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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 12543 | . 2 ⊢ 2 ∈ ℕ0 | |
| 2 | zexpcl 14117 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 2 ∈ ℕ0) → (𝐴↑2) ∈ ℤ) | |
| 3 | 1, 2 | mpan2 691 | 1 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7431 2c2 12321 ℕ0cn0 12526 ℤcz 12613 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: zsqcl2 14178 zesq 14265 sqoddm1div8 14282 sqrt2irrlem 16284 dvdssqim 16591 dvdssq 16604 nn0gcdsq 16789 numdensq 16791 pythagtriplem3 16856 prmreclem1 16954 4sqlem8 16983 4sqlem10 16985 4sqlem11 16993 4sqlem12 16994 4sqlem14 16996 4sqlem15 16997 4sqlem16 16998 odadd2 19867 muval1 27176 dvdssqf 27181 mumullem1 27222 lgsmulsqcoprm 27387 lgsqrlem2 27391 lgsqrlem4 27393 lgsqr 27395 lgsqrmod 27396 lgsqrmodndvds 27397 2lgsoddprmlem2 27453 2sqlem3 27464 2sqlem4 27465 2sqlem8 27470 2sqblem 27475 2sqcoprm 27479 2sqmod 27480 aks4d1p1p2 42071 pellexlem5 42844 rmspecnonsq 42918 rmspecfund 42920 jm2.18 43000 jm2.22 43007 jm2.20nn 43009 jm2.27a 43017 jm2.27c 43019 jm3.1lem3 43031 sfprmdvdsmersenne 47590 |
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