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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 14121 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ+) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7431 ℤcz 12613 ℝ+crp 13034 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: bitsfzolem 16471 bitsfzo 16472 bitsmod 16473 bitsinv1 16479 sadasslem 16507 sadeq 16509 plyeq0lem 26249 aalioulem4 26377 aalioulem5 26378 aalioulem6 26379 aaliou 26380 aaliou3lem8 26387 nnlogbexp 26824 lgamgulmlem3 27074 ftalem5 27120 basellem3 27126 2sqmod 27480 rplogsumlem2 27529 rpvmasumlem 27531 pntlemh 27643 pntlemq 27645 pntlemr 27646 pntlemj 27647 pntlemf 27649 padicabv 27674 ostth2lem3 27679 dya2ub 34272 dya2iocress 34276 dya2iocbrsiga 34277 dya2icobrsiga 34278 sxbrsigalem2 34288 omssubadd 34302 signsply0 34566 hgt750leme 34673 tgoldbachgtde 34675 faclim 35746 iprodfac 35747 knoppndvlem17 36529 knoppndvlem18 36530 geomcau 37766 lcmineqlem21 42050 3lexlogpow5ineq5 42061 aks4d1p1p7 42075 aks4d1p1 42077 aks4d1p8d2 42086 aks4d1p8 42088 fltltc 42671 fltnlta 42673 pellfund14 42909 dvdivbd 45938 stirlinglem1 46089 stirlinglem2 46090 stirlinglem4 46092 stirlinglem8 46096 stirlinglem10 46098 stirlinglem11 46099 stirlinglem13 46101 stirlinglem15 46103 stirlingr 46105 sge0ad2en 46446 ovnsubaddlem1 46585 fllog2 48489 dignn0flhalflem1 48536 dignn0flhalflem2 48537 |
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