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Mirrors > Home > MPE Home > Th. List > reexpcl | Structured version Visualization version GIF version |
Description: Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.) |
Ref | Expression |
---|---|
reexpcl | ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-resscn 10185 | . 2 ⊢ ℝ ⊆ ℂ | |
2 | remulcl 10213 | . 2 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) | |
3 | 1re 10231 | . 2 ⊢ 1 ∈ ℝ | |
4 | 1, 2, 3 | expcllem 13065 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2139 (class class class)co 6813 ℝcr 10127 ℕ0cn0 11484 ↑cexp 13054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-n0 11485 df-z 11570 df-uz 11880 df-seq 12996 df-exp 13055 |
This theorem is referenced by: expgt1 13092 leexp2r 13112 leexp1a 13113 resqcl 13125 bernneq 13184 bernneq3 13186 expnbnd 13187 expnlbnd 13188 expmulnbnd 13190 digit2 13191 digit1 13192 reexpcld 13219 faclbnd 13271 faclbnd2 13272 faclbnd3 13273 faclbnd4lem1 13274 faclbnd5 13279 faclbnd6 13280 geomulcvg 14806 reeftcl 15004 ege2le3 15019 eftlub 15038 eflegeo 15050 resin4p 15067 recos4p 15068 ef01bndlem 15113 sin01bnd 15114 cos01bnd 15115 sin01gt0 15119 rpnnen2lem2 15143 rpnnen2lem4 15145 rpnnen2lem11 15152 powm2modprm 15710 prmreclem6 15827 mbfi1fseqlem6 23686 aaliou3lem8 24299 radcnvlem1 24366 abelthlem5 24388 abelthlem7 24391 tangtx 24456 advlogexp 24600 logtayllem 24604 leibpilem2 24867 leibpi 24868 leibpisum 24869 basellem3 25008 chtublem 25135 logexprlim 25149 dchrisum0flblem1 25396 pntlem3 25497 ostth2lem1 25506 ostth2lem3 25523 ostth3 25526 hgt750lem 31038 tgoldbachgnn 31046 subfacval2 31476 nn0prpw 32624 mblfinlem1 33759 mblfinlem2 33760 bfplem1 33934 rpexpmord 38015 tgoldbach 42215 tgoldbachOLD 42222 dignn0fr 42905 digexp 42911 dig2bits 42918 |
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