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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 2734 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
2 | 1 | cardfz 14007 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘𝑁)) |
3 | 2 | fveq2d 6910 | . 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 14010 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin) | |
5 | 1 | hashgval 14368 | . . 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 14008 | . . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 |
8 | f1ocnvfv2 7296 | . . 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 690 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘𝑁)) = 𝑁) |
10 | 3, 6, 9 | 3eqtr3d 2782 | 1 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ↦ cmpt 5230 ◡ccnv 5687 ↾ cres 5690 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 ωcom 7886 reccrdg 8447 Fincfn 8983 cardccrd 9972 0cc0 11152 1c1 11153 + caddc 11155 ℕ0cn0 12523 ...cfz 13543 ♯chash 14365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-hash 14366 |
This theorem is referenced by: fz1eqb 14389 isfinite4 14397 hasheq0 14398 hashsng 14404 fseq1hash 14411 hashdom 14414 hashfz 14462 ishashinf 14498 isercolllem2 15698 isercoll 15700 summolem3 15746 summolem2a 15747 o1fsum 15845 climcndslem1 15881 climcndslem2 15882 harmonic 15891 mertenslem1 15916 prodmolem3 15965 prodmolem2a 15966 risefallfac 16056 bpolylem 16080 phicl2 16801 phibnd 16804 hashdvds 16808 phiprmpw 16809 eulerth 16816 pcfac 16932 prmreclem2 16950 prmreclem3 16951 prmreclem5 16953 4sqlem11 16988 vdwlem12 17025 ramub2 17047 ramlb 17052 0ram 17053 ram0 17055 dfod2 19596 gsumval3 19939 uniioombllem4 25634 birthdaylem2 27009 birthdaylem3 27010 basellem4 27141 basellem5 27142 basellem8 27145 ppiltx 27234 vmasum 27274 logfac2 27275 chpval2 27276 chpchtsum 27277 chpub 27278 logfaclbnd 27280 bposlem1 27342 lgsqrlem4 27407 gausslemma2dlem6 27430 lgseisenlem4 27436 lgsquadlem1 27438 lgsquadlem2 27439 lgsquadlem3 27440 dchrmusum2 27552 dchrisum0lem2a 27575 mudivsum 27588 mulogsumlem 27589 selberglem2 27604 ballotlem1 34467 ballotlemfmpn 34475 derangen2 35158 subfaclefac 35160 subfacp1lem1 35163 erdszelem10 35184 erdsze2lem1 35187 snmlff 35313 bcprod 35717 bj-finsumval0 37267 hashscontpow 42103 sticksstones2 42128 sticksstones5 42131 sticksstones10 42136 sticksstones12a 42138 fz1sumconst 42321 eldioph2lem1 42747 rp-isfinite5 43506 rp-isfinite6 43507 stoweidlem38 45993 dirkertrigeq 46056 etransclem32 46221 nn0mulfsum 48473 aacllem 49031 |
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