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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 15884 | . 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (ℤ≥‘2)(𝑥 ∥ 𝑃 → 𝑥 = 𝑃))) | |
2 | 1 | simplbi 490 | 1 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ∀wral 3088 class class class wbr 4929 ‘cfv 6188 2c2 11495 ℤ≥cuz 12058 ∥ cdvds 15467 ℙcprime 15871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-seq 13185 df-exp 13245 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-dvds 15468 df-prm 15872 |
This theorem is referenced by: prmgt1 15897 prmm2nn0 15898 oddprmgt2 15899 sqnprm 15902 isprm5 15907 isprm7 15908 prmrp 15912 isprm6 15914 prmdvdsexpb 15916 prmdiv 15978 prmdiveq 15979 modprm1div 15990 oddprm 16003 pcpremul 16036 pceulem 16038 pczpre 16040 pczcl 16041 pc1 16048 pczdvds 16055 pczndvds 16057 pczndvds2 16059 pcidlem 16064 pcmpt 16084 pcfaclem 16090 pcfac 16091 pockthlem 16097 pockthg 16098 prmunb 16106 prmreclem2 16109 prmgapprmolem 16253 odcau 18490 sylow3lem6 18518 gexexlem 18728 znfld 20409 logbprmirr 25075 wilthlem1 25347 wilthlem3 25349 wilth 25350 ppisval 25383 ppisval2 25384 chtge0 25391 isppw 25393 ppiprm 25430 chtprm 25432 chtwordi 25435 vma1 25445 fsumvma2 25492 chpval2 25496 chpchtsum 25497 chpub 25498 mersenne 25505 perfect1 25506 bposlem1 25562 lgslem1 25575 lgsval2lem 25585 lgsdirprm 25609 lgsne0 25613 lgsqrlem2 25625 gausslemma2dlem0b 25635 gausslemma2dlem4 25647 lgseisenlem1 25653 lgseisenlem3 25655 lgseisen 25657 lgsquadlem3 25660 m1lgs 25666 2sqblem 25709 chtppilimlem1 25751 rplogsumlem2 25763 rpvmasumlem 25765 dchrisum0flblem2 25787 padicabvcxp 25910 ostth3 25916 umgrhashecclwwlk 27602 fmtnoprmfac1 43101 fmtnoprmfac2lem1 43102 lighneallem2 43145 lighneallem4 43149 gbowgt5 43301 ztprmneprm 43765 |
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