| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > prmuz2 | Structured version Visualization version GIF version | ||
| Description: A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
| Ref | Expression |
|---|---|
| prmuz2 | ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isprm4 16721 | . 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (ℤ≥‘2)(𝑥 ∥ 𝑃 → 𝑥 = 𝑃))) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 ‘cfv 6561 2c2 12321 ℤ≥cuz 12878 ∥ cdvds 16290 ℙcprime 16708 |
| 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 ax-pre-sup 11233 |
| 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-rmo 3380 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-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-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-prm 16709 |
| This theorem is referenced by: prmgt1 16734 prmm2nn0 16735 oddprmgt2 16736 sqnprm 16739 isprm5 16744 isprm7 16745 prmrp 16749 isprm6 16751 prmdvdsexpb 16753 prmdvdsncoprmbd 16764 prmdiv 16822 prmdiveq 16823 modprm1div 16835 oddprm 16848 pcpremul 16881 pceulem 16883 pczpre 16885 pczcl 16886 pc1 16893 pczdvds 16901 pczndvds 16903 pczndvds2 16905 pcidlem 16910 pcmpt 16930 pcfaclem 16936 pcfac 16937 pockthlem 16943 pockthg 16944 prmunb 16952 prmreclem2 16955 prmgapprmolem 17099 odcau 19622 sylow3lem6 19650 gexexlem 19870 znfld 21579 logbprmirr 26839 wilthlem1 27111 wilthlem3 27113 wilth 27114 ppisval 27147 ppisval2 27148 chtge0 27155 isppw 27157 ppiprm 27194 chtprm 27196 chtwordi 27199 vma1 27209 fsumvma2 27258 chpval2 27262 chpchtsum 27263 chpub 27264 mersenne 27271 perfect1 27272 bposlem1 27328 lgslem1 27341 lgsval2lem 27351 lgsdirprm 27375 lgsne0 27379 lgsqrlem2 27391 gausslemma2dlem0b 27401 gausslemma2dlem4 27413 lgseisenlem1 27419 lgseisenlem3 27421 lgseisen 27423 lgsquadlem3 27426 m1lgs 27432 2sqblem 27475 chtppilimlem1 27517 rplogsumlem2 27529 rpvmasumlem 27531 dchrisum0flblem2 27553 padicabvcxp 27676 ostth3 27682 umgrhashecclwwlk 30097 aks4d1p6 42082 aks6d1c7 42185 fmtnoprmfac1 47552 fmtnoprmfac2lem1 47553 lighneallem2 47593 lighneallem4 47597 gbowgt5 47749 ztprmneprm 48263 |
| Copyright terms: Public domain | W3C validator |