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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 12621 | . 2 ⊢ ℤ ⊆ ℂ | |
| 2 | zmulcl 12666 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 · 𝑦) ∈ ℤ) | |
| 3 | 1z 12647 | . 2 ⊢ 1 ∈ ℤ | |
| 4 | 1, 2, 3 | expcllem 14113 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 (class class class)co 7431 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-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: zexpcld 14128 zsqcl 14169 modexp 14277 climcndslem1 15885 iddvdsexp 16317 dvdsexp2im 16364 dvdsexp 16365 3dvds 16368 dvdsexpim 16592 zexpgcd 16602 prmdvdsexp 16752 rpexp 16759 rpexp12i 16761 numdenexp 16797 phiprmpw 16813 eulerthlem2 16819 fermltl 16821 prmdiv 16822 prmdiveq 16823 odzcllem 16830 odzdvds 16833 odzphi 16834 vfermltlALT 16840 powm2modprm 16841 pcneg 16912 pcprmpw 16921 prmpwdvds 16942 pockthlem 16943 dyaddisjlem 25630 aalioulem1 26374 aaliou3lem6 26390 muf 27183 dvdsppwf1o 27229 mersenne 27271 lgslem1 27341 lgsval2lem 27351 lgsvalmod 27360 lgsmod 27367 lgsdirprm 27375 lgsne0 27379 lgsqrlem1 27390 gausslemma2dlem7 27417 gausslemma2d 27418 lgseisenlem2 27420 lgseisenlem4 27422 m1lgs 27432 2sqreultlem 27491 2sqreunnltlem 27494 znfermltl 33394 mdetlap 33831 oddpwdc 34356 nn0prpwlem 36323 nn0prpw 36324 knoppndvlem2 36514 aks4d1p3 42079 aks4d1p6 42082 aks6d1c2p2 42120 jm2.18 43000 jm2.22 43007 jm2.23 43008 jm2.20nn 43009 inductionexd 44168 etransclem3 46252 etransclem7 46256 etransclem10 46259 etransclem24 46273 etransclem27 46276 etransclem35 46284 2pwp1prm 47576 sfprmdvdsmersenne 47590 lighneallem4b 47596 lighneallem4 47597 proththd 47601 41prothprmlem2 47605 nnpw2evenALTV 47689 fpprmod 47714 fppr2odd 47718 dfwppr 47725 fpprwppr 47726 fpprwpprb 47727 pw2m1lepw2m1 48437 nnpw2blenfzo 48502 dignn0fr 48522 digexp 48528 dignn0flhalflem1 48536 |
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