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Mirrors > Home > MPE Home > Th. List > rpexpcl | Structured version Visualization version GIF version |
Description: Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.) |
Ref | Expression |
---|---|
rpexpcl | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 474 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℝ+) | |
2 | rpne0 12041 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
3 | 2 | adantr 472 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ≠ 0) |
4 | simpr 479 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
5 | rpssre 12036 | . . . 4 ⊢ ℝ+ ⊆ ℝ | |
6 | ax-resscn 10185 | . . . 4 ⊢ ℝ ⊆ ℂ | |
7 | 5, 6 | sstri 3753 | . . 3 ⊢ ℝ+ ⊆ ℂ |
8 | rpmulcl 12048 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑥 · 𝑦) ∈ ℝ+) | |
9 | 1rp 12029 | . . 3 ⊢ 1 ∈ ℝ+ | |
10 | rpreccl 12050 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+) | |
11 | 10 | adantr 472 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℝ+) |
12 | 7, 8, 9, 11 | expcl2lem 13066 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
13 | 1, 3, 4, 12 | syl3anc 1477 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2139 ≠ wne 2932 (class class class)co 6813 ℂcc 10126 ℝcr 10127 0cc0 10128 1c1 10129 / cdiv 10876 ℤcz 11569 ℝ+crp 12025 ↑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-rmo 3058 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-div 10877 df-nn 11213 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-seq 12996 df-exp 13055 |
This theorem is referenced by: expgt0 13087 ltexp2a 13106 expcan 13107 ltexp2 13108 leexp2a 13110 ltexp2r 13111 expnlbnd2 13189 rpexpcld 13226 expcnv 14795 effsumlt 15040 ef01bndlem 15113 rpnnen2lem11 15152 iscmet3lem3 23288 iscmet3lem1 23289 iscmet3lem2 23290 iscmet3 23291 minveclem3 23400 pjthlem1 23408 aaliou3lem1 24296 aaliou3lem2 24297 aaliou3lem3 24298 aaliou3lem8 24299 aaliou3lem5 24301 aaliou3lem6 24302 aaliou3lem7 24303 aaliou3lem9 24304 tanregt0 24484 asinlem3 24797 cxp2limlem 24901 ftalem5 25002 basellem3 25008 basellem4 25009 basellem8 25013 chebbnd1lem3 25359 dchrisum0lem1a 25374 dchrisum0lem1b 25403 dchrisum0lem1 25404 dchrisum0lem2a 25405 dchrisum0lem2 25406 dchrisum0lem3 25407 pntlemd 25482 pntlema 25484 pntlemb 25485 pntlemh 25487 pntlemr 25490 pntlemi 25492 pntlemf 25493 pntlemo 25495 pntlem3 25497 pntleml 25499 ostth2lem1 25506 ostth3 25526 minvecolem3 28041 pjhthlem1 28559 dpexpp1 29925 dya2icoseg 30648 faclimlem3 31938 geomcau 33868 dignnld 42907 |
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