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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 14087. (Contributed by NM, 26-May-2005.) |
Ref | Expression |
---|---|
expcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 4002 | . 2 ⊢ ℂ ⊆ ℂ | |
2 | mulcl 11228 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
3 | ax-1cn 11202 | . 2 ⊢ 1 ∈ ℂ | |
4 | 1, 2, 3 | expcllem 14075 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 (class class class)co 7424 ℂcc 11142 ℕ0cn0 12508 ↑cexp 14064 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-n0 12509 df-z 12595 df-uz 12859 df-seq 14005 df-exp 14065 |
This theorem is referenced by: expeq0 14095 expnegz 14099 mulexp 14104 mulexpz 14105 expadd 14107 expaddzlem 14108 expaddz 14109 expmul 14110 expmulz 14111 expdiv 14116 expcld 14148 binom3 14224 digit2 14236 digit1 14237 faclbnd2 14288 faclbnd4lem4 14293 faclbnd6 14296 cjexp 15135 absexp 15289 ackbijnn 15812 binomlem 15813 binom1p 15815 binom1dif 15817 expcnv 15848 geolim 15854 geolim2 15855 geo2sum 15857 geomulcvg 15860 geoisum 15861 geoisumr 15862 geoisum1 15863 geoisum1c 15864 0.999... 15865 fallrisefac 16007 0risefac 16020 binomrisefac 16024 bpolysum 16035 bpolydiflem 16036 fsumkthpow 16038 bpoly3 16040 bpoly4 16041 fsumcube 16042 eftcl 16055 eftabs 16057 efcllem 16059 efcj 16074 efaddlem 16075 eflegeo 16103 efi4p 16119 prmreclem6 16895 karatsuba 17058 expmhm 21374 expcn 24808 mbfi1fseqlem6 25668 itg0 25727 itgz 25728 itgcl 25731 itgcnlem 25737 itgsplit 25783 dvexp 25903 dvexp3 25928 plyf 26150 ply1termlem 26155 plypow 26157 plyeq0lem 26162 plypf1 26164 plyaddlem1 26165 plymullem1 26166 coeeulem 26176 coeidlem 26189 coeid3 26192 plyco 26193 dgrcolem2 26227 plycjlem 26229 plyrecj 26232 vieta1 26265 elqaalem3 26274 aareccl 26279 aalioulem1 26285 geolim3 26292 psergf 26366 dvradcnv 26375 psercn2 26377 psercn2OLD 26378 pserdvlem2 26383 pserdv2 26385 abelthlem4 26389 abelthlem5 26390 abelthlem6 26391 abelthlem7 26393 abelthlem9 26395 advlogexp 26607 logtayllem 26611 logtayl 26612 logtaylsum 26613 logtayl2 26614 cxpeq 26710 dcubic1lem 26793 dcubic2 26794 dcubic1 26795 dcubic 26796 mcubic 26797 cubic2 26798 cubic 26799 binom4 26800 dquartlem2 26802 dquart 26803 quart1cl 26804 quart1lem 26805 quart1 26806 quartlem1 26807 quartlem2 26808 quart 26811 atantayl 26887 atantayl2 26888 atantayl3 26889 leibpi 26892 log2cnv 26894 log2tlbnd 26895 log2ublem3 26898 ftalem1 27023 ftalem4 27026 ftalem5 27027 basellem3 27033 musum 27141 1sgmprm 27150 perfect 27182 lgsquadlem1 27331 rplogsumlem2 27436 ostth2lem2 27585 numclwwlk3lem1 30210 ipval2 30535 dipcl 30540 dipcn 30548 subfacval2 34802 lcmineqlem1 41504 lcmineqlem2 41505 lcmineqlem8 41511 lcmineqlem10 41513 jm2.23 42420 lhe4.4ex1a 43769 perfectALTV 47065 altgsumbc 47467 altgsumbcALT 47468 nn0digval 47724 ackval42 47820 |
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