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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 16016 | . 2 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (ℤ≥‘2)(𝑥 ∥ 𝑃 → 𝑥 = 𝑃))) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 class class class wbr 5057 ‘cfv 6348 2c2 11680 ℤ≥cuz 12231 ∥ cdvds 15595 ℙcprime 16003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-prm 16004 |
This theorem is referenced by: prmgt1 16029 prmm2nn0 16030 oddprmgt2 16031 sqnprm 16034 isprm5 16039 isprm7 16040 prmrp 16044 isprm6 16046 prmdvdsexpb 16048 prmdiv 16110 prmdiveq 16111 modprm1div 16122 oddprm 16135 pcpremul 16168 pceulem 16170 pczpre 16172 pczcl 16173 pc1 16180 pczdvds 16187 pczndvds 16189 pczndvds2 16191 pcidlem 16196 pcmpt 16216 pcfaclem 16222 pcfac 16223 pockthlem 16229 pockthg 16230 prmunb 16238 prmreclem2 16241 prmgapprmolem 16385 odcau 18658 sylow3lem6 18686 gexexlem 18901 znfld 20635 logbprmirr 25301 wilthlem1 25572 wilthlem3 25574 wilth 25575 ppisval 25608 ppisval2 25609 chtge0 25616 isppw 25618 ppiprm 25655 chtprm 25657 chtwordi 25660 vma1 25670 fsumvma2 25717 chpval2 25721 chpchtsum 25722 chpub 25723 mersenne 25730 perfect1 25731 bposlem1 25787 lgslem1 25800 lgsval2lem 25810 lgsdirprm 25834 lgsne0 25838 lgsqrlem2 25850 gausslemma2dlem0b 25860 gausslemma2dlem4 25872 lgseisenlem1 25878 lgseisenlem3 25880 lgseisen 25882 lgsquadlem3 25885 m1lgs 25891 2sqblem 25934 chtppilimlem1 25976 rplogsumlem2 25988 rpvmasumlem 25990 dchrisum0flblem2 26012 padicabvcxp 26135 ostth3 26141 umgrhashecclwwlk 27784 fmtnoprmfac1 43604 fmtnoprmfac2lem1 43605 lighneallem2 43648 lighneallem4 43652 gbowgt5 43804 ztprmneprm 44323 |
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