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| Mirrors > Home > MPE Home > Th. List > expcl | Structured version Visualization version GIF version | ||
| Description: Closure law for nonnegative integer exponentiation. For integer exponents, see expclz 14125. (Contributed by NM, 26-May-2005.) |
| Ref | Expression |
|---|---|
| expcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssid 4006 | . 2 ⊢ ℂ ⊆ ℂ | |
| 2 | mulcl 11239 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
| 3 | ax-1cn 11213 | . 2 ⊢ 1 ∈ ℂ | |
| 4 | 1, 2, 3 | expcllem 14113 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 ℕ0cn0 12526 ↑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: expeq0 14133 expnegz 14137 mulexp 14142 mulexpz 14143 expadd 14145 expaddzlem 14146 expaddz 14147 expmul 14148 expmulz 14149 expdiv 14154 expcld 14186 binom3 14263 digit2 14275 digit1 14276 faclbnd2 14330 faclbnd4lem4 14335 faclbnd6 14338 cjexp 15189 absexp 15343 ackbijnn 15864 binomlem 15865 binom1p 15867 binom1dif 15869 expcnv 15900 geolim 15906 geolim2 15907 geo2sum 15909 geomulcvg 15912 geoisum 15913 geoisumr 15914 geoisum1 15915 geoisum1c 15916 0.999... 15917 fallrisefac 16061 0risefac 16074 binomrisefac 16078 bpolysum 16089 bpolydiflem 16090 fsumkthpow 16092 bpoly3 16094 bpoly4 16095 fsumcube 16096 eftcl 16109 eftabs 16111 efcllem 16113 efcj 16128 efaddlem 16129 eflegeo 16157 efi4p 16173 prmreclem6 16959 karatsuba 17121 expmhm 21454 expcn 24896 mbfi1fseqlem6 25755 itg0 25815 itgz 25816 itgcl 25819 itgcnlem 25825 itgsplit 25871 dvexp 25991 dvexp3 26016 plyf 26237 ply1termlem 26242 plypow 26244 plyeq0lem 26249 plypf1 26251 plyaddlem1 26252 plymullem1 26253 coeeulem 26263 coeidlem 26276 coeid3 26279 plyco 26280 dgrcolem2 26314 plycjlem 26316 plyrecj 26321 vieta1 26354 elqaalem3 26363 aareccl 26368 aalioulem1 26374 geolim3 26381 psergf 26455 dvradcnv 26464 psercn2 26466 psercn2OLD 26467 pserdvlem2 26472 pserdv2 26474 abelthlem4 26478 abelthlem5 26479 abelthlem6 26480 abelthlem7 26482 abelthlem9 26484 advlogexp 26697 logtayllem 26701 logtayl 26702 logtaylsum 26703 logtayl2 26704 cxpeq 26800 dcubic1lem 26886 dcubic2 26887 dcubic1 26888 dcubic 26889 mcubic 26890 cubic2 26891 cubic 26892 binom4 26893 dquartlem2 26895 dquart 26896 quart1cl 26897 quart1lem 26898 quart1 26899 quartlem1 26900 quartlem2 26901 quart 26904 atantayl 26980 atantayl2 26981 atantayl3 26982 leibpi 26985 log2cnv 26987 log2tlbnd 26988 log2ublem3 26991 ftalem1 27116 ftalem4 27119 ftalem5 27120 basellem3 27126 musum 27234 1sgmprm 27243 perfect 27275 lgsquadlem1 27424 rplogsumlem2 27529 ostth2lem2 27678 numclwwlk3lem1 30401 ipval2 30726 dipcl 30731 dipcn 30739 subfacval2 35192 lcmineqlem1 42030 lcmineqlem2 42031 lcmineqlem8 42037 lcmineqlem10 42039 jm2.23 43008 lhe4.4ex1a 44348 perfectALTV 47710 altgsumbc 48268 altgsumbcALT 48269 nn0digval 48521 ackval42 48617 |
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