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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hashrepr | Structured version Visualization version GIF version |
Description: Develop the number of representations of an integer π as a sum of nonnegative integers in set π΄. (Contributed by Thierry Arnoux, 14-Dec-2021.) |
Ref | Expression |
---|---|
hashrepr.a | β’ (π β π΄ β β) |
hashrepr.m | β’ (π β π β β0) |
hashrepr.s | β’ (π β π β β0) |
Ref | Expression |
---|---|
hashrepr | β’ (π β (β―β(π΄(reprβπ)π)) = Ξ£π β (β(reprβπ)π)βπ β (0..^π)(((πββ)βπ΄)β(πβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashrepr.a | . . 3 β’ (π β π΄ β β) | |
2 | hashrepr.m | . . . 4 β’ (π β π β β0) | |
3 | 2 | nn0zd 12566 | . . 3 β’ (π β π β β€) |
4 | hashrepr.s | . . 3 β’ (π β π β β0) | |
5 | fzfid 13920 | . . 3 β’ (π β (1...π) β Fin) | |
6 | fz1ssnn 13514 | . . . 4 β’ (1...π) β β | |
7 | 6 | a1i 11 | . . 3 β’ (π β (1...π) β β) |
8 | 1, 3, 4, 5, 7 | hashreprin 33463 | . 2 β’ (π β (β―β((π΄ β© (1...π))(reprβπ)π)) = Ξ£π β ((1...π)(reprβπ)π)βπ β (0..^π)(((πββ)βπ΄)β(πβπ))) |
9 | 2, 4, 1 | reprinfz1 33465 | . . 3 β’ (π β (π΄(reprβπ)π) = ((π΄ β© (1...π))(reprβπ)π)) |
10 | 9 | fveq2d 6882 | . 2 β’ (π β (β―β(π΄(reprβπ)π)) = (β―β((π΄ β© (1...π))(reprβπ)π))) |
11 | 2, 4 | reprfz1 33467 | . . 3 β’ (π β (β(reprβπ)π) = ((1...π)(reprβπ)π)) |
12 | 11 | sumeq1d 15629 | . 2 β’ (π β Ξ£π β (β(reprβπ)π)βπ β (0..^π)(((πββ)βπ΄)β(πβπ)) = Ξ£π β ((1...π)(reprβπ)π)βπ β (0..^π)(((πββ)βπ΄)β(πβπ))) |
13 | 8, 10, 12 | 3eqtr4d 2781 | 1 β’ (π β (β―β(π΄(reprβπ)π)) = Ξ£π β (β(reprβπ)π)βπ β (0..^π)(((πββ)βπ΄)β(πβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β© cin 3943 β wss 3944 βcfv 6532 (class class class)co 7393 0cc0 11092 1c1 11093 βcn 12194 β0cn0 12454 ...cfz 13466 ..^cfzo 13609 β―chash 14272 Ξ£csu 15614 βcprod 15831 πcind 32839 reprcrepr 33451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-pm 8806 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-n0 12455 df-z 12541 df-uz 12805 df-rp 12957 df-ico 13312 df-fz 13467 df-fzo 13610 df-seq 13949 df-exp 14010 df-hash 14273 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-clim 15414 df-sum 15615 df-prod 15832 df-ind 32840 df-repr 33452 |
This theorem is referenced by: circlemethnat 33484 |
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