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Mirrors > Home > ILE Home > Th. List > prmz | GIF version |
Description: A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.) |
Ref | Expression |
---|---|
prmz | ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmnn 12104 | . 2 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
2 | 1 | nnzd 9372 | 1 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ℤcz 9251 ℙcprime 12101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-n0 9175 df-z 9252 df-prm 12102 |
This theorem is referenced by: dvdsprime 12116 prm2orodd 12120 oddprmge3 12129 exprmfct 12132 prmdvdsfz 12133 isprm5lem 12135 isprm5 12136 coprm 12138 prmrp 12139 euclemma 12140 prmdvdsexpb 12143 prmexpb 12145 prmfac1 12146 rpexp 12147 prmndvdsfaclt 12150 cncongrprm 12151 phiprmpw 12216 phiprm 12217 fermltl 12228 prmdiv 12229 prmdiveq 12230 vfermltl 12245 reumodprminv 12247 modprm0 12248 oddprm 12253 prm23lt5 12257 prm23ge5 12258 pcneg 12318 pcprmpw2 12326 pcprmpw 12327 difsqpwdvds 12331 pcmpt 12335 pcmptdvds 12337 pcprod 12338 prmpwdvds 12347 prmunb 12354 1arithlem4 12358 1arith 12359 lgslem1 14294 lgsval2lem 14304 lgsvalmod 14313 lgsmod 14320 lgsdirprm 14328 lgsdir 14329 lgsdilem2 14330 lgsdi 14331 lgsne0 14332 lgsprme0 14336 2sqlem3 14346 2sqlem4 14347 2sqlem6 14349 2sqlem8 14352 |
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