expcl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > expcl		Structured version Visualization version GIF version

Theorem expcl 14045

Description: Closure law for nonnegative integer exponentiation. For integer exponents, see expclz 14050. (Contributed by NM, 26-May-2005.)

**Assertion**
Ref	Expression
expcl	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝐴↑𝑁) ∈ ℂ)

Proof of Theorem expcl Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 4005	. 2 ⊢ ℂ ⊆ ℂ
2		mulcl 11194	. 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
3		ax-1cn 11168	. 2 ⊢ 1 ∈ ℂ
4	1, 2, 3	expcllem 14038	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝐴↑𝑁) ∈ ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 (class class class)co 7409 ℂcc 11108 ℕ₀cn0 12472 ↑cexp 14027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-seq 13967 df-exp 14028

This theorem is referenced by: expeq0 14058 expnegz 14062 mulexp 14067 mulexpz 14068 expadd 14070 expaddzlem 14071 expaddz 14072 expmul 14073 expmulz 14074 expdiv 14079 expcld 14111 binom3 14187 digit2 14199 digit1 14200 faclbnd2 14251 faclbnd4lem4 14256 faclbnd6 14259 cjexp 15097 absexp 15251 ackbijnn 15774 binomlem 15775 binom1p 15777 binom1dif 15779 expcnv 15810 geolim 15816 geolim2 15817 geo2sum 15819 geomulcvg 15822 geoisum 15823 geoisumr 15824 geoisum1 15825 geoisum1c 15826 0.999... 15827 fallrisefac 15969 0risefac 15982 binomrisefac 15986 bpolysum 15997 bpolydiflem 15998 fsumkthpow 16000 bpoly3 16002 bpoly4 16003 fsumcube 16004 eftcl 16017 eftabs 16019 efcllem 16021 efcj 16035 efaddlem 16036 eflegeo 16064 efi4p 16080 prmreclem6 16854 karatsuba 17017 expmhm 21014 mbfi1fseqlem6 25238 itg0 25297 itgz 25298 itgcl 25301 itgcnlem 25307 itgsplit 25353 dvexp 25470 dvexp3 25495 plyf 25712 ply1termlem 25717 plypow 25719 plyeq0lem 25724 plypf1 25726 plyaddlem1 25727 plymullem1 25728 coeeulem 25738 coeidlem 25751 coeid3 25754 plyco 25755 dgrcolem2 25788 plycjlem 25790 plyrecj 25793 vieta1 25825 elqaalem3 25834 aareccl 25839 aalioulem1 25845 geolim3 25852 psergf 25924 dvradcnv 25933 psercn2 25935 pserdvlem2 25940 pserdv2 25942 abelthlem4 25946 abelthlem5 25947 abelthlem6 25948 abelthlem7 25950 abelthlem9 25952 advlogexp 26163 logtayllem 26167 logtayl 26168 logtaylsum 26169 logtayl2 26170 cxpeq 26265 dcubic1lem 26348 dcubic2 26349 dcubic1 26350 dcubic 26351 mcubic 26352 cubic2 26353 cubic 26354 binom4 26355 dquartlem2 26357 dquart 26358 quart1cl 26359 quart1lem 26360 quart1 26361 quartlem1 26362 quartlem2 26363 quart 26366 atantayl 26442 atantayl2 26443 atantayl3 26444 leibpi 26447 log2cnv 26449 log2tlbnd 26450 log2ublem3 26453 ftalem1 26577 ftalem4 26580 ftalem5 26581 basellem3 26587 musum 26695 1sgmprm 26702 perfect 26734 lgsquadlem1 26883 rplogsumlem2 26988 ostth2lem2 27137 numclwwlk3lem1 29635 ipval2 29960 dipcl 29965 dipcn 29973 subfacval2 34178 gg-expcn 35164 gg-psercn2 35178 lcmineqlem1 40894 lcmineqlem2 40895 lcmineqlem8 40901 lcmineqlem10 40903 jm2.23 41735 lhe4.4ex1a 43088 perfectALTV 46391 altgsumbc 47028 altgsumbcALT 47029 nn0digval 47286 ackval42 47382

W3C validator