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Mirrors > Home > MPE Home > Th. List > rpexpcl | Structured version Visualization version GIF version |
Description: Closure law for integer 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 482 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℝ+) | |
2 | rpne0 13048 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
3 | 2 | adantr 480 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ≠ 0) |
4 | simpr 484 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
5 | rpssre 13039 | . . . 4 ⊢ ℝ+ ⊆ ℝ | |
6 | ax-resscn 11209 | . . . 4 ⊢ ℝ ⊆ ℂ | |
7 | 5, 6 | sstri 4004 | . . 3 ⊢ ℝ+ ⊆ ℂ |
8 | rpmulcl 13055 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑥 · 𝑦) ∈ ℝ+) | |
9 | 1rp 13035 | . . 3 ⊢ 1 ∈ ℝ+ | |
10 | rpreccl 13058 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+) | |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℝ+) |
12 | 7, 8, 9, 11 | expcl2lem 14110 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
13 | 1, 3, 4, 12 | syl3anc 1370 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ≠ wne 2937 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 / cdiv 11917 ℤcz 12610 ℝ+crp 13031 ↑cexp 14098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 |
This theorem is referenced by: expgt0 14132 ltexp2a 14202 expcan 14205 ltexp2 14206 leexp2a 14208 ltexp2r 14209 expnlbnd2 14269 rpexpcld 14282 expcnv 15896 effsumlt 16143 ef01bndlem 16216 rpnnen2lem11 16256 iscmet3lem3 25337 iscmet3lem1 25338 iscmet3lem2 25339 iscmet3 25340 minveclem3 25476 pjthlem1 25484 aaliou3lem1 26398 aaliou3lem2 26399 aaliou3lem3 26400 aaliou3lem8 26401 aaliou3lem5 26403 aaliou3lem6 26404 aaliou3lem7 26405 aaliou3lem9 26406 tanregt0 26595 asinlem3 26928 cxp2limlem 27033 ftalem5 27134 basellem3 27140 basellem4 27141 basellem8 27145 chebbnd1lem3 27529 dchrisum0lem1a 27544 dchrisum0lem1b 27573 dchrisum0lem1 27574 dchrisum0lem2a 27575 dchrisum0lem2 27576 dchrisum0lem3 27577 pntlemd 27652 pntlema 27654 pntlemb 27655 pntlemh 27657 pntlemr 27660 pntlemi 27662 pntlemf 27663 pntlemo 27665 pntlem3 27667 pntleml 27669 ostth2lem1 27676 ostth3 27696 minvecolem3 30904 pjhthlem1 31419 dpexpp1 32874 dya2icoseg 34258 faclimlem3 35724 geomcau 37745 dignnld 48452 |
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