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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 2728 | . . . 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 13975 | . . 3 β’ (π β β0 β (cardβ(1...π)) = (β‘(rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)βπ)) |
3 | 2 | fveq2d 6906 | . 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 13978 | . . 3 β’ (π β β0 β (1...π) β Fin) | |
5 | 1 | hashgval 14332 | . . 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 13976 | . . 3 β’ (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο):Οβ1-1-ontoββ0 |
8 | f1ocnvfv2 7292 | . . 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 688 | . 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 2776 | 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 3473 β¦ cmpt 5235 β‘ccnv 5681 βΎ cres 5684 β1-1-ontoβwf1o 6552 βcfv 6553 (class class class)co 7426 Οcom 7876 reccrdg 8436 Fincfn 8970 cardccrd 9966 0cc0 11146 1c1 11147 + caddc 11149 β0cn0 12510 ...cfz 13524 β―chash 14329 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-hash 14330 |
This theorem is referenced by: fz1eqb 14353 isfinite4 14361 hasheq0 14362 hashsng 14368 fseq1hash 14375 hashdom 14378 hashfz 14426 ishashinf 14464 isercolllem2 15652 isercoll 15654 summolem3 15700 summolem2a 15701 o1fsum 15799 climcndslem1 15835 climcndslem2 15836 harmonic 15845 mertenslem1 15870 prodmolem3 15917 prodmolem2a 15918 risefallfac 16008 bpolylem 16032 phicl2 16744 phibnd 16747 hashdvds 16751 phiprmpw 16752 eulerth 16759 pcfac 16875 prmreclem2 16893 prmreclem3 16894 prmreclem5 16896 4sqlem11 16931 vdwlem12 16968 ramub2 16990 ramlb 16995 0ram 16996 ram0 16998 dfod2 19526 gsumval3 19869 uniioombllem4 25535 birthdaylem2 26904 birthdaylem3 26905 basellem4 27036 basellem5 27037 basellem8 27040 ppiltx 27129 vmasum 27169 logfac2 27170 chpval2 27171 chpchtsum 27172 chpub 27173 logfaclbnd 27175 bposlem1 27237 lgsqrlem4 27302 gausslemma2dlem6 27325 lgseisenlem4 27331 lgsquadlem1 27333 lgsquadlem2 27334 lgsquadlem3 27335 dchrmusum2 27447 dchrisum0lem2a 27470 mudivsum 27483 mulogsumlem 27484 selberglem2 27499 ballotlem1 34139 ballotlemfmpn 34147 derangen2 34817 subfaclefac 34819 subfacp1lem1 34822 erdszelem10 34843 erdsze2lem1 34846 snmlff 34972 bcprod 35365 bj-finsumval0 36797 hashscontpow 41625 sticksstones2 41651 sticksstones5 41654 sticksstones10 41659 sticksstones12a 41661 fz1sumconst 41900 eldioph2lem1 42211 rp-isfinite5 42978 rp-isfinite6 42979 stoweidlem38 45455 dirkertrigeq 45518 etransclem32 45683 nn0mulfsum 47775 aacllem 48312 |
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