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Mirrors > Home > MPE Home > Th. List > reexpcl | Structured version Visualization version GIF version |
Description: Closure of exponentiation of reals. For integer exponents, see reexpclz 14030. (Contributed by NM, 14-Dec-2005.) |
Ref | Expression |
---|---|
reexpcl | ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-resscn 11149 | . 2 ⊢ ℝ ⊆ ℂ | |
2 | remulcl 11177 | . 2 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) | |
3 | 1re 11196 | . 2 ⊢ 1 ∈ ℝ | |
4 | 1, 2, 3 | expcllem 14020 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 (class class class)co 7393 ℝcr 11091 ℕ0cn0 12454 ↑cexp 14009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-n0 12455 df-z 12541 df-uz 12805 df-seq 13949 df-exp 14010 |
This theorem is referenced by: expgt1 14048 resqcl 14071 reexpcld 14110 rpexpmord 14115 leexp2r 14121 leexp1a 14122 bernneq 14174 bernneq3 14176 expnbnd 14177 expnlbnd 14178 expmulnbnd 14180 digit2 14181 digit1 14182 expnngt1 14186 faclbnd 14232 faclbnd2 14233 faclbnd3 14234 faclbnd4lem1 14235 faclbnd5 14240 faclbnd6 14241 geomulcvg 15804 reeftcl 16000 ege2le3 16015 eftlub 16034 eflegeo 16046 resin4p 16063 recos4p 16064 ef01bndlem 16109 sin01bnd 16110 cos01bnd 16111 sin01gt0 16115 rpnnen2lem2 16140 rpnnen2lem4 16142 rpnnen2lem11 16149 powm2modprm 16718 prmreclem6 16836 mbfi1fseqlem6 25167 aaliou3lem8 25787 radcnvlem1 25854 abelthlem5 25876 abelthlem7 25879 tangtx 25944 advlogexp 26092 logtayllem 26096 leibpilem2 26373 leibpi 26374 leibpisum 26375 basellem3 26514 chtublem 26641 logexprlim 26655 dchrisum0flblem1 26938 pntlem3 27039 ostth2lem1 27048 ostth2lem3 27065 ostth3 27068 hgt750lem 33492 tgoldbachgnn 33500 subfacval2 34007 nn0prpw 35010 mblfinlem1 36327 mblfinlem2 36328 bfplem1 36493 lcmineqlem20 40716 3lexlogpow5ineq1 40722 tgoldbach 46255 dignn0fr 46933 digexp 46939 dig2bits 46946 |
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