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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 11410 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≠ 0) | |
2 | hashcl 13462 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
3 | elnn0 11644 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) | |
4 | 2, 3 | sylib 210 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) |
5 | 4 | ord 853 | . . . 4 ⊢ (𝐴 ∈ Fin → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0)) |
6 | 5 | necon1ad 2986 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ)) |
7 | 1, 6 | impbid2 218 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ (♯‘𝐴) ≠ 0)) |
8 | hasheq0 13469 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
9 | 8 | necon3bid 3013 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 ↔ 𝐴 ≠ ∅)) |
10 | 7, 9 | bitrd 271 | 1 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∅c0 4141 ‘cfv 6135 Fincfn 8241 0cc0 10272 ℕcn 11374 ℕ0cn0 11642 ♯chash 13435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-hash 13436 |
This theorem is referenced by: hashge1 13493 lennncl 13622 lswlgt0cl 13659 wrdind 13842 wrdindOLD 13843 wrd2ind 13844 wrd2indOLD 13845 incexc 14973 incexc2 14974 ramub1 16136 gsumwmhm 17769 psgnunilem5 18297 psgnunilem5OLD 18298 psgnunilem4 18301 gexcl2 18388 sylow1lem3 18399 sylow1lem5 18401 pgpfi 18404 pgpfi2 18405 sylow2alem2 18417 sylow2blem3 18421 slwhash 18423 fislw 18424 sylow3lem3 18428 sylow3lem4 18429 efgsp1 18534 efgsres 18535 efgsresOLD 18536 efgredlem 18545 efgredlemOLD 18546 lt6abl 18682 ablfacrp2 18853 ablfac1lem 18854 ablfac1b 18856 ablfac1c 18857 ablfac1eu 18859 pgpfac1lem2 18861 pgpfac1lem3a 18862 pgpfaclem2 18868 ablfaclem3 18873 lebnumlem3 23170 birthdaylem3 25132 birthday 25133 amgmlem 25168 amgm 25169 musum 25369 dchrabs 25437 dchrisum0flblem1 25649 cusgrrusgr 26929 wlkiswwlksupgr2 27226 frgrreg 27826 tgoldbachgtda 31341 derangfmla 31771 erdszelem2 31773 rrndstprj2 34256 rrncmslem 34257 rrnequiv 34260 isnumbasgrplem3 38638 fzisoeu 40427 fourierdlem54 41308 fourierdlem103 41357 fourierdlem104 41358 qndenserrnbllem 41442 ovnhoilem1 41746 hoiqssbllem1 41767 hoiqssbllem2 41768 hoiqssbllem3 41769 vonsn 41836 amgmlemALT 43659 |
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