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Mirrors > Home > MPE Home > Th. List > rpexpcld | Structured version Visualization version GIF version |
Description: Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpexpcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
rpexpcld.2 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Ref | Expression |
---|---|
rpexpcld | ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpexpcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | rpexpcld.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | rpexpcl 13444 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 (class class class)co 7135 ℤcz 11969 ℝ+crp 12377 ↑cexp 13425 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 |
This theorem is referenced by: bitsfzolem 15773 bitsfzo 15774 bitsmod 15775 bitsinv1 15781 sadasslem 15809 sadeq 15811 plyeq0lem 24807 aalioulem4 24931 aalioulem5 24932 aalioulem6 24933 aaliou 24934 aaliou3lem8 24941 nnlogbexp 25367 lgamgulmlem3 25616 ftalem5 25662 basellem3 25668 2sqmod 26020 rplogsumlem2 26069 rpvmasumlem 26071 pntlemh 26183 pntlemq 26185 pntlemr 26186 pntlemj 26187 pntlemf 26189 padicabv 26214 ostth2lem3 26219 dya2ub 31638 dya2iocress 31642 dya2iocbrsiga 31643 dya2icobrsiga 31644 sxbrsigalem2 31654 omssubadd 31668 signsply0 31931 hgt750leme 32039 tgoldbachgtde 32041 faclim 33091 iprodfac 33092 knoppndvlem17 33980 knoppndvlem18 33981 geomcau 35197 lcmineqlem21 39337 fltltc 39617 fltnlta 39619 pellfund14 39839 dvdivbd 42565 stirlinglem1 42716 stirlinglem2 42717 stirlinglem4 42719 stirlinglem8 42723 stirlinglem10 42725 stirlinglem11 42726 stirlinglem13 42728 stirlinglem15 42730 stirlingr 42732 sge0ad2en 43070 ovnsubaddlem1 43209 fllog2 44982 dignn0flhalflem1 45029 dignn0flhalflem2 45030 |
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