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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 12599 | . 2 ⊢ ℤ ⊆ ℂ | |
2 | zmulcl 12644 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 · 𝑦) ∈ ℤ) | |
3 | 1z 12625 | . 2 ⊢ 1 ∈ ℤ | |
4 | 1, 2, 3 | expcllem 14073 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 (class class class)co 7419 ℕ0cn0 12505 ℤcz 12591 ↑cexp 14062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-seq 14003 df-exp 14063 |
This theorem is referenced by: zexpcld 14088 zsqcl 14129 modexp 14236 climcndslem1 15831 iddvdsexp 16260 dvdsexp2im 16307 dvdsexp 16308 3dvds 16311 prmdvdsexp 16689 rpexp 16697 rpexp12i 16699 phiprmpw 16748 eulerthlem2 16754 fermltl 16756 prmdiv 16757 prmdiveq 16758 odzcllem 16764 odzdvds 16767 odzphi 16768 vfermltlALT 16774 powm2modprm 16775 pcneg 16846 pcprmpw 16855 prmpwdvds 16876 pockthlem 16877 dyaddisjlem 25568 aalioulem1 26312 aaliou3lem6 26328 muf 27117 dvdsppwf1o 27163 mersenne 27205 lgslem1 27275 lgsval2lem 27285 lgsvalmod 27294 lgsmod 27301 lgsdirprm 27309 lgsne0 27313 lgsqrlem1 27324 gausslemma2dlem7 27351 gausslemma2d 27352 lgseisenlem2 27354 lgseisenlem4 27356 m1lgs 27366 2sqreultlem 27425 2sqreunnltlem 27428 znfermltl 33177 mdetlap 33561 oddpwdc 34102 nn0prpwlem 35934 nn0prpw 35935 knoppndvlem2 36116 aks4d1p3 41678 aks4d1p6 41681 aks6d1c2p2 41719 dvdsexpim 42020 zexpgcd 42028 numdenexp 42029 jm2.18 42548 jm2.22 42555 jm2.23 42556 jm2.20nn 42557 inductionexd 43724 etransclem3 45760 etransclem7 45764 etransclem10 45767 etransclem24 45781 etransclem27 45784 etransclem35 45792 2pwp1prm 47063 sfprmdvdsmersenne 47077 lighneallem4b 47083 lighneallem4 47084 proththd 47088 41prothprmlem2 47092 nnpw2evenALTV 47176 fpprmod 47201 fppr2odd 47205 dfwppr 47212 fpprwppr 47213 fpprwpprb 47214 pw2m1lepw2m1 47771 nnpw2blenfzo 47837 dignn0fr 47857 digexp 47863 dignn0flhalflem1 47871 |
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