Step | Hyp | Ref
| Expression |
1 | | nnre 12216 |
. . . . . . 7
β’ (π΄ β β β π΄ β
β) |
2 | | chtval 26604 |
. . . . . . 7
β’ (π΄ β β β
(ΞΈβπ΄) =
Ξ£π β ((0[,]π΄) β© β)(logβπ)) |
3 | 1, 2 | syl 17 |
. . . . . 6
β’ (π΄ β β β
(ΞΈβπ΄) =
Ξ£π β ((0[,]π΄) β© β)(logβπ)) |
4 | | 2eluzge1 12875 |
. . . . . . . . . 10
β’ 2 β
(β€β₯β1) |
5 | | ppisval2 26599 |
. . . . . . . . . 10
β’ ((π΄ β β β§ 2 β
(β€β₯β1)) β ((0[,]π΄) β© β) =
((1...(ββπ΄))
β© β)) |
6 | 1, 4, 5 | sylancl 587 |
. . . . . . . . 9
β’ (π΄ β β β
((0[,]π΄) β© β) =
((1...(ββπ΄))
β© β)) |
7 | | nnz 12576 |
. . . . . . . . . . . 12
β’ (π΄ β β β π΄ β
β€) |
8 | | flid 13770 |
. . . . . . . . . . . 12
β’ (π΄ β β€ β
(ββπ΄) = π΄) |
9 | 7, 8 | syl 17 |
. . . . . . . . . . 11
β’ (π΄ β β β
(ββπ΄) = π΄) |
10 | 9 | oveq2d 7422 |
. . . . . . . . . 10
β’ (π΄ β β β
(1...(ββπ΄)) =
(1...π΄)) |
11 | 10 | ineq1d 4211 |
. . . . . . . . 9
β’ (π΄ β β β
((1...(ββπ΄))
β© β) = ((1...π΄)
β© β)) |
12 | 6, 11 | eqtrd 2773 |
. . . . . . . 8
β’ (π΄ β β β
((0[,]π΄) β© β) =
((1...π΄) β©
β)) |
13 | 12 | sumeq1d 15644 |
. . . . . . 7
β’ (π΄ β β β
Ξ£π β ((0[,]π΄) β© β)(logβπ) = Ξ£π β ((1...π΄) β© β)(logβπ)) |
14 | | inss1 4228 |
. . . . . . . 8
β’
((1...π΄) β©
β) β (1...π΄) |
15 | | elinel1 4195 |
. . . . . . . . . 10
β’ (π β ((1...π΄) β© β) β π β (1...π΄)) |
16 | | elfznn 13527 |
. . . . . . . . . . . . . 14
β’ (π β (1...π΄) β π β β) |
17 | 16 | adantl 483 |
. . . . . . . . . . . . 13
β’ ((π΄ β β β§ π β (1...π΄)) β π β β) |
18 | 17 | nnrpd 13011 |
. . . . . . . . . . . 12
β’ ((π΄ β β β§ π β (1...π΄)) β π β β+) |
19 | 18 | relogcld 26123 |
. . . . . . . . . . 11
β’ ((π΄ β β β§ π β (1...π΄)) β (logβπ) β β) |
20 | 19 | recnd 11239 |
. . . . . . . . . 10
β’ ((π΄ β β β§ π β (1...π΄)) β (logβπ) β β) |
21 | 15, 20 | sylan2 594 |
. . . . . . . . 9
β’ ((π΄ β β β§ π β ((1...π΄) β© β)) β (logβπ) β
β) |
22 | 21 | ralrimiva 3147 |
. . . . . . . 8
β’ (π΄ β β β
βπ β
((1...π΄) β©
β)(logβπ)
β β) |
23 | | fzfi 13934 |
. . . . . . . . . 10
β’
(1...π΄) β
Fin |
24 | 23 | olci 865 |
. . . . . . . . 9
β’
((1...π΄) β
(β€β₯β1) β¨ (1...π΄) β Fin) |
25 | | sumss2 15669 |
. . . . . . . . 9
β’
(((((1...π΄) β©
β) β (1...π΄)
β§ βπ β
((1...π΄) β©
β)(logβπ)
β β) β§ ((1...π΄) β (β€β₯β1)
β¨ (1...π΄) β Fin))
β Ξ£π β
((1...π΄) β©
β)(logβπ) =
Ξ£π β (1...π΄)if(π β ((1...π΄) β© β), (logβπ), 0)) |
26 | 24, 25 | mpan2 690 |
. . . . . . . 8
β’
((((1...π΄) β©
β) β (1...π΄)
β§ βπ β
((1...π΄) β©
β)(logβπ)
β β) β Ξ£π β ((1...π΄) β© β)(logβπ) = Ξ£π β (1...π΄)if(π β ((1...π΄) β© β), (logβπ), 0)) |
27 | 14, 22, 26 | sylancr 588 |
. . . . . . 7
β’ (π΄ β β β
Ξ£π β ((1...π΄) β© β)(logβπ) = Ξ£π β (1...π΄)if(π β ((1...π΄) β© β), (logβπ), 0)) |
28 | 13, 27 | eqtrd 2773 |
. . . . . 6
β’ (π΄ β β β
Ξ£π β ((0[,]π΄) β© β)(logβπ) = Ξ£π β (1...π΄)if(π β ((1...π΄) β© β), (logβπ), 0)) |
29 | 3, 28 | eqtrd 2773 |
. . . . 5
β’ (π΄ β β β
(ΞΈβπ΄) =
Ξ£π β (1...π΄)if(π β ((1...π΄) β© β), (logβπ), 0)) |
30 | | elin 3964 |
. . . . . . . 8
β’ (π β ((1...π΄) β© β) β (π β (1...π΄) β§ π β β)) |
31 | 30 | baibr 538 |
. . . . . . 7
β’ (π β (1...π΄) β (π β β β π β ((1...π΄) β© β))) |
32 | 31 | ifbid 4551 |
. . . . . 6
β’ (π β (1...π΄) β if(π β β, (logβπ), 0) = if(π β ((1...π΄) β© β), (logβπ), 0)) |
33 | 32 | sumeq2i 15642 |
. . . . 5
β’
Ξ£π β
(1...π΄)if(π β β,
(logβπ), 0) =
Ξ£π β (1...π΄)if(π β ((1...π΄) β© β), (logβπ), 0) |
34 | 29, 33 | eqtr4di 2791 |
. . . 4
β’ (π΄ β β β
(ΞΈβπ΄) =
Ξ£π β (1...π΄)if(π β β, (logβπ), 0)) |
35 | | eleq1w 2817 |
. . . . . . . 8
β’ (π = π β (π β β β π β β)) |
36 | | fveq2 6889 |
. . . . . . . 8
β’ (π = π β (logβπ) = (logβπ)) |
37 | 35, 36 | ifbieq1d 4552 |
. . . . . . 7
β’ (π = π β if(π β β, (logβπ), 0) = if(π β β, (logβπ), 0)) |
38 | | eqid 2733 |
. . . . . . 7
β’ (π β β β¦ if(π β β,
(logβπ), 0)) = (π β β β¦ if(π β β,
(logβπ),
0)) |
39 | | fvex 6902 |
. . . . . . . 8
β’
(logβπ) β
V |
40 | | 0cn 11203 |
. . . . . . . . 9
β’ 0 β
β |
41 | 40 | elexi 3494 |
. . . . . . . 8
β’ 0 β
V |
42 | 39, 41 | ifex 4578 |
. . . . . . 7
β’ if(π β β,
(logβπ), 0) β
V |
43 | 37, 38, 42 | fvmpt 6996 |
. . . . . 6
β’ (π β β β ((π β β β¦ if(π β β,
(logβπ),
0))βπ) = if(π β β,
(logβπ),
0)) |
44 | 17, 43 | syl 17 |
. . . . 5
β’ ((π΄ β β β§ π β (1...π΄)) β ((π β β β¦ if(π β β, (logβπ), 0))βπ) = if(π β β, (logβπ), 0)) |
45 | | elnnuz 12863 |
. . . . . 6
β’ (π΄ β β β π΄ β
(β€β₯β1)) |
46 | 45 | biimpi 215 |
. . . . 5
β’ (π΄ β β β π΄ β
(β€β₯β1)) |
47 | | ifcl 4573 |
. . . . . 6
β’
(((logβπ)
β β β§ 0 β β) β if(π β β, (logβπ), 0) β
β) |
48 | 20, 40, 47 | sylancl 587 |
. . . . 5
β’ ((π΄ β β β§ π β (1...π΄)) β if(π β β, (logβπ), 0) β
β) |
49 | 44, 46, 48 | fsumser 15673 |
. . . 4
β’ (π΄ β β β
Ξ£π β (1...π΄)if(π β β, (logβπ), 0) = (seq1( + , (π β β β¦ if(π β β,
(logβπ),
0)))βπ΄)) |
50 | 34, 49 | eqtrd 2773 |
. . 3
β’ (π΄ β β β
(ΞΈβπ΄) = (seq1(
+ , (π β β
β¦ if(π β
β, (logβπ),
0)))βπ΄)) |
51 | 50 | fveq2d 6893 |
. 2
β’ (π΄ β β β
(expβ(ΞΈβπ΄)) = (expβ(seq1( + , (π β β β¦ if(π β β,
(logβπ),
0)))βπ΄))) |
52 | | addcl 11189 |
. . . 4
β’ ((π β β β§ π β β) β (π + π) β β) |
53 | 52 | adantl 483 |
. . 3
β’ ((π΄ β β β§ (π β β β§ π β β)) β (π + π) β β) |
54 | 44, 48 | eqeltrd 2834 |
. . 3
β’ ((π΄ β β β§ π β (1...π΄)) β ((π β β β¦ if(π β β, (logβπ), 0))βπ) β β) |
55 | | efadd 16034 |
. . . 4
β’ ((π β β β§ π β β) β
(expβ(π + π)) = ((expβπ) Β· (expβπ))) |
56 | 55 | adantl 483 |
. . 3
β’ ((π΄ β β β§ (π β β β§ π β β)) β
(expβ(π + π)) = ((expβπ) Β· (expβπ))) |
57 | | 1nn 12220 |
. . . . . . 7
β’ 1 β
β |
58 | | ifcl 4573 |
. . . . . . 7
β’ ((π β β β§ 1 β
β) β if(π β
β, π, 1) β
β) |
59 | 17, 57, 58 | sylancl 587 |
. . . . . 6
β’ ((π΄ β β β§ π β (1...π΄)) β if(π β β, π, 1) β β) |
60 | 59 | nnrpd 13011 |
. . . . 5
β’ ((π΄ β β β§ π β (1...π΄)) β if(π β β, π, 1) β
β+) |
61 | 60 | reeflogd 26124 |
. . . 4
β’ ((π΄ β β β§ π β (1...π΄)) β (expβ(logβif(π β β, π, 1))) = if(π β β, π, 1)) |
62 | | fvif 6905 |
. . . . . . 7
β’
(logβif(π
β β, π, 1)) =
if(π β β,
(logβπ),
(logβ1)) |
63 | | log1 26086 |
. . . . . . . 8
β’
(logβ1) = 0 |
64 | | ifeq2 4533 |
. . . . . . . 8
β’
((logβ1) = 0 β if(π β β, (logβπ), (logβ1)) = if(π β β,
(logβπ),
0)) |
65 | 63, 64 | ax-mp 5 |
. . . . . . 7
β’ if(π β β,
(logβπ),
(logβ1)) = if(π
β β, (logβπ), 0) |
66 | 62, 65 | eqtri 2761 |
. . . . . 6
β’
(logβif(π
β β, π, 1)) =
if(π β β,
(logβπ),
0) |
67 | 44, 66 | eqtr4di 2791 |
. . . . 5
β’ ((π΄ β β β§ π β (1...π΄)) β ((π β β β¦ if(π β β, (logβπ), 0))βπ) = (logβif(π β β, π, 1))) |
68 | 67 | fveq2d 6893 |
. . . 4
β’ ((π΄ β β β§ π β (1...π΄)) β (expβ((π β β β¦ if(π β β, (logβπ), 0))βπ)) = (expβ(logβif(π β β, π, 1)))) |
69 | | id 22 |
. . . . . . 7
β’ (π = π β π = π) |
70 | 35, 69 | ifbieq1d 4552 |
. . . . . 6
β’ (π = π β if(π β β, π, 1) = if(π β β, π, 1)) |
71 | | prmorcht.1 |
. . . . . 6
β’ πΉ = (π β β β¦ if(π β β, π, 1)) |
72 | | vex 3479 |
. . . . . . 7
β’ π β V |
73 | 57 | elexi 3494 |
. . . . . . 7
β’ 1 β
V |
74 | 72, 73 | ifex 4578 |
. . . . . 6
β’ if(π β β, π, 1) β V |
75 | 70, 71, 74 | fvmpt 6996 |
. . . . 5
β’ (π β β β (πΉβπ) = if(π β β, π, 1)) |
76 | 17, 75 | syl 17 |
. . . 4
β’ ((π΄ β β β§ π β (1...π΄)) β (πΉβπ) = if(π β β, π, 1)) |
77 | 61, 68, 76 | 3eqtr4d 2783 |
. . 3
β’ ((π΄ β β β§ π β (1...π΄)) β (expβ((π β β β¦ if(π β β, (logβπ), 0))βπ)) = (πΉβπ)) |
78 | 53, 54, 46, 56, 77 | seqhomo 14012 |
. 2
β’ (π΄ β β β
(expβ(seq1( + , (π
β β β¦ if(π
β β, (logβπ), 0)))βπ΄)) = (seq1( Β· , πΉ)βπ΄)) |
79 | 51, 78 | eqtrd 2773 |
1
β’ (π΄ β β β
(expβ(ΞΈβπ΄)) = (seq1( Β· , πΉ)βπ΄)) |