| 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 12300 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≠ 0) | |
| 2 | hashcl 14395 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
| 3 | elnn0 12528 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) | |
| 4 | 2, 3 | sylib 218 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) |
| 5 | 4 | ord 865 | . . . 4 ⊢ (𝐴 ∈ Fin → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0)) |
| 6 | 5 | necon1ad 2957 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ)) |
| 7 | 1, 6 | impbid2 226 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ (♯‘𝐴) ≠ 0)) |
| 8 | hasheq0 14402 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
| 9 | 8 | necon3bid 2985 | . 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 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 ‘cfv 6561 Fincfn 8985 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 ♯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: hashge1 14428 lennncl 14572 lswlgt0cl 14607 wrdind 14760 wrd2ind 14761 incexc 15873 incexc2 15874 ramub1 17066 gsumwmhm 18858 psgnunilem5 19512 psgnunilem4 19515 gexcl2 19607 sylow1lem3 19618 sylow1lem5 19620 pgpfi 19623 pgpfi2 19624 sylow2alem2 19636 sylow2blem3 19640 slwhash 19642 fislw 19643 sylow3lem3 19647 sylow3lem4 19648 efgsres 19756 efgredlem 19765 lt6abl 19913 ablfacrp2 20087 ablfac1lem 20088 ablfac1b 20090 ablfac1c 20091 ablfac1eu 20093 pgpfac1lem2 20095 pgpfac1lem3a 20096 pgpfaclem2 20102 ablfaclem3 20107 lebnumlem3 24995 birthdaylem3 26996 birthday 26997 amgmlem 27033 amgm 27034 musum 27234 dchrabs 27304 dchrisum0flblem1 27552 cusgrrusgr 29599 frgrreg 30413 tgoldbachgtda 34676 derangfmla 35195 erdszelem2 35197 rrndstprj2 37838 rrncmslem 37839 rrnequiv 37842 sticksstones21 42168 sticksstones22 42169 isnumbasgrplem3 43117 fzisoeu 45312 fourierdlem54 46175 fourierdlem103 46224 fourierdlem104 46225 qndenserrnbllem 46309 ovnhoilem1 46616 hoiqssbllem1 46637 hoiqssbllem2 46638 hoiqssbllem3 46639 vonsn 46706 amgmlemALT 49322 |
| Copyright terms: Public domain | W3C validator |