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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 11762 | . 2 ⊢ 2 ∈ ℕ0 | |
2 | zexpcl 13294 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 2 ∈ ℕ0) → (𝐴↑2) ∈ ℤ) | |
3 | 1, 2 | mpan2 687 | 1 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 (class class class)co 7016 2c2 11540 ℕ0cn0 11745 ℤcz 11829 ↑cexp 13279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-z 11830 df-uz 12094 df-seq 13220 df-exp 13280 |
This theorem is referenced by: zsqcl2 13352 zesq 13437 sqoddm1div8 13454 sqrt2irrlem 15434 dvdssqim 15733 dvdssq 15740 nn0gcdsq 15921 numdensq 15923 pythagtriplem3 15984 prmreclem1 16081 4sqlem8 16110 4sqlem10 16112 4sqlem11 16120 4sqlem12 16121 4sqlem14 16123 4sqlem15 16124 4sqlem16 16125 odadd2 18692 muval1 25392 dvdssqf 25397 mumullem1 25438 lgsmulsqcoprm 25601 lgsqrlem2 25605 lgsqrlem4 25607 lgsqr 25609 lgsqrmod 25610 lgsqrmodndvds 25611 2lgsoddprmlem2 25667 2sqlem3 25678 2sqlem4 25679 2sqlem8 25684 2sqblem 25689 2sqcoprm 25693 2sqmod 25694 pellexlem5 38934 rmspecnonsq 39008 rmspecfund 39010 jm2.18 39089 jm2.22 39096 jm2.20nn 39098 jm2.27a 39106 jm2.27c 39108 jm3.1lem3 39120 sfprmdvdsmersenne 43270 |
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