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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 2737 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
| 2 | 1 | cardfz 14011 | . . 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 14014 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin) | |
| 5 | 1 | hashgval 14372 | . . 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 14012 | . . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 |
| 8 | f1ocnvfv2 7297 | . . 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 2785 | 1 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ◡ccnv 5684 ↾ cres 5687 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ωcom 7887 reccrdg 8449 Fincfn 8985 cardccrd 9975 0cc0 11155 1c1 11156 + caddc 11158 ℕ0cn0 12526 ...cfz 13547 ♯chash 14369 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 |
| This theorem is referenced by: fz1eqb 14393 isfinite4 14401 hasheq0 14402 hashsng 14408 fseq1hash 14415 hashdom 14418 hashfz 14466 ishashinf 14502 isercolllem2 15702 isercoll 15704 summolem3 15750 summolem2a 15751 o1fsum 15849 climcndslem1 15885 climcndslem2 15886 harmonic 15895 mertenslem1 15920 prodmolem3 15969 prodmolem2a 15970 risefallfac 16060 bpolylem 16084 phicl2 16805 phibnd 16808 hashdvds 16812 phiprmpw 16813 eulerth 16820 pcfac 16937 prmreclem2 16955 prmreclem3 16956 prmreclem5 16958 4sqlem11 16993 vdwlem12 17030 ramub2 17052 ramlb 17057 0ram 17058 ram0 17060 dfod2 19582 gsumval3 19925 uniioombllem4 25621 birthdaylem2 26995 birthdaylem3 26996 basellem4 27127 basellem5 27128 basellem8 27131 ppiltx 27220 vmasum 27260 logfac2 27261 chpval2 27262 chpchtsum 27263 chpub 27264 logfaclbnd 27266 bposlem1 27328 lgsqrlem4 27393 gausslemma2dlem6 27416 lgseisenlem4 27422 lgsquadlem1 27424 lgsquadlem2 27425 lgsquadlem3 27426 dchrmusum2 27538 dchrisum0lem2a 27561 mudivsum 27574 mulogsumlem 27575 selberglem2 27590 cyclnumvtx 29820 ballotlem1 34489 ballotlemfmpn 34497 derangen2 35179 subfaclefac 35181 subfacp1lem1 35184 erdszelem10 35205 erdsze2lem1 35208 snmlff 35334 bcprod 35738 bj-finsumval0 37286 hashscontpow 42123 sticksstones2 42148 sticksstones5 42151 sticksstones10 42156 sticksstones12a 42158 fz1sumconst 42343 eldioph2lem1 42771 rp-isfinite5 43530 rp-isfinite6 43531 stoweidlem38 46053 dirkertrigeq 46116 etransclem32 46281 nn0mulfsum 48545 aacllem 49320 |
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