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Mirrors > Home > MPE Home > Th. List > hashfz1 | Structured version Visualization version GIF version |
Description: The set (1...𝑁) has 𝑁 elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
hashfz1 | ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
2 | 1 | cardfz 13971 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘𝑁)) |
3 | 2 | fveq2d 6900 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...𝑁))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘𝑁))) |
4 | fzfid 13974 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin) | |
5 | 1 | hashgval 14328 | . . 3 ⊢ ((1...𝑁) ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...𝑁))) = (♯‘(1...𝑁))) |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...𝑁))) = (♯‘(1...𝑁))) |
7 | 1 | hashgf1o 13972 | . . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 |
8 | f1ocnvfv2 7286 | . . 3 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 ∧ 𝑁 ∈ ℕ0) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘𝑁)) = 𝑁) | |
9 | 7, 8 | mpan 688 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘𝑁)) = 𝑁) |
10 | 3, 6, 9 | 3eqtr3d 2773 | 1 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ↦ cmpt 5232 ◡ccnv 5677 ↾ cres 5680 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ωcom 7871 reccrdg 8430 Fincfn 8964 cardccrd 9960 0cc0 11140 1c1 11141 + caddc 11143 ℕ0cn0 12505 ...cfz 13519 ♯chash 14325 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-hash 14326 |
This theorem is referenced by: fz1eqb 14349 isfinite4 14357 hasheq0 14358 hashsng 14364 fseq1hash 14371 hashdom 14374 hashfz 14422 ishashinf 14460 isercolllem2 15648 isercoll 15650 summolem3 15696 summolem2a 15697 o1fsum 15795 climcndslem1 15831 climcndslem2 15832 harmonic 15841 mertenslem1 15866 prodmolem3 15913 prodmolem2a 15914 risefallfac 16004 bpolylem 16028 phicl2 16740 phibnd 16743 hashdvds 16747 phiprmpw 16748 eulerth 16755 pcfac 16871 prmreclem2 16889 prmreclem3 16890 prmreclem5 16892 4sqlem11 16927 vdwlem12 16964 ramub2 16986 ramlb 16991 0ram 16992 ram0 16994 dfod2 19531 gsumval3 19874 uniioombllem4 25559 birthdaylem2 26929 birthdaylem3 26930 basellem4 27061 basellem5 27062 basellem8 27065 ppiltx 27154 vmasum 27194 logfac2 27195 chpval2 27196 chpchtsum 27197 chpub 27198 logfaclbnd 27200 bposlem1 27262 lgsqrlem4 27327 gausslemma2dlem6 27350 lgseisenlem4 27356 lgsquadlem1 27358 lgsquadlem2 27359 lgsquadlem3 27360 dchrmusum2 27472 dchrisum0lem2a 27495 mudivsum 27508 mulogsumlem 27509 selberglem2 27524 ballotlem1 34237 ballotlemfmpn 34245 derangen2 34915 subfaclefac 34917 subfacp1lem1 34920 erdszelem10 34941 erdsze2lem1 34944 snmlff 35070 bcprod 35463 bj-finsumval0 36895 hashscontpow 41725 sticksstones2 41750 sticksstones5 41753 sticksstones10 41758 sticksstones12a 41760 fz1sumconst 42004 eldioph2lem1 42322 rp-isfinite5 43089 rp-isfinite6 43090 stoweidlem38 45564 dirkertrigeq 45627 etransclem32 45792 nn0mulfsum 47883 aacllem 48420 |
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