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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 2737 | . . . 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 13882 | . . 3 β’ (π β β0 β (cardβ(1...π)) = (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ)) |
3 | 2 | fveq2d 6851 | . 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 13885 | . . 3 β’ (π β β0 β (1...π) β Fin) | |
5 | 1 | hashgval 14240 | . . 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 13883 | . . 3 β’ (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο):Οβ1-1-ontoββ0 |
8 | f1ocnvfv2 7228 | . . 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 689 | . 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 2785 | 1 β’ (π β β0 β (β―β(1...π)) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3448 β¦ cmpt 5193 β‘ccnv 5637 βΎ cres 5640 β1-1-ontoβwf1o 6500 βcfv 6501 (class class class)co 7362 Οcom 7807 reccrdg 8360 Fincfn 8890 cardccrd 9878 0cc0 11058 1c1 11059 + caddc 11061 β0cn0 12420 ...cfz 13431 β―chash 14237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 |
This theorem is referenced by: fz1eqb 14261 isfinite4 14269 hasheq0 14270 hashsng 14276 fseq1hash 14283 hashdom 14286 hashfz 14334 ishashinf 14369 isercolllem2 15557 isercoll 15559 summolem3 15606 summolem2a 15607 o1fsum 15705 climcndslem1 15741 climcndslem2 15742 harmonic 15751 mertenslem1 15776 prodmolem3 15823 prodmolem2a 15824 risefallfac 15914 bpolylem 15938 phicl2 16647 phibnd 16650 hashdvds 16654 phiprmpw 16655 eulerth 16662 pcfac 16778 prmreclem2 16796 prmreclem3 16797 prmreclem5 16799 4sqlem11 16834 vdwlem12 16871 ramub2 16893 ramlb 16898 0ram 16899 ram0 16901 dfod2 19353 gsumval3 19691 uniioombllem4 24966 birthdaylem2 26318 birthdaylem3 26319 basellem4 26449 basellem5 26450 basellem8 26453 ppiltx 26542 vmasum 26580 logfac2 26581 chpval2 26582 chpchtsum 26583 chpub 26584 logfaclbnd 26586 bposlem1 26648 lgsqrlem4 26713 gausslemma2dlem6 26736 lgseisenlem4 26742 lgsquadlem1 26744 lgsquadlem2 26745 lgsquadlem3 26746 dchrmusum2 26858 dchrisum0lem2a 26881 mudivsum 26894 mulogsumlem 26895 selberglem2 26910 ballotlem1 33126 ballotlemfmpn 33134 derangen2 33808 subfaclefac 33810 subfacp1lem1 33813 erdszelem10 33834 erdsze2lem1 33837 snmlff 33963 bcprod 34350 bj-finsumval0 35785 sticksstones2 40584 sticksstones5 40587 sticksstones10 40592 sticksstones12a 40594 eldioph2lem1 41112 rp-isfinite5 41863 rp-isfinite6 41864 stoweidlem38 44353 dirkertrigeq 44416 etransclem32 44581 nn0mulfsum 46784 aacllem 47322 |
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