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Mirrors > Home > MPE Home > Th. List > expcl | Structured version Visualization version GIF version |
Description: Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.) |
Ref | Expression |
---|---|
expcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3842 | . 2 ⊢ ℂ ⊆ ℂ | |
2 | mulcl 10356 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
3 | ax-1cn 10330 | . 2 ⊢ 1 ∈ ℂ | |
4 | 1, 2, 3 | expcllem 13189 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 ℕ0cn0 11642 ↑cexp 13178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-seq 13120 df-exp 13179 |
This theorem is referenced by: expeq0 13208 expnegz 13212 mulexp 13217 mulexpz 13218 expadd 13220 expaddzlem 13221 expaddz 13222 expmul 13223 expmulz 13224 expdiv 13229 binom3 13304 digit2 13316 digit1 13317 expcld 13327 faclbnd2 13396 faclbnd4lem4 13401 faclbnd6 13404 cjexp 14297 absexp 14451 ackbijnn 14964 binomlem 14965 binom1p 14967 binom1dif 14969 expcnv 15000 geolim 15005 geolim2 15006 geo2sum 15008 geomulcvg 15011 geoisum 15012 geoisumr 15013 geoisum1 15014 geoisum1c 15015 0.999... 15016 fallrisefac 15158 0risefac 15171 binomrisefac 15175 bpolysum 15186 bpolydiflem 15187 fsumkthpow 15189 bpoly3 15191 bpoly4 15192 fsumcube 15193 eftcl 15206 eftabs 15208 efcllem 15210 efcj 15224 efaddlem 15225 eflegeo 15253 efi4p 15269 prmreclem6 16029 karatsuba 16192 expmhm 20211 mbfi1fseqlem6 23924 itg0 23983 itgz 23984 itgcl 23987 itgcnlem 23993 itgsplit 24039 dvexp 24153 dvexp3 24178 plyf 24391 ply1termlem 24396 plypow 24398 plyeq0lem 24403 plypf1 24405 plyaddlem1 24406 plymullem1 24407 coeeulem 24417 coeidlem 24430 coeid3 24433 plyco 24434 dgrcolem2 24467 plycjlem 24469 plyrecj 24472 vieta1 24504 elqaalem3 24513 aareccl 24518 aalioulem1 24524 geolim3 24531 psergf 24603 dvradcnv 24612 psercn2 24614 pserdvlem2 24619 pserdv2 24621 abelthlem4 24625 abelthlem5 24626 abelthlem6 24627 abelthlem7 24629 abelthlem9 24631 advlogexp 24838 logtayllem 24842 logtayl 24843 logtaylsum 24844 logtayl2 24845 cxpeq 24938 dcubic1lem 25021 dcubic2 25022 dcubic1 25023 dcubic 25024 mcubic 25025 cubic2 25026 cubic 25027 binom4 25028 dquartlem2 25030 dquart 25031 quart1cl 25032 quart1lem 25033 quart1 25034 quartlem1 25035 quartlem2 25036 quart 25039 atantayl 25115 atantayl2 25116 atantayl3 25117 leibpi 25121 log2cnv 25123 log2tlbnd 25124 log2ublem3 25127 ftalem1 25251 ftalem4 25254 ftalem5 25255 basellem3 25261 musum 25369 1sgmprm 25376 perfect 25408 lgsquadlem1 25557 rplogsumlem2 25626 ostth2lem2 25775 numclwwlk3lem1 27814 ipval2 28134 dipcl 28139 dipcn 28147 subfacval2 31768 jm2.23 38526 lhe4.4ex1a 39488 perfectALTV 42661 altgsumbc 43149 altgsumbcALT 43150 nn0digval 43413 |
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