . . 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 11264 . . . . 5 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (2↑𝑛) ∈ β„‚)
38 oveq2 7422 . . . . . 6 (𝑖 = 𝑛 β†’ (2↑𝑖) = (2↑𝑛))
3938sumsn 15716 . . . . 5 ((𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)) ∧ (2↑𝑛) ∈ β„‚) β†’ Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
404, 37, 39syl2anc 583 . . . 4 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛))
41 bitsinv1 16408 . . . . 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 5173 . . 3 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ (𝑑 ∈ β„• ∧ 𝑛 ∈ (bitsβ€˜(π΄β€˜π‘‘)))) β†’ (2↑𝑛) ≀ (π΄β€˜π‘‘))
44 lemul1a 12090 . . 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 1371 . 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 13962 . . . . . . 7 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (1...(π‘†β€˜π΄))) β†’ (1...(π‘†β€˜π΄)) ∈ Fin)
47 elfznn 13554 . . . . . . . . . . 11 (π‘˜ ∈ (1...(π‘†β€˜π΄)) β†’ π‘˜ ∈ β„•)
48 ffvelcdm 7085 . . . . . . . . . . 11 ((𝐴:β„•βŸΆβ„•0 ∧ π‘˜ ∈ β„•) β†’ (π΄β€˜π‘˜) ∈ β„•0)
4913, 47, 48syl2an 595 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ (π΄β€˜π‘˜) ∈ β„•0)
5049nn0red 12555 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ (π΄β€˜π‘˜) ∈ ℝ)
5147adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ π‘˜ ∈ β„•)
5251nnred 12249 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ π‘˜ ∈ ℝ)
5350, 52remulcld 11266 . . . . . . . 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 12557 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 ≀ (π΄β€˜π‘˜))
56 0red 11239 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 ∈ ℝ)
5751nngt0d 12283 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 < π‘˜)
5856, 52, 57ltled 11384 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ π‘˜ ∈ (1...(π‘†β€˜π΄))) β†’ 0 ≀ π‘˜)
5950, 52, 55, 58mulge0d 11813 . . . . . . . 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 6891 . . . . . . . 8 (π‘˜ = 𝑑 β†’ (π΄β€˜π‘˜) = (π΄β€˜π‘‘))
62 id 22 . . . . . . . 8 (π‘˜ = 𝑑 β†’ π‘˜ = 𝑑)
6361, 62oveq12d 7432 . . . . . . 7 (π‘˜ = 𝑑 β†’ ((π΄β€˜π‘˜) Β· π‘˜) = ((π΄β€˜π‘‘) Β· 𝑑))
64 simpr 484 . . . . . . 7 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (1...(π‘†β€˜π΄))) β†’ 𝑑 ∈ (1...(π‘†β€˜π΄)))
6546, 54, 60, 63, 64fsumge1 15767 . . . . . 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 3954 . . . . . . 7 (𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄))) ↔ (𝑑 ∈ β„• ∧ Β¬ 𝑑 ∈ (1...(π‘†β€˜π΄))))
68 nndiffz1 32538 . . . . . . . . . . . . . 14 ((π‘†β€˜π΄) ∈ β„•0 β†’ (β„• βˆ– (1...(π‘†β€˜π΄))) = (β„€β‰₯β€˜((π‘†β€˜π΄) + 1)))
6968eleq2d 2814 . . . . . . . . . . . . 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 574 . . . . . . . . . . 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 33916 . . . . . . . . . . 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 7429 . . . . . . . . 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 484 . . . . . . . . . . . 12 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄))))
7675eldifad 3956 . . . . . . . . . . 11 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ 𝑑 ∈ β„•)
7776nncnd 12250 . . . . . . . . . 10 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ 𝑑 ∈ β„‚)
7877mul02d 11434 . . . . . . . . 9 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ (0 Β· 𝑑) = 0)
7974, 78eqtrd 2767 . . . . . . . 8 ((𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) ∧ 𝑑 ∈ (β„• βˆ– (1...(π‘†β€˜π΄)))) β†’ ((π΄β€˜π‘‘) Β· 𝑑) = 0)
80 fzfid 13962 . . . . . . . . . 10 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (1...(π‘†β€˜π΄)) ∈ Fin)
8180, 53, 59fsumge0 15765 . . . . . . . . 9 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ 0 ≀ Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8281adantr 480 . . . . . . . 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 5164 . . . . . . 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 594 . . . . . 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 467 . . . . 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 33917 . . . . 5 (𝐴 ∈ ((β„•0 ↑m β„•) ∩ 𝑅) β†’ (π‘†β€˜π΄) = Ξ£π‘˜ ∈ (1...(π‘†β€˜π΄))((π΄β€˜π‘˜) Β· π‘˜))
8887adantr 480 . . . 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 5170 . . 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 11393 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 395   = wceq 1534   ∈ wcel 2099  {cab 2704   βˆ– cdif 3941   ∩ cin 3943   βŠ† wss 3944  {csn 4624   class class class wbr 5142   ↦ cmpt 5225  β—‘ccnv 5671   β€œ cima 5675  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ↑m cmap 8836  Fincfn 8955  β„‚cc 11128  β„cr 11129  0cc0 11130  1c1 11131   + caddc 11133   Β· cmul 11135   ≀ cle 11271  β„•cn 12234  2c2 12289  β„•0cn0 12494  β„€β‰₯cuz 12844  ...cfz 13508  β†‘cexp 14050  Ξ£csu 15656  bitscbits 16385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-rp 12999  df-ico 13354  df-fz 13509  df-fzo 13652  df-fl 13781  df-mod 13859  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-rlim 15457  df-sum 15657  df-dvds 16223  df-bits 16388
This theorem is referenced by: (None)
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