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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 3587 | . 2 ⊢ ℂ ⊆ ℂ | |
2 | mulcl 9899 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
3 | ax-1cn 9873 | . 2 ⊢ 1 ∈ ℂ | |
4 | 1, 2, 3 | expcllem 12733 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 ℕ0cn0 11169 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: expeq0 12752 expnegz 12756 mulexp 12761 mulexpz 12762 expadd 12764 expaddzlem 12765 expaddz 12766 expmul 12767 expmulz 12768 expdiv 12773 binom3 12847 digit2 12859 digit1 12860 expcld 12870 faclbnd2 12940 faclbnd4lem4 12945 faclbnd6 12948 cjexp 13738 absexp 13892 ackbijnn 14399 binomlem 14400 binom1p 14402 binom1dif 14404 expcnv 14435 geolim 14440 geolim2 14441 geo2sum 14443 geomulcvg 14446 geoisum 14447 geoisumr 14448 geoisum1 14449 geoisum1c 14450 0.999... 14451 0.999...OLD 14452 fallrisefac 14595 0risefac 14608 binomrisefac 14612 bpolysum 14623 bpolydiflem 14624 fsumkthpow 14626 bpoly3 14628 bpoly4 14629 fsumcube 14630 eftcl 14643 eftabs 14645 efcllem 14647 efcj 14661 efaddlem 14662 eflegeo 14690 efi4p 14706 prmreclem6 15463 decsplitOLD 15629 karatsuba 15630 karatsubaOLD 15631 expmhm 19634 mbfi1fseqlem6 23293 itg0 23352 itgz 23353 itgcl 23356 itgcnlem 23362 itgsplit 23408 dvexp 23522 dvexp3 23545 plyf 23758 ply1termlem 23763 plypow 23765 plyeq0lem 23770 plypf1 23772 plyaddlem1 23773 plymullem1 23774 coeeulem 23784 coeidlem 23797 coeid3 23800 plyco 23801 dgrcolem2 23834 plycjlem 23836 plyrecj 23839 vieta1 23871 elqaalem3 23880 aareccl 23885 aalioulem1 23891 geolim3 23898 psergf 23970 dvradcnv 23979 psercn2 23981 pserdvlem2 23986 pserdv2 23988 abelthlem4 23992 abelthlem5 23993 abelthlem6 23994 abelthlem7 23996 abelthlem9 23998 advlogexp 24201 logtayllem 24205 logtayl 24206 logtaylsum 24207 logtayl2 24208 cxpeq 24298 dcubic1lem 24370 dcubic2 24371 dcubic1 24372 dcubic 24373 mcubic 24374 cubic2 24375 cubic 24376 binom4 24377 dquartlem2 24379 dquart 24380 quart1cl 24381 quart1lem 24382 quart1 24383 quartlem1 24384 quartlem2 24385 quart 24388 atantayl 24464 atantayl2 24465 atantayl3 24466 leibpi 24469 log2cnv 24471 log2tlbnd 24472 log2ublem3 24475 ftalem1 24599 ftalem4 24602 ftalem5 24603 basellem3 24609 musum 24717 1sgmprm 24724 perfect 24756 lgsquadlem1 24905 rplogsumlem2 24974 ostth2lem2 25123 numclwlk3lem3 26600 ipval2 26946 dipcl 26951 dipcn 26959 subfacval2 30423 jm2.23 36581 lhe4.4ex1a 37550 perfectALTV 40166 av-numclwlk3lem3 41506 altgsumbc 41923 altgsumbcALT 41924 nn0digval 42192 |
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