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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 16662 | . 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (ℤ≥‘2)(𝑥 ∥ 𝑃 → 𝑥 = 𝑃))) | |
2 | 1 | simplbi 496 | 1 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3058 class class class wbr 5152 ‘cfv 6553 2c2 12305 ℤ≥cuz 12860 ∥ cdvds 16238 ℙcprime 16649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-dvds 16239 df-prm 16650 |
This theorem is referenced by: prmgt1 16675 prmm2nn0 16676 oddprmgt2 16677 sqnprm 16680 isprm5 16685 isprm7 16686 prmrp 16690 isprm6 16692 prmdvdsexpb 16694 prmdvdsncoprmbd 16706 prmdiv 16761 prmdiveq 16762 modprm1div 16773 oddprm 16786 pcpremul 16819 pceulem 16821 pczpre 16823 pczcl 16824 pc1 16831 pczdvds 16839 pczndvds 16841 pczndvds2 16843 pcidlem 16848 pcmpt 16868 pcfaclem 16874 pcfac 16875 pockthlem 16881 pockthg 16882 prmunb 16890 prmreclem2 16893 prmgapprmolem 17037 odcau 19566 sylow3lem6 19594 gexexlem 19814 znfld 21501 logbprmirr 26748 wilthlem1 27020 wilthlem3 27022 wilth 27023 ppisval 27056 ppisval2 27057 chtge0 27064 isppw 27066 ppiprm 27103 chtprm 27105 chtwordi 27108 vma1 27118 fsumvma2 27167 chpval2 27171 chpchtsum 27172 chpub 27173 mersenne 27180 perfect1 27181 bposlem1 27237 lgslem1 27250 lgsval2lem 27260 lgsdirprm 27284 lgsne0 27288 lgsqrlem2 27300 gausslemma2dlem0b 27310 gausslemma2dlem4 27322 lgseisenlem1 27328 lgseisenlem3 27330 lgseisen 27332 lgsquadlem3 27335 m1lgs 27341 2sqblem 27384 chtppilimlem1 27426 rplogsumlem2 27438 rpvmasumlem 27440 dchrisum0flblem2 27462 padicabvcxp 27585 ostth3 27591 umgrhashecclwwlk 29908 aks4d1p6 41584 aks6d1c7 41688 fmtnoprmfac1 46934 fmtnoprmfac2lem1 46935 lighneallem2 46975 lighneallem4 46979 gbowgt5 47131 ztprmneprm 47489 |
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