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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 2818 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
2 | 1 | cardfz 13326 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘𝑁)) |
3 | 2 | fveq2d 6667 | . 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 13329 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin) | |
5 | 1 | hashgval 13681 | . . 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 13327 | . . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 |
8 | f1ocnvfv2 7025 | . . 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 686 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘𝑁)) = 𝑁) |
10 | 3, 6, 9 | 3eqtr3d 2861 | 1 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ↦ cmpt 5137 ◡ccnv 5547 ↾ cres 5550 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ωcom 7569 reccrdg 8034 Fincfn 8497 cardccrd 9352 0cc0 10525 1c1 10526 + caddc 10528 ℕ0cn0 11885 ...cfz 12880 ♯chash 13678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 |
This theorem is referenced by: fz1eqb 13703 isfinite4 13711 hasheq0 13712 hashsng 13718 fseq1hash 13725 hashdom 13728 hashfz 13776 ishashinf 13809 isercolllem2 15010 isercoll 15012 summolem3 15059 summolem2a 15060 o1fsum 15156 climcndslem1 15192 climcndslem2 15193 harmonic 15202 mertenslem1 15228 prodmolem3 15275 prodmolem2a 15276 risefallfac 15366 bpolylem 15390 phicl2 16093 phibnd 16096 hashdvds 16100 phiprmpw 16101 eulerth 16108 pcfac 16223 prmreclem2 16241 prmreclem3 16242 prmreclem5 16244 4sqlem11 16279 vdwlem12 16316 ramub2 16338 ramlb 16343 0ram 16344 ram0 16346 dfod2 18620 gsumval3 18956 uniioombllem4 24114 birthdaylem2 25457 birthdaylem3 25458 basellem4 25588 basellem5 25589 basellem8 25592 ppiltx 25681 vmasum 25719 logfac2 25720 chpval2 25721 chpchtsum 25722 chpub 25723 logfaclbnd 25725 bposlem1 25787 lgsqrlem4 25852 gausslemma2dlem6 25875 lgseisenlem4 25881 lgsquadlem1 25883 lgsquadlem2 25884 lgsquadlem3 25885 dchrmusum2 25997 dchrisum0lem2a 26020 mudivsum 26033 mulogsumlem 26034 selberglem2 26049 ballotlem1 31643 ballotlemfmpn 31651 derangen2 32318 subfaclefac 32320 subfacp1lem1 32323 erdszelem10 32344 erdsze2lem1 32347 snmlff 32473 bcprod 32867 bj-finsumval0 34455 eldioph2lem1 39235 rp-isfinite5 39761 rp-isfinite6 39762 stoweidlem38 42200 dirkertrigeq 42263 etransclem32 42428 nn0mulfsum 44612 aacllem 44830 |
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