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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 12192 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≠ 0) | |
2 | hashcl 14262 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
3 | elnn0 12420 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) | |
4 | 2, 3 | sylib 217 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) |
5 | 4 | ord 863 | . . . 4 ⊢ (𝐴 ∈ Fin → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0)) |
6 | 5 | necon1ad 2957 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ≠ 0 → (♯‘𝐴) ∈ ℕ)) |
7 | 1, 6 | impbid2 225 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ (♯‘𝐴) ≠ 0)) |
8 | hasheq0 14269 | . . 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 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∅c0 4283 ‘cfv 6497 Fincfn 8886 0cc0 11056 ℕcn 12158 ℕ0cn0 12418 ♯chash 14236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 |
This theorem is referenced by: hashge1 14295 lennncl 14428 lswlgt0cl 14463 wrdind 14616 wrd2ind 14617 incexc 15727 incexc2 15728 ramub1 16905 gsumwmhm 18660 psgnunilem5 19281 psgnunilem4 19284 gexcl2 19376 sylow1lem3 19387 sylow1lem5 19389 pgpfi 19392 pgpfi2 19393 sylow2alem2 19405 sylow2blem3 19409 slwhash 19411 fislw 19412 sylow3lem3 19416 sylow3lem4 19417 efgsres 19525 efgredlem 19534 lt6abl 19677 ablfacrp2 19851 ablfac1lem 19852 ablfac1b 19854 ablfac1c 19855 ablfac1eu 19857 pgpfac1lem2 19859 pgpfac1lem3a 19860 pgpfaclem2 19866 ablfaclem3 19871 lebnumlem3 24342 birthdaylem3 26319 birthday 26320 amgmlem 26355 amgm 26356 musum 26556 dchrabs 26624 dchrisum0flblem1 26872 cusgrrusgr 28571 frgrreg 29380 tgoldbachgtda 33331 derangfmla 33841 erdszelem2 33843 rrndstprj2 36336 rrncmslem 36337 rrnequiv 36340 sticksstones21 40621 sticksstones22 40622 isnumbasgrplem3 41475 fzisoeu 43621 fourierdlem54 44487 fourierdlem103 44536 fourierdlem104 44537 qndenserrnbllem 44621 ovnhoilem1 44928 hoiqssbllem1 44949 hoiqssbllem2 44950 hoiqssbllem3 44951 vonsn 45018 amgmlemALT 47336 |
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