Step | Hyp | Ref
| Expression |
1 | | 1red 11163 |
. 2
β’ (π β 1 β
β) |
2 | | fzfid 13885 |
. . . . . 6
β’ ((π β§ π₯ β β+) β
(1...(ββπ₯))
β Fin) |
3 | | rpvmasum.g |
. . . . . . . 8
β’ πΊ = (DChrβπ) |
4 | | rpvmasum.z |
. . . . . . . 8
β’ π =
(β€/nβ€βπ) |
5 | | rpvmasum.d |
. . . . . . . 8
β’ π· = (BaseβπΊ) |
6 | | rpvmasum.l |
. . . . . . . 8
β’ πΏ = (β€RHomβπ) |
7 | | dchrisum.b |
. . . . . . . . 9
β’ (π β π β π·) |
8 | 7 | ad2antrr 725 |
. . . . . . . 8
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β π β π·) |
9 | | elfzelz 13448 |
. . . . . . . . 9
β’ (π β
(1...(ββπ₯))
β π β
β€) |
10 | 9 | adantl 483 |
. . . . . . . 8
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β π β
β€) |
11 | 3, 4, 5, 6, 8, 10 | dchrzrhcl 26609 |
. . . . . . 7
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β (πβ(πΏβπ)) β β) |
12 | | elfznn 13477 |
. . . . . . . . . . . 12
β’ (π β
(1...(ββπ₯))
β π β
β) |
13 | 12 | adantl 483 |
. . . . . . . . . . 11
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β π β
β) |
14 | | mucl 26506 |
. . . . . . . . . . 11
β’ (π β β β
(ΞΌβπ) β
β€) |
15 | 13, 14 | syl 17 |
. . . . . . . . . 10
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β (ΞΌβπ)
β β€) |
16 | 15 | zred 12614 |
. . . . . . . . 9
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β (ΞΌβπ)
β β) |
17 | 16, 13 | nndivred 12214 |
. . . . . . . 8
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β ((ΞΌβπ) /
π) β
β) |
18 | 17 | recnd 11190 |
. . . . . . 7
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β ((ΞΌβπ) /
π) β
β) |
19 | 11, 18 | mulcld 11182 |
. . . . . 6
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β ((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) β β) |
20 | 2, 19 | fsumcl 15625 |
. . . . 5
β’ ((π β§ π₯ β β+) β
Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) β β) |
21 | | dchrvmasumif.s |
. . . . . . 7
β’ (π β seq1( + , πΉ) β π) |
22 | | climcl 15388 |
. . . . . . 7
β’ (seq1( +
, πΉ) β π β π β β) |
23 | 21, 22 | syl 17 |
. . . . . 6
β’ (π β π β β) |
24 | 23 | adantr 482 |
. . . . 5
β’ ((π β§ π₯ β β+) β π β
β) |
25 | 20, 24 | mulcld 11182 |
. . . 4
β’ ((π β§ π₯ β β+) β
(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) β β) |
26 | | 0cnd 11155 |
. . . . . 6
β’ ((π β§ π = 0) β 0 β
β) |
27 | | df-ne 2945 |
. . . . . . 7
β’ (π β 0 β Β¬ π = 0) |
28 | | dchrvmasumif.t |
. . . . . . . . . 10
β’ (π β seq1( + , πΎ) β π) |
29 | | climcl 15388 |
. . . . . . . . . 10
β’ (seq1( +
, πΎ) β π β π β β) |
30 | 28, 29 | syl 17 |
. . . . . . . . 9
β’ (π β π β β) |
31 | 30 | adantr 482 |
. . . . . . . 8
β’ ((π β§ π β 0) β π β β) |
32 | 23 | adantr 482 |
. . . . . . . 8
β’ ((π β§ π β 0) β π β β) |
33 | | simpr 486 |
. . . . . . . 8
β’ ((π β§ π β 0) β π β 0) |
34 | 31, 32, 33 | divcld 11938 |
. . . . . . 7
β’ ((π β§ π β 0) β (π / π) β β) |
35 | 27, 34 | sylan2br 596 |
. . . . . 6
β’ ((π β§ Β¬ π = 0) β (π / π) β β) |
36 | 26, 35 | ifclda 4526 |
. . . . 5
β’ (π β if(π = 0, 0, (π / π)) β β) |
37 | 36 | adantr 482 |
. . . 4
β’ ((π β§ π₯ β β+) β if(π = 0, 0, (π / π)) β β) |
38 | | rpvmasum.a |
. . . . 5
β’ (π β π β β) |
39 | | rpvmasum.1 |
. . . . 5
β’ 1 =
(0gβπΊ) |
40 | | dchrisum.n1 |
. . . . 5
β’ (π β π β 1 ) |
41 | | dchrvmasumif.f |
. . . . 5
β’ πΉ = (π β β β¦ ((πβ(πΏβπ)) / π)) |
42 | | dchrvmasumif.c |
. . . . 5
β’ (π β πΆ β (0[,)+β)) |
43 | | dchrvmasumif.1 |
. . . . 5
β’ (π β βπ¦ β (1[,)+β)(absβ((seq1( + ,
πΉ)β(ββπ¦)) β π)) β€ (πΆ / π¦)) |
44 | 4, 6, 38, 3, 5, 39, 7, 40, 41, 42, 21, 43 | dchrmusum2 26858 |
. . . 4
β’ (π β (π₯ β β+ β¦
(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π)) β π(1)) |
45 | | rpssre 12929 |
. . . . 5
β’
β+ β β |
46 | | o1const 15509 |
. . . . 5
β’
((β+ β β β§ if(π = 0, 0, (π / π)) β β) β (π₯ β β+
β¦ if(π = 0, 0, (π / π))) β π(1)) |
47 | 45, 36, 46 | sylancr 588 |
. . . 4
β’ (π β (π₯ β β+ β¦ if(π = 0, 0, (π / π))) β π(1)) |
48 | 25, 37, 44, 47 | o1mul2 15514 |
. . 3
β’ (π β (π₯ β β+ β¦
((Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· if(π = 0, 0, (π / π)))) β π(1)) |
49 | | fzfid 13885 |
. . . . . . 7
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β (1...(ββ(π₯ / π))) β Fin) |
50 | 8 | adantr 482 |
. . . . . . . . 9
β’ ((((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β§ π β
(1...(ββ(π₯ /
π)))) β π β π·) |
51 | | elfzelz 13448 |
. . . . . . . . . 10
β’ (π β
(1...(ββ(π₯ /
π))) β π β
β€) |
52 | 51 | adantl 483 |
. . . . . . . . 9
β’ ((((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β§ π β
(1...(ββ(π₯ /
π)))) β π β
β€) |
53 | 3, 4, 5, 6, 50, 52 | dchrzrhcl 26609 |
. . . . . . . 8
β’ ((((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β§ π β
(1...(ββ(π₯ /
π)))) β (πβ(πΏβπ)) β β) |
54 | | simpr 486 |
. . . . . . . . . . . . 13
β’ ((π β§ π₯ β β+) β π₯ β
β+) |
55 | 12 | nnrpd 12962 |
. . . . . . . . . . . . 13
β’ (π β
(1...(ββπ₯))
β π β
β+) |
56 | | rpdivcl 12947 |
. . . . . . . . . . . . 13
β’ ((π₯ β β+
β§ π β
β+) β (π₯ / π) β
β+) |
57 | 54, 55, 56 | syl2an 597 |
. . . . . . . . . . . 12
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β (π₯ / π) β
β+) |
58 | | elfznn 13477 |
. . . . . . . . . . . . 13
β’ (π β
(1...(ββ(π₯ /
π))) β π β
β) |
59 | 58 | nnrpd 12962 |
. . . . . . . . . . . 12
β’ (π β
(1...(ββ(π₯ /
π))) β π β
β+) |
60 | | ifcl 4536 |
. . . . . . . . . . . 12
β’ (((π₯ / π) β β+ β§ π β β+)
β if(π = 0, (π₯ / π), π) β
β+) |
61 | 57, 59, 60 | syl2an 597 |
. . . . . . . . . . 11
β’ ((((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β§ π β
(1...(ββ(π₯ /
π)))) β if(π = 0, (π₯ / π), π) β
β+) |
62 | 61 | relogcld 25994 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β§ π β
(1...(ββ(π₯ /
π)))) β
(logβif(π = 0, (π₯ / π), π)) β β) |
63 | 58 | adantl 483 |
. . . . . . . . . 10
β’ ((((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β§ π β
(1...(ββ(π₯ /
π)))) β π β
β) |
64 | 62, 63 | nndivred 12214 |
. . . . . . . . 9
β’ ((((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β§ π β
(1...(ββ(π₯ /
π)))) β
((logβif(π = 0,
(π₯ / π), π)) / π) β β) |
65 | 64 | recnd 11190 |
. . . . . . . 8
β’ ((((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β§ π β
(1...(ββ(π₯ /
π)))) β
((logβif(π = 0,
(π₯ / π), π)) / π) β β) |
66 | 53, 65 | mulcld 11182 |
. . . . . . 7
β’ ((((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β§ π β
(1...(ββ(π₯ /
π)))) β ((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)) β β) |
67 | 49, 66 | fsumcl 15625 |
. . . . . 6
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β Ξ£π β
(1...(ββ(π₯ /
π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)) β β) |
68 | 19, 67 | mulcld 11182 |
. . . . 5
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β (((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β β) |
69 | 2, 68 | fsumcl 15625 |
. . . 4
β’ ((π β§ π₯ β β+) β
Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β β) |
70 | 25, 37 | mulcld 11182 |
. . . 4
β’ ((π β§ π₯ β β+) β
((Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· if(π = 0, 0, (π / π))) β β) |
71 | | 0cn 11154 |
. . . . . . . . . 10
β’ 0 β
β |
72 | 30 | ad2antrr 725 |
. . . . . . . . . 10
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β π β
β) |
73 | | ifcl 4536 |
. . . . . . . . . 10
β’ ((0
β β β§ π
β β) β if(π
= 0, 0, π) β
β) |
74 | 71, 72, 73 | sylancr 588 |
. . . . . . . . 9
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β if(π = 0, 0, π) β
β) |
75 | 19, 67, 74 | subdid 11618 |
. . . . . . . 8
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β (((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· (Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)) β if(π = 0, 0, π))) = ((((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β (((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π)))) |
76 | 75 | sumeq2dv 15595 |
. . . . . . 7
β’ ((π β§ π₯ β β+) β
Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· (Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)) β if(π = 0, 0, π))) = Ξ£π β (1...(ββπ₯))((((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β (((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π)))) |
77 | 19, 74 | mulcld 11182 |
. . . . . . . 8
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β (((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π)) β β) |
78 | 2, 68, 77 | fsumsub 15680 |
. . . . . . 7
β’ ((π β§ π₯ β β+) β
Ξ£π β
(1...(ββπ₯))((((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β (((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π))) = (Ξ£π β (1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β Ξ£π β (1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π)))) |
79 | 20, 24, 37 | mulassd 11185 |
. . . . . . . . 9
β’ ((π β§ π₯ β β+) β
((Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· if(π = 0, 0, (π / π))) = (Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· (π Β· if(π = 0, 0, (π / π))))) |
80 | | ovif2 7460 |
. . . . . . . . . . . 12
β’ (π Β· if(π = 0, 0, (π / π))) = if(π = 0, (π Β· 0), (π Β· (π / π))) |
81 | 23 | mul01d 11361 |
. . . . . . . . . . . . . 14
β’ (π β (π Β· 0) = 0) |
82 | 81 | ifeq1d 4510 |
. . . . . . . . . . . . 13
β’ (π β if(π = 0, (π Β· 0), (π Β· (π / π))) = if(π = 0, 0, (π Β· (π / π)))) |
83 | 31, 32, 33 | divcan2d 11940 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β 0) β (π Β· (π / π)) = π) |
84 | 27, 83 | sylan2br 596 |
. . . . . . . . . . . . . 14
β’ ((π β§ Β¬ π = 0) β (π Β· (π / π)) = π) |
85 | 84 | ifeq2da 4523 |
. . . . . . . . . . . . 13
β’ (π β if(π = 0, 0, (π Β· (π / π))) = if(π = 0, 0, π)) |
86 | 82, 85 | eqtrd 2777 |
. . . . . . . . . . . 12
β’ (π β if(π = 0, (π Β· 0), (π Β· (π / π))) = if(π = 0, 0, π)) |
87 | 80, 86 | eqtrid 2789 |
. . . . . . . . . . 11
β’ (π β (π Β· if(π = 0, 0, (π / π))) = if(π = 0, 0, π)) |
88 | 87 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ π₯ β β+) β (π Β· if(π = 0, 0, (π / π))) = if(π = 0, 0, π)) |
89 | 88 | oveq2d 7378 |
. . . . . . . . 9
β’ ((π β§ π₯ β β+) β
(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· (π Β· if(π = 0, 0, (π / π)))) = (Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π))) |
90 | 71, 30, 73 | sylancr 588 |
. . . . . . . . . . 11
β’ (π β if(π = 0, 0, π) β β) |
91 | 90 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ π₯ β β+) β if(π = 0, 0, π) β β) |
92 | 2, 91, 19 | fsummulc1 15677 |
. . . . . . . . 9
β’ ((π β§ π₯ β β+) β
(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π)) = Ξ£π β (1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π))) |
93 | 79, 89, 92 | 3eqtrrd 2782 |
. . . . . . . 8
β’ ((π β§ π₯ β β+) β
Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π)) = ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· if(π = 0, 0, (π / π)))) |
94 | 93 | oveq2d 7378 |
. . . . . . 7
β’ ((π β§ π₯ β β+) β
(Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β Ξ£π β (1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· if(π = 0, 0, π))) = (Ξ£π β (1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· if(π = 0, 0, (π / π))))) |
95 | 76, 78, 94 | 3eqtrd 2781 |
. . . . . 6
β’ ((π β§ π₯ β β+) β
Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· (Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)) β if(π = 0, 0, π))) = (Ξ£π β (1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· if(π = 0, 0, (π / π))))) |
96 | 95 | mpteq2dva 5210 |
. . . . 5
β’ (π β (π₯ β β+ β¦
Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· (Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)) β if(π = 0, 0, π)))) = (π₯ β β+ β¦
(Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· if(π = 0, 0, (π / π)))))) |
97 | | dchrvmasumif.g |
. . . . . 6
β’ πΎ = (π β β β¦ ((πβ(πΏβπ)) Β· ((logβπ) / π))) |
98 | | dchrvmasumif.e |
. . . . . 6
β’ (π β πΈ β (0[,)+β)) |
99 | | dchrvmasumif.2 |
. . . . . 6
β’ (π β βπ¦ β (3[,)+β)(absβ((seq1( + ,
πΎ)β(ββπ¦)) β π)) β€ (πΈ Β· ((logβπ¦) / π¦))) |
100 | 4, 6, 38, 3, 5, 39, 7, 40, 41, 42, 21, 43, 97, 98, 28, 99 | dchrvmasumiflem1 26865 |
. . . . 5
β’ (π β (π₯ β β+ β¦
Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· (Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)) β if(π = 0, 0, π)))) β π(1)) |
101 | 96, 100 | eqeltrrd 2839 |
. . . 4
β’ (π β (π₯ β β+ β¦
(Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))) β ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· if(π = 0, 0, (π / π))))) β π(1)) |
102 | 69, 70, 101 | o1dif 15519 |
. . 3
β’ (π β ((π₯ β β+ β¦
Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)))) β π(1) β (π₯ β β+
β¦ ((Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· if(π = 0, 0, (π / π)))) β π(1))) |
103 | 48, 102 | mpbird 257 |
. 2
β’ (π β (π₯ β β+ β¦
Ξ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)))) β π(1)) |
104 | 7 | ad2antrr 725 |
. . . . . 6
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β π β π·) |
105 | | elfzelz 13448 |
. . . . . . 7
β’ (π β
(1...(ββπ₯))
β π β
β€) |
106 | 105 | adantl 483 |
. . . . . 6
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β π β
β€) |
107 | 3, 4, 5, 6, 104, 106 | dchrzrhcl 26609 |
. . . . 5
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β (πβ(πΏβπ)) β β) |
108 | | elfznn 13477 |
. . . . . . . 8
β’ (π β
(1...(ββπ₯))
β π β
β) |
109 | 108 | adantl 483 |
. . . . . . 7
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β π β
β) |
110 | | vmacl 26483 |
. . . . . . . 8
β’ (π β β β
(Ξβπ) β
β) |
111 | | nndivre 12201 |
. . . . . . . 8
β’
(((Ξβπ)
β β β§ π
β β) β ((Ξβπ) / π) β β) |
112 | 110, 111 | mpancom 687 |
. . . . . . 7
β’ (π β β β
((Ξβπ) / π) β
β) |
113 | 109, 112 | syl 17 |
. . . . . 6
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β ((Ξβπ)
/ π) β
β) |
114 | 113 | recnd 11190 |
. . . . 5
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β ((Ξβπ)
/ π) β
β) |
115 | 107, 114 | mulcld 11182 |
. . . 4
β’ (((π β§ π₯ β β+) β§ π β
(1...(ββπ₯)))
β ((πβ(πΏβπ)) Β· ((Ξβπ) / π)) β β) |
116 | 2, 115 | fsumcl 15625 |
. . 3
β’ ((π β§ π₯ β β+) β
Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((Ξβπ) / π)) β β) |
117 | | relogcl 25947 |
. . . . . 6
β’ (π₯ β β+
β (logβπ₯) β
β) |
118 | 117 | adantl 483 |
. . . . 5
β’ ((π β§ π₯ β β+) β
(logβπ₯) β
β) |
119 | 118 | recnd 11190 |
. . . 4
β’ ((π β§ π₯ β β+) β
(logβπ₯) β
β) |
120 | | ifcl 4536 |
. . . 4
β’
(((logβπ₯)
β β β§ 0 β β) β if(π = 0, (logβπ₯), 0) β β) |
121 | 119, 71, 120 | sylancl 587 |
. . 3
β’ ((π β§ π₯ β β+) β if(π = 0, (logβπ₯), 0) β
β) |
122 | 116, 121 | addcld 11181 |
. 2
β’ ((π β§ π₯ β β+) β
(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((Ξβπ) / π)) + if(π = 0, (logβπ₯), 0)) β β) |
123 | 122 | abscld 15328 |
. . . 4
β’ ((π β§ π₯ β β+) β
(absβ(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((Ξβπ) / π)) + if(π = 0, (logβπ₯), 0))) β β) |
124 | 123 | adantrr 716 |
. . 3
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(absβ(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((Ξβπ) / π)) + if(π = 0, (logβπ₯), 0))) β β) |
125 | 38 | adantr 482 |
. . . . 5
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β π β
β) |
126 | 7 | adantr 482 |
. . . . 5
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β π β π·) |
127 | 40 | adantr 482 |
. . . . 5
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β π β 1 ) |
128 | | simprl 770 |
. . . . 5
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β π₯ β
β+) |
129 | | simprr 772 |
. . . . 5
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β 1 β€ π₯) |
130 | 4, 6, 125, 3, 5, 39, 126, 127, 128, 129 | dchrvmasum2if 26861 |
. . . 4
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((Ξβπ) / π)) + if(π = 0, (logβπ₯), 0)) = Ξ£π β (1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π)))) |
131 | 130 | fveq2d 6851 |
. . 3
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(absβ(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((Ξβπ) / π)) + if(π = 0, (logβπ₯), 0))) = (absβΞ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))))) |
132 | 124, 131 | eqled 11265 |
. 2
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(absβ(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((Ξβπ) / π)) + if(π = 0, (logβπ₯), 0))) β€ (absβΞ£π β
(1...(ββπ₯))(((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· Ξ£π β (1...(ββ(π₯ / π)))((πβ(πΏβπ)) Β· ((logβif(π = 0, (π₯ / π), π)) / π))))) |
133 | 1, 103, 69, 122, 132 | o1le 15544 |
1
β’ (π β (π₯ β β+ β¦
(Ξ£π β
(1...(ββπ₯))((πβ(πΏβπ)) Β· ((Ξβπ) / π)) + if(π = 0, (logβπ₯), 0))) β π(1)) |