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Mirrors > Home > MPE Home > Th. List > hashfz1 | Structured version Visualization version GIF version |
Description: The set (1...π) has π elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
hashfz1 | β’ (π β β0 β (β―β(1...π)) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 β’ (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) | |
2 | 1 | cardfz 13941 | . . 3 β’ (π β β0 β (cardβ(1...π)) = (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ)) |
3 | 2 | fveq2d 6889 | . 2 β’ (π β β0 β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβ(1...π))) = ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ))) |
4 | fzfid 13944 | . . 3 β’ (π β β0 β (1...π) β Fin) | |
5 | 1 | hashgval 14298 | . . 3 β’ ((1...π) β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβ(1...π))) = (β―β(1...π))) |
6 | 4, 5 | syl 17 | . 2 β’ (π β β0 β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβ(1...π))) = (β―β(1...π))) |
7 | 1 | hashgf1o 13942 | . . 3 β’ (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο):Οβ1-1-ontoββ0 |
8 | f1ocnvfv2 7271 | . . 3 β’ (((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο):Οβ1-1-ontoββ0 β§ π β β0) β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ)) = π) | |
9 | 7, 8 | mpan 687 | . 2 β’ (π β β0 β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ)) = π) |
10 | 3, 6, 9 | 3eqtr3d 2774 | 1 β’ (π β β0 β (β―β(1...π)) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 β¦ cmpt 5224 β‘ccnv 5668 βΎ cres 5671 β1-1-ontoβwf1o 6536 βcfv 6537 (class class class)co 7405 Οcom 7852 reccrdg 8410 Fincfn 8941 cardccrd 9932 0cc0 11112 1c1 11113 + caddc 11115 β0cn0 12476 ...cfz 13490 β―chash 14295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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-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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-hash 14296 |
This theorem is referenced by: fz1eqb 14319 isfinite4 14327 hasheq0 14328 hashsng 14334 fseq1hash 14341 hashdom 14344 hashfz 14392 ishashinf 14430 isercolllem2 15618 isercoll 15620 summolem3 15666 summolem2a 15667 o1fsum 15765 climcndslem1 15801 climcndslem2 15802 harmonic 15811 mertenslem1 15836 prodmolem3 15883 prodmolem2a 15884 risefallfac 15974 bpolylem 15998 phicl2 16710 phibnd 16713 hashdvds 16717 phiprmpw 16718 eulerth 16725 pcfac 16841 prmreclem2 16859 prmreclem3 16860 prmreclem5 16862 4sqlem11 16897 vdwlem12 16934 ramub2 16956 ramlb 16961 0ram 16962 ram0 16964 dfod2 19484 gsumval3 19827 uniioombllem4 25470 birthdaylem2 26839 birthdaylem3 26840 basellem4 26971 basellem5 26972 basellem8 26975 ppiltx 27064 vmasum 27104 logfac2 27105 chpval2 27106 chpchtsum 27107 chpub 27108 logfaclbnd 27110 bposlem1 27172 lgsqrlem4 27237 gausslemma2dlem6 27260 lgseisenlem4 27266 lgsquadlem1 27268 lgsquadlem2 27269 lgsquadlem3 27270 dchrmusum2 27382 dchrisum0lem2a 27405 mudivsum 27418 mulogsumlem 27419 selberglem2 27434 ballotlem1 34015 ballotlemfmpn 34023 derangen2 34693 subfaclefac 34695 subfacp1lem1 34698 erdszelem10 34719 erdsze2lem1 34722 snmlff 34848 bcprod 35241 bj-finsumval0 36673 hashscontpow 41499 sticksstones2 41521 sticksstones5 41524 sticksstones10 41529 sticksstones12a 41531 fz1sumconst 41762 eldioph2lem1 42073 rp-isfinite5 42841 rp-isfinite6 42842 stoweidlem38 45323 dirkertrigeq 45386 etransclem32 45551 nn0mulfsum 47582 aacllem 48119 |
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