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Mirrors > Home > MPE Home > Th. List > zsqcl | Structured version Visualization version GIF version |
Description: Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
zsqcl | ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 11347 | . 2 ⊢ 2 ∈ ℕ0 | |
2 | zexpcl 12915 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 2 ∈ ℕ0) → (𝐴↑2) ∈ ℤ) | |
3 | 1, 2 | mpan2 707 | 1 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 (class class class)co 6690 2c2 11108 ℕ0cn0 11330 ℤcz 11415 ↑cexp 12900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-uz 11726 df-seq 12842 df-exp 12901 |
This theorem is referenced by: zsqcl2 12981 zesq 13027 sqoddm1div8 13068 sqrt2irrlem 15021 sqrt2irrlemOLD 15022 dvdssqim 15320 dvdssq 15327 nn0gcdsq 15507 numdensq 15509 pythagtriplem3 15570 prmreclem1 15667 4sqlem8 15696 4sqlem10 15698 4sqlem11 15706 4sqlem12 15707 4sqlem14 15709 4sqlem15 15710 4sqlem16 15711 odadd2 18298 muval1 24904 dvdssqf 24909 mumullem1 24950 lgsmulsqcoprm 25113 lgsqrlem2 25117 lgsqrlem4 25119 lgsqr 25121 lgsqrmod 25122 lgsqrmodndvds 25123 2lgsoddprmlem2 25179 2sqlem3 25190 2sqlem4 25191 2sqlem8 25196 2sqblem 25201 2sqcoprm 29775 2sqmod 29776 pellexlem5 37714 rmspecnonsq 37789 rmspecfund 37791 jm2.18 37872 jm2.22 37879 jm2.20nn 37881 jm2.27a 37889 jm2.27c 37891 jm3.1lem3 37903 sfprmdvdsmersenne 41845 |
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