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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 13051 | . . 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 13042 | . . . 4 ⊢ ℝ+ ⊆ ℝ | |
| 6 | ax-resscn 11212 | . . . 4 ⊢ ℝ ⊆ ℂ | |
| 7 | 5, 6 | sstri 3993 | . . 3 ⊢ ℝ+ ⊆ ℂ |
| 8 | rpmulcl 13058 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑥 · 𝑦) ∈ ℝ+) | |
| 9 | 1rp 13038 | . . 3 ⊢ 1 ∈ ℝ+ | |
| 10 | rpreccl 13061 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+) | |
| 11 | 10 | adantr 480 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℝ+) |
| 12 | 7, 8, 9, 11 | expcl2lem 14114 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
| 13 | 1, 3, 4, 12 | syl3anc 1373 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 / cdiv 11920 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: expgt0 14136 ltexp2a 14206 expcan 14209 ltexp2 14210 leexp2a 14212 ltexp2r 14213 expnlbnd2 14273 rpexpcld 14286 expcnv 15900 effsumlt 16147 ef01bndlem 16220 rpnnen2lem11 16260 iscmet3lem3 25324 iscmet3lem1 25325 iscmet3lem2 25326 iscmet3 25327 minveclem3 25463 pjthlem1 25471 aaliou3lem1 26384 aaliou3lem2 26385 aaliou3lem3 26386 aaliou3lem8 26387 aaliou3lem5 26389 aaliou3lem6 26390 aaliou3lem7 26391 aaliou3lem9 26392 tanregt0 26581 asinlem3 26914 cxp2limlem 27019 ftalem5 27120 basellem3 27126 basellem4 27127 basellem8 27131 chebbnd1lem3 27515 dchrisum0lem1a 27530 dchrisum0lem1b 27559 dchrisum0lem1 27560 dchrisum0lem2a 27561 dchrisum0lem2 27562 dchrisum0lem3 27563 pntlemd 27638 pntlema 27640 pntlemb 27641 pntlemh 27643 pntlemr 27646 pntlemi 27648 pntlemf 27649 pntlemo 27651 pntlem3 27653 pntleml 27655 ostth2lem1 27662 ostth3 27682 minvecolem3 30895 pjhthlem1 31410 dpexpp1 32890 dya2icoseg 34279 faclimlem3 35745 geomcau 37766 dignnld 48524 |
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