![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12297 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≠ 0) | |
2 | hashcl 14391 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
3 | elnn0 12525 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) | |
4 | 2, 3 | sylib 218 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) |
5 | 4 | ord 864 | . . . 4 ⊢ (𝐴 ∈ Fin → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0)) |
6 | 5 | necon1ad 2954 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ)) |
7 | 1, 6 | impbid2 226 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ (♯‘𝐴) ≠ 0)) |
8 | hasheq0 14398 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
9 | 8 | necon3bid 2982 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 ↔ 𝐴 ≠ ∅)) |
10 | 7, 9 | bitrd 279 | 1 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∅c0 4338 ‘cfv 6562 Fincfn 8983 0cc0 11152 ℕcn 12263 ℕ0cn0 12523 ♯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: hashge1 14424 lennncl 14568 lswlgt0cl 14603 wrdind 14756 wrd2ind 14757 incexc 15869 incexc2 15870 ramub1 17061 gsumwmhm 18870 psgnunilem5 19526 psgnunilem4 19529 gexcl2 19621 sylow1lem3 19632 sylow1lem5 19634 pgpfi 19637 pgpfi2 19638 sylow2alem2 19650 sylow2blem3 19654 slwhash 19656 fislw 19657 sylow3lem3 19661 sylow3lem4 19662 efgsres 19770 efgredlem 19779 lt6abl 19927 ablfacrp2 20101 ablfac1lem 20102 ablfac1b 20104 ablfac1c 20105 ablfac1eu 20107 pgpfac1lem2 20109 pgpfac1lem3a 20110 pgpfaclem2 20116 ablfaclem3 20121 lebnumlem3 25008 birthdaylem3 27010 birthday 27011 amgmlem 27047 amgm 27048 musum 27248 dchrabs 27318 dchrisum0flblem1 27566 cusgrrusgr 29613 frgrreg 30422 tgoldbachgtda 34654 derangfmla 35174 erdszelem2 35176 rrndstprj2 37817 rrncmslem 37818 rrnequiv 37821 sticksstones21 42148 sticksstones22 42149 isnumbasgrplem3 43093 fzisoeu 45250 fourierdlem54 46115 fourierdlem103 46164 fourierdlem104 46165 qndenserrnbllem 46249 ovnhoilem1 46556 hoiqssbllem1 46577 hoiqssbllem2 46578 hoiqssbllem3 46579 vonsn 46646 amgmlemALT 49033 |
Copyright terms: Public domain | W3C validator |