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Mirrors > Home > MPE Home > Th. List > zexpcl | Structured version Visualization version GIF version |
Description: Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.) |
Ref | Expression |
---|---|
zexpcl | ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsscn 11712 | . 2 ⊢ ℤ ⊆ ℂ | |
2 | zmulcl 11754 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 · 𝑦) ∈ ℤ) | |
3 | 1z 11735 | . 2 ⊢ 1 ∈ ℤ | |
4 | 1, 2, 3 | expcllem 13165 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 (class class class)co 6905 ℕ0cn0 11618 ℤcz 11704 ↑cexp 13154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-seq 13096 df-exp 13155 |
This theorem is referenced by: zsqcl 13228 modexp 13293 climcndslem1 14955 iddvdsexp 15382 dvdsexp 15426 3dvds 15429 prmdvdsexp 15798 rpexp 15803 rpexp12i 15805 phiprmpw 15852 eulerthlem2 15858 fermltl 15860 prmdiv 15861 prmdiveq 15862 odzcllem 15868 odzdvds 15871 odzphi 15872 vfermltlALT 15878 powm2modprm 15879 pcneg 15949 pcprmpw 15958 prmpwdvds 15979 pockthlem 15980 dyaddisjlem 23761 aalioulem1 24486 aaliou3lem6 24502 muf 25279 dvdsppwf1o 25325 mersenne 25365 lgslem1 25435 lgsval2lem 25445 lgsvalmod 25454 lgsmod 25461 lgsdirprm 25469 lgsne0 25473 lgsqrlem1 25484 gausslemma2dlem7 25511 gausslemma2d 25512 lgseisenlem2 25514 lgseisenlem4 25516 m1lgs 25526 mdetlap 30432 oddpwdc 30950 dvdspw 32167 nn0prpwlem 32844 nn0prpw 32845 knoppndvlem2 33025 jm2.18 38391 jm2.22 38398 jm2.23 38399 jm2.20nn 38400 inductionexd 39286 etransclem3 41241 etransclem7 41245 etransclem10 41248 etransclem24 41262 etransclem27 41265 etransclem35 41273 2pwp1prm 42326 sfprmdvdsmersenne 42343 lighneallem4b 42349 lighneallem4 42350 proththd 42354 41prothprmlem2 42358 nnpw2evenALTV 42434 pw2m1lepw2m1 43150 nnpw2blenfzo 43215 dignn0fr 43235 digexp 43241 dignn0flhalflem1 43249 |
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