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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 2821 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
2 | 1 | cardfz 13337 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘𝑁)) |
3 | 2 | fveq2d 6673 | . 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 13340 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin) | |
5 | 1 | hashgval 13692 | . . 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 13338 | . . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 |
8 | f1ocnvfv2 7033 | . . 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 2864 | 1 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ↦ cmpt 5145 ◡ccnv 5553 ↾ cres 5556 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 ωcom 7579 reccrdg 8044 Fincfn 8508 cardccrd 9363 0cc0 10536 1c1 10537 + caddc 10539 ℕ0cn0 11896 ...cfz 12891 ♯chash 13689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-hash 13690 |
This theorem is referenced by: fz1eqb 13714 isfinite4 13722 hasheq0 13723 hashsng 13729 fseq1hash 13736 hashdom 13739 hashfz 13787 ishashinf 13820 isercolllem2 15021 isercoll 15023 summolem3 15070 summolem2a 15071 o1fsum 15167 climcndslem1 15203 climcndslem2 15204 harmonic 15213 mertenslem1 15239 prodmolem3 15286 prodmolem2a 15287 risefallfac 15377 bpolylem 15401 phicl2 16104 phibnd 16107 hashdvds 16111 phiprmpw 16112 eulerth 16119 pcfac 16234 prmreclem2 16252 prmreclem3 16253 prmreclem5 16255 4sqlem11 16290 vdwlem12 16327 ramub2 16349 ramlb 16354 0ram 16355 ram0 16357 dfod2 18690 gsumval3 19026 uniioombllem4 24186 birthdaylem2 25529 birthdaylem3 25530 basellem4 25660 basellem5 25661 basellem8 25664 ppiltx 25753 vmasum 25791 logfac2 25792 chpval2 25793 chpchtsum 25794 chpub 25795 logfaclbnd 25797 bposlem1 25859 lgsqrlem4 25924 gausslemma2dlem6 25947 lgseisenlem4 25953 lgsquadlem1 25955 lgsquadlem2 25956 lgsquadlem3 25957 dchrmusum2 26069 dchrisum0lem2a 26092 mudivsum 26105 mulogsumlem 26106 selberglem2 26121 ballotlem1 31744 ballotlemfmpn 31752 derangen2 32421 subfaclefac 32423 subfacp1lem1 32426 erdszelem10 32447 erdsze2lem1 32450 snmlff 32576 bcprod 32970 bj-finsumval0 34566 eldioph2lem1 39355 rp-isfinite5 39881 rp-isfinite6 39882 stoweidlem38 42322 dirkertrigeq 42385 etransclem32 42550 nn0mulfsum 44683 aacllem 44901 |
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