Nearby theorems
|
|
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > breprexp | Structured version Visualization version Unicode version | ||
Description: Express the  th power of the finite
series in terms of the number
of representations of integers  as sums of terms. This is
a general formulation which allows logarithmic weighting of the sums
(see https://mathoverflow.net/questions/253246) and a mix of different
smoothing functions taken into account in  . See breprexpnat 30712
for the simple case presented in the proposition of [Nathanson]
p. 123. (Contributed by Thierry Arnoux,
6-Dec-2021.) |
| Ref | Expression |
|---|---|
| breprexp.n |
        |
| breprexp.s |
|
| breprexp.z |
  |
| breprexp.h |
  ..^   |
| Ref | Expression |
|---|---|
| breprexp |
  ..^ 
       repr
..^ |
,, ,,, ,,, , ,,,, ,, , ,, ,,,
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | breprexp.s |
. 2
| |
| 2 | nn0ssre 11296 |
. . . . . 6
  | |
| 3 | 2 | a1i 11 |
. . . . 5
|
| 4 | 3 | sselda 3603 |
. . . 4
![/\](wedge.gif "/\"){kind=small}
|
| 5 | leid 10133 |
. . . 4
 | |
| 6 | 4, 5 | syl 17 |
. . 3
|
| 7 | breq1 4656 |
. . . . 5
| |
| 8 | oveq2 6658 |
. . . . . . 7
..^ ..^ | |
| 9 | 8 | prodeq1d 14651 |
. . . . . 6
..^ ..^
|
| 10 | oveq1 6657 |
. . . . . . . 8
| |
| 11 | 10 | oveq2d 6666 |
. . . . . . 7
|
| 12 | fveq2 6191 |
. . . . . . . . . 10
repr repr | |
| 13 | 12 | oveqd 6667 |
. . . . . . . . 9
repr repr |
| 14 | 8 | prodeq1d 14651 |
. . . . . . . . . . 11
..^ ..^ |
| 15 | 14 | oveq1d 6665 |
. . . . . . . . . 10
..^
..^ |
| 16 | 15 | adantr 481 |
. . . . . . . . 9
repr ..^
..^ |
| 17 | 13, 16 | sumeq12dv 14437 |
. . . . . . . 8
repr ..^
repr ..^ |
| 18 | 17 | adantr 481 |
. . . . . . 7
repr
..^
repr ..^ |
| 19 | 11, 18 | sumeq12dv 14437 |
. . . . . 6
repr
..^ repr ..^ |
| 20 | 9, 19 | eqeq12d 2637 |
. . . . 5
..^ repr ..^
..^ repr
..^ |
| 21 | 7, 20 | imbi12d 334 |
. . . 4
..^
repr
..^
..^ repr
..^ |
| 22 | breq1 4656 |
. . . . 5
| |
| 23 | oveq2 6658 |
. . . . . . 7
..^ ..^ | |
| 24 | 23 | prodeq1d 14651 |
. . . . . 6
..^ ..^
|
| 25 | oveq1 6657 |
. . . . . . . 8
| |
| 26 | 25 | oveq2d 6666 |
. . . . . . 7
|
| 27 | fveq2 6191 |
. . . . . . . . . 10
repr repr | |
| 28 | 27 | oveqd 6667 |
. . . . . . . . 9
repr repr |
| 29 | 23 | prodeq1d 14651 |
. . . . . . . . . . 11
..^ ..^ |
| 30 | 29 | oveq1d 6665 |
. . . . . . . . . 10
..^
..^ |
| 31 | 30 | adantr 481 |
. . . . . . . . 9
repr
..^ ..^ |
| 32 | 28, 31 | sumeq12dv 14437 |
. . . . . . . 8
repr ..^
repr ..^ |
| 33 | 32 | adantr 481 |
. . . . . . 7
repr
..^
repr ..^ |
| 34 | 26, 33 | sumeq12dv 14437 |
. . . . . 6
repr
..^ repr ..^ |
| 35 | 24, 34 | eqeq12d 2637 |
. . . . 5
..^ repr ..^
..^ repr
..^ |
| 36 | 22, 35 | imbi12d 334 |
. . . 4
..^
repr
..^
..^ repr
..^ |
| 37 | breq1 4656 |
. . . . 5
| |
| 38 | oveq2 6658 |
. . . . . . 7
..^ ..^ | |
| 39 | 38 | prodeq1d 14651 |
. . . . . 6
..^ ..^
|
| 40 | oveq1 6657 |
. . . . . . . 8
| |
| 41 | 40 | oveq2d 6666 |
. . . . . . 7
|
| 42 | fveq2 6191 |
. . . . . . . . . 10
repr repr | |
| 43 | 42 | oveqd 6667 |
. . . . . . . . 9
repr repr |
| 44 | 38 | prodeq1d 14651 |
. . . . . . . . . . 11
..^ ..^ |
| 45 | 44 | oveq1d 6665 |
. . . . . . . . . 10
..^
..^ |
| 46 | 45 | adantr 481 |
. . . . . . . . 9
repr
..^ ..^ |
| 47 | 43, 46 | sumeq12dv 14437 |
. . . . . . . 8
repr ..^
repr ..^ |
| 48 | 47 | adantr 481 |
. . . . . . 7
repr
..^
repr ..^ |
| 49 | 41, 48 | sumeq12dv 14437 |
. . . . . 6
repr
..^ repr ..^ |
| 50 | 39, 49 | eqeq12d 2637 |
. . . . 5
..^ repr ..^
..^ repr
..^ |
| 51 | 37, 50 | imbi12d 334 |
. . . 4
..^
repr
..^
..^ repr
..^ |
| 52 | breq1 4656 |
. . . . 5
| |
| 53 | oveq2 6658 |
. . . . . . 7
..^ ..^ | |
| 54 | 53 | prodeq1d 14651 |
. . . . . 6
..^ ..^
|
| 55 | oveq1 6657 |
. . . . . . . 8
| |
| 56 | 55 | oveq2d 6666 |
. . . . . . 7
|
| 57 | fveq2 6191 |
. . . . . . . . . 10
repr repr | |
| 58 | 57 | oveqd 6667 |
. . . . . . . . 9
repr repr |
| 59 | 53 | prodeq1d 14651 |
. . . . . . . . . . 11
..^ ..^ |
| 60 | 59 | oveq1d 6665 |
. . . . . . . . . 10
..^
..^ |
| 61 | 60 | adantr 481 |
. . . . . . . . 9
repr
..^ ..^ |
| 62 | 58, 61 | sumeq12dv 14437 |
. . . . . . . 8
repr ..^
repr ..^ |
| 63 | 62 | adantr 481 |
. . . . . . 7
repr
..^
repr ..^ |
| 64 | 56, 63 | sumeq12dv 14437 |
. . . . . 6
repr
..^ repr ..^ |
| 65 | 54, 64 | eqeq12d 2637 |
. . . . 5
..^ repr ..^
..^ repr
..^ |
| 66 | 52, 65 | imbi12d 334 |
. . . 4
..^
repr
..^
..^ repr
..^ |
| 67 | 0nn0 11307 |
. . . . . . . 8
| |
| 68 | fz1ssnn 12372 |
. . . . . . . . . . . . 13
| |
| 69 | 68 | a1i 11 |
. . . . . . . . . . . 12
|
| 70 | 0zd 11389 |
. . . . . . . . . . . 12
 | |
| 71 | breprexp.n |
. . . . . . . . . . . 12
| |
| 72 | 69, 70, 71 | repr0 30689 |
. . . . . . . . . . 11
repr      |
| 73 | eqid 2622 |
. . . . . . . . . . . 12
| |
| 74 | 73 | iftruei 4093 |
. . . . . . . . . . 11
|
| 75 | 72, 74 | syl6eq 2672 |
. . . . . . . . . 10
repr |
| 76 | snfi 8038 |
. . . . . . . . . 10
 | |
| 77 | 75, 76 | syl6eqel 2709 |
. . . . . . . . 9
repr |
| 78 | fzo0 12492 |
. . . . . . . . . . . . . . . 16
..^ | |
| 79 | 78 | prodeq1i 14648 |
. . . . . . . . . . . . . . 15
..^ |
| 80 | prod0 14673 |
. . . . . . . . . . . . . . 15
| |
| 81 | 79, 80 | eqtri 2644 |
. . . . . . . . . . . . . 14
..^ |
| 82 | 81 | a1i 11 |
. . . . . . . . . . . . 13
..^ |
| 83 | breprexp.z |
. . . . . . . . . . . . . 14
| |
| 84 | exp0 12864 |
. . . . . . . . . . . . . 14
| |
| 85 | 83, 84 | syl 17 |
. . . . . . . . . . . . 13
|
| 86 | 82, 85 | oveq12d 6668 |
. . . . . . . . . . . 12
..^ |
| 87 | ax-1cn 9994 |
. . . . . . . . . . . . 13
| |
| 88 | 87 | mulid1i 10042 |
. . . . . . . . . . . 12
|
| 89 | 86, 88 | syl6eq 2672 |
. . . . . . . . . . 11
..^ |
| 90 | 89, 87 | syl6eqel 2709 |
. . . . . . . . . 10
..^ |
| 91 | 90 | adantr 481 |
. . . . . . . . 9
repr
..^ |
| 92 | 77, 91 | fsumcl 14464 |
. . . . . . . 8
repr
..^ |
| 93 | oveq2 6658 |
. . . . . . . . . 10
repr repr | |
| 94 | simpl 473 |
. . . . . . . . . . . 12
repr
| |
| 95 | 94 | oveq2d 6666 |
. . . . . . . . . . 11
repr |
| 96 | 95 | oveq2d 6666 |
. . . . . . . . . 10
repr ..^
..^ |
| 97 | 93, 96 | sumeq12dv 14437 |
. . . . . . . . 9
repr ..^
repr ..^ |
| 98 | 97 | sumsn 14475 |
. . . . . . . 8
repr ..^ repr ..^
repr ..^ |
| 99 | 67, 92, 98 | sylancr 695 |
. . . . . . 7
repr
..^
repr ..^ |
| 100 | 75 | sumeq1d 14431 |
. . . . . . 7
repr
..^
..^ |
| 101 | 0ex 4790 |
. . . . . . . . 9
 | |
| 102 | 78 | prodeq1i 14648 |
. . . . . . . . . . . . 13
..^
|
| 103 | prod0 14673 |
. . . . . . . . . . . . 13
| |
| 104 | 102, 103 | eqtri 2644 |
. . . . . . . . . . . 12
..^ |
| 105 | 104 | a1i 11 |
. . . . . . . . . . 11
..^ |
| 106 | 105, 87 | syl6eqel 2709 |
. . . . . . . . . 10
..^ |
| 107 | 85, 87 | syl6eqel 2709 |
. . . . . . . . . 10
|
| 108 | 106, 107 | mulcld 10060 |
. . . . . . . . 9
..^ |
| 109 | fveq1 6190 |
. . . . . . . . . . . . . 14
| |
| 110 | 109 | fveq2d 6195 |
. . . . . . . . . . . . 13
|
| 111 | 110 | ralrimivw 2967 |
. . . . . . . . . . . 12
 ..^ |
| 112 | 111 | prodeq2d 14652 |
. . . . . . . . . . 11
..^ ..^ |
| 113 | 112 | oveq1d 6665 |
. . . . . . . . . 10
..^
..^ |
| 114 | 113 | sumsn 14475 |
. . . . . . . . 9
..^ ..^
..^ |
| 115 | 101, 108, 114 | sylancr 695 |
. . . . . . . 8
..^ ..^ |
| 116 | 105, 85 | oveq12d 6668 |
. . . . . . . . 9
..^ |
| 117 | 116, 86, 89 | 3eqtr2d 2662 |
. . . . . . . 8
..^ |
| 118 | 115, 117 | eqtrd 2656 |
. . . . . . 7
..^ |
| 119 | 99, 100, 118 | 3eqtrd 2660 |
. . . . . 6
repr
..^ |
| 120 | 71 | nn0cnd 11353 |
. . . . . . . . . 10
|
| 121 | 120 | mul02d 10234 |
. . . . . . . . 9
|
| 122 | 121 | oveq2d 6666 |
. . . . . . . 8
|
| 123 | fz0sn 12439 |
. . . . . . . 8
| |
| 124 | 122, 123 | syl6eq 2672 |
. . . . . . 7
|
| 125 | 124 | sumeq1d 14431 |
. . . . . 6
repr ..^ repr ..^ |
| 126 | 78 | prodeq1i 14648 |
. . . . . . . 8
..^ |
| 127 | prod0 14673 |
. . . . . . . 8
| |
| 128 | 126, 127 | eqtri 2644 |
. . . . . . 7
..^ |
| 129 | 128 | a1i 11 |
. . . . . 6
..^
|
| 130 | 119, 125, 129 | 3eqtr4rd 2667 |
. . . . 5
..^
repr
..^ |
| 131 | 130 | a1d 25 |
. . . 4
..^
repr
..^ |
| 132 | simpll 790 |
. . . . . 6
..^
repr
..^
| |
| 133 | simplr 792 |
. . . . . . 7
..^
repr
..^
..^ repr
..^ | |
| 134 | oveq2 6658 |
. . . . . . . . . . . 12
repr repr | |
| 135 | oveq2 6658 |
. . . . . . . . . . . . . 14
| |
| 136 | 135 | oveq2d 6666 |
. . . . . . . . . . . . 13
..^
..^ |
| 137 | 136 | adantr 481 |
. . . . . . . . . . . 12
repr
..^ ..^ |
| 138 | 134, 137 | sumeq12dv 14437 |
. . . . . . . . . . 11
repr ..^
repr ..^ |
| 139 | 138 | cbvsumv 14426 |
. . . . . . . . . 10
repr
..^ repr ..^ |
| 140 | 139 | eqeq2i 2634 |
. . . . . . . . 9
..^ repr ..^
..^ repr
..^ |
| 141 | simpl 473 |
. . . . . . . . . . . . . . . 16
| |
| 142 | 141 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
|
| 143 | 142 | fveq1d 6193 |
. . . . . . . . . . . . . 14
|
| 144 | 143 | oveq1d 6665 |
. . . . . . . . . . . . 13
|
| 145 | 144 | sumeq2dv 14433 |
. . . . . . . . . . . 12
|
| 146 | 145 | cbvprodv 14646 |
. . . . . . . . . . 11
..^ ..^
|
| 147 | fveq2 6191 |
. . . . . . . . . . . . . . 15
| |
| 148 | oveq2 6658 |
. . . . . . . . . . . . . . 15
| |
| 149 | 147, 148 | oveq12d 6668 |
. . . . . . . . . . . . . 14
|
| 150 | 149 | cbvsumv 14426 |
. . . . . . . . . . . . 13
|
| 151 | 150 | a1i 11 |
. . . . . . . . . . . 12
..^
|
| 152 | 151 | prodeq2i 14649 |
. . . . . . . . . . 11
..^ ..^
|
| 153 | 146, 152 | eqtri 2644 |
. . . . . . . . . 10
..^ ..^
|
| 154 | fveq2 6191 |
. . . . . . . . . . . . . . . . . 18
| |
| 155 | fveq2 6191 |
. . . . . . . . . . . . . . . . . 18
| |
| 156 | 154, 155 | fveq12d 6197 |
. . . . . . . . . . . . . . . . 17
|
| 157 | 156 | cbvprodv 14646 |
. . . . . . . . . . . . . . . 16
..^ ..^ |
| 158 | 157 | oveq1i 6660 |
. . . . . . . . . . . . . . 15
..^
..^ |
| 159 | 158 | a1i 11 |
. . . . . . . . . . . . . 14
repr ..^ ..^ |
| 160 | 159 | sumeq2i 14429 |
. . . . . . . . . . . . 13
repr ..^
repr ..^ |
| 161 | simpl 473 |
. . . . . . . . . . . . . . . . . 18
..^ | |
| 162 | 161 | fveq1d 6193 |
. . . . . . . . . . . . . . . . 17
..^ |
| 163 | 162 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
..^ |
| 164 | 163 | prodeq2dv 14653 |
. . . . . . . . . . . . . . 15
..^ ..^ |
| 165 | 164 | oveq1d 6665 |
. . . . . . . . . . . . . 14
..^
..^ |
| 166 | 165 | cbvsumv 14426 |
. . . . . . . . . . . . 13
repr ..^
repr ..^ |
| 167 | 160, 166 | eqtri 2644 |
. . . . . . . . . . . 12
repr ..^
repr ..^ |
| 168 | 167 | a1i 11 |
. . . . . . . . . . 11
repr ..^
repr ..^ |
| 169 | 168 | sumeq2i 14429 |
. . . . . . . . . 10
repr
..^ repr ..^ |
| 170 | 153, 169 | eqeq12i 2636 |
. . . . . . . . 9
..^ repr ..^
..^ repr
..^ |
| 171 | 140, 170 | bitri 264 |
. . . . . . . 8
..^ repr ..^
..^ repr
..^ |
| 172 | 171 | imbi2i 326 |
. . . . . . 7
..^ repr ..^
..^ repr
..^ |
| 173 | 133, 172 | sylib 208 |
. . . . . 6
..^
repr
..^
..^ repr
..^ |
| 174 | simpr 477 |
. . . . . 6
..^
repr
..^
| |
| 175 | 71 | ad3antrrr 766 |
. . . . . . 7
..^
repr
..^
|
| 176 | 1 | ad3antrrr 766 |
. . . . . . 7
..^
repr
..^
|
| 177 | 83 | ad3antrrr 766 |
. . . . . . 7
..^
repr
..^
|
| 178 | breprexp.h |
. . . . . . . 8
..^ | |
| 179 | 178 | ad3antrrr 766 |
. . . . . . 7
..^
repr
..^
..^ |
| 180 | simpllr 799 |
. . . . . . 7
..^
repr
..^
| |
| 181 | simpr 477 |
. . . . . . 7
..^
repr
..^
| |
| 182 | 2, 180 | sseldi 3601 |
. . . . . . . . 9
..^
repr
..^
|
| 183 | 1red 10055 |
. . . . . . . . . 10
..^
repr
..^
| |
| 184 | 182, 183 | readdcld 10069 |
. . . . . . . . 9
..^
repr
..^
|
| 185 | 2, 176 | sseldi 3601 |
. . . . . . . . 9
..^
repr
..^
|
| 186 | 182 | ltp1d 10954 |
. . . . . . . . . 10
..^
repr
..^
 |
| 187 | 182, 184, 186 | ltled 10185 |
. . . . . . . . 9
..^
repr
..^
|
| 188 | 182, 184, 185, 187, 181 | letrd 10194 |
. . . . . . . 8
..^
repr
..^
|
| 189 | simplr 792 |
. . . . . . . . 9
..^
repr
..^
..^ repr
..^ | |
| 190 | 189, 172 | sylibr 224 |
. . . . . . . 8
..^
repr
..^
..^ repr
..^ |
| 191 | 188, 190 | mpd 15 |
. . . . . . 7
..^
repr
..^
..^ repr
..^ |
| 192 | 175, 176, 177, 179, 180, 181, 191 | breprexplemc 30710 |
. . . . . 6
..^
repr
..^
..^ repr
..^ |
| 193 | 132, 173, 174, 192 | syl21anc 1325 |
. . . . 5
..^
repr
..^
..^ repr
..^ |
| 194 | 193 | ex 450 |
. . . 4
..^ repr ..^ ..^
repr
..^ |
| 195 | 21, 36, 51, 66, 131, 194 | nn0indd 11474 |
. . 3
..^ repr
..^ |
| 196 | 6, 195 | mpd 15 |
. 2
..^ repr
..^ |
| 197 | 1, 196 | mpdan 702 |
1
..^
repr
..^ |
| Colors of variables: wff setvar class |
| Syntax hints: wi 4
wa 384
wceq 1483
wcel 1990
cvv 3200
wss 3574
c0 3915
cif 4086
csn 4177
class class class wbr 4653 wf 5884 cfv 5888 (class class class)co 6650
cmap 7857
cfn 7955
cc 9934
cr 9935
cc0 9936
c1 9937
caddc 9939 cmul 9941 cle 10075 cn 11020 cn0 11292 cfz 12326 ..^cfzo 12465
cexp 12860
csu 14416
cprod 14635 reprcrepr 30686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-prod 14636 df-repr 30687 |
| This theorem is referenced by: breprexpnat 30712 vtsprod 30717 |
| Copyright terms: Public domain | W3C validator |