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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 486 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℝ+) | |
2 | rpne0 12602 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
3 | 2 | adantr 484 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ≠ 0) |
4 | simpr 488 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
5 | rpssre 12593 | . . . 4 ⊢ ℝ+ ⊆ ℝ | |
6 | ax-resscn 10786 | . . . 4 ⊢ ℝ ⊆ ℂ | |
7 | 5, 6 | sstri 3910 | . . 3 ⊢ ℝ+ ⊆ ℂ |
8 | rpmulcl 12609 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑥 · 𝑦) ∈ ℝ+) | |
9 | 1rp 12590 | . . 3 ⊢ 1 ∈ ℝ+ | |
10 | rpreccl 12612 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+) | |
11 | 10 | adantr 484 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℝ+) |
12 | 7, 8, 9, 11 | expcl2lem 13647 | . 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 399 ∈ wcel 2110 ≠ wne 2940 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 / cdiv 11489 ℤcz 12176 ℝ+crp 12586 ↑cexp 13635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-seq 13575 df-exp 13636 |
This theorem is referenced by: expgt0 13668 ltexp2a 13736 expcan 13739 ltexp2 13740 leexp2a 13742 ltexp2r 13743 expnlbnd2 13801 rpexpcld 13814 expcnv 15428 effsumlt 15672 ef01bndlem 15745 rpnnen2lem11 15785 iscmet3lem3 24187 iscmet3lem1 24188 iscmet3lem2 24189 iscmet3 24190 minveclem3 24326 pjthlem1 24334 aaliou3lem1 25235 aaliou3lem2 25236 aaliou3lem3 25237 aaliou3lem8 25238 aaliou3lem5 25240 aaliou3lem6 25241 aaliou3lem7 25242 aaliou3lem9 25243 tanregt0 25428 asinlem3 25754 cxp2limlem 25858 ftalem5 25959 basellem3 25965 basellem4 25966 basellem8 25970 chebbnd1lem3 26352 dchrisum0lem1a 26367 dchrisum0lem1b 26396 dchrisum0lem1 26397 dchrisum0lem2a 26398 dchrisum0lem2 26399 dchrisum0lem3 26400 pntlemd 26475 pntlema 26477 pntlemb 26478 pntlemh 26480 pntlemr 26483 pntlemi 26485 pntlemf 26486 pntlemo 26488 pntlem3 26490 pntleml 26492 ostth2lem1 26499 ostth3 26519 minvecolem3 28957 pjhthlem1 29472 dpexpp1 30902 dya2icoseg 31956 faclimlem3 33429 geomcau 35654 dignnld 45622 |
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