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Mirrors > Home > MPE Home > Th. List > hashnncl | Structured version Visualization version GIF version |
Description: Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Ref | Expression |
---|---|
hashnncl | ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnne0 11659 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≠ 0) | |
2 | hashcl 13713 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
3 | elnn0 11887 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) | |
4 | 2, 3 | sylib 221 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) |
5 | 4 | ord 861 | . . . 4 ⊢ (𝐴 ∈ Fin → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0)) |
6 | 5 | necon1ad 3004 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ)) |
7 | 1, 6 | impbid2 229 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ (♯‘𝐴) ≠ 0)) |
8 | hasheq0 13720 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
9 | 8 | necon3bid 3031 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 ↔ 𝐴 ≠ ∅)) |
10 | 7, 9 | bitrd 282 | 1 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ‘cfv 6324 Fincfn 8492 0cc0 10526 ℕcn 11625 ℕ0cn0 11885 ♯chash 13686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 |
This theorem is referenced by: hashge1 13746 lennncl 13877 lswlgt0cl 13912 wrdind 14075 wrd2ind 14076 incexc 15184 incexc2 15185 ramub1 16354 gsumwmhm 18002 psgnunilem5 18614 psgnunilem4 18617 gexcl2 18706 sylow1lem3 18717 sylow1lem5 18719 pgpfi 18722 pgpfi2 18723 sylow2alem2 18735 sylow2blem3 18739 slwhash 18741 fislw 18742 sylow3lem3 18746 sylow3lem4 18747 efgsres 18856 efgredlem 18865 lt6abl 19008 ablfacrp2 19182 ablfac1lem 19183 ablfac1b 19185 ablfac1c 19186 ablfac1eu 19188 pgpfac1lem2 19190 pgpfac1lem3a 19191 pgpfaclem2 19197 ablfaclem3 19202 lebnumlem3 23568 birthdaylem3 25539 birthday 25540 amgmlem 25575 amgm 25576 musum 25776 dchrabs 25844 dchrisum0flblem1 26092 cusgrrusgr 27371 frgrreg 28179 tgoldbachgtda 32042 derangfmla 32550 erdszelem2 32552 rrndstprj2 35269 rrncmslem 35270 rrnequiv 35273 isnumbasgrplem3 40049 fzisoeu 41932 fourierdlem54 42802 fourierdlem103 42851 fourierdlem104 42852 qndenserrnbllem 42936 ovnhoilem1 43240 hoiqssbllem1 43261 hoiqssbllem2 43262 hoiqssbllem3 43263 vonsn 43330 amgmlemALT 45331 |
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