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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 481 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℝ+) | |
2 | rpne0 13025 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
3 | 2 | adantr 479 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ≠ 0) |
4 | simpr 483 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
5 | rpssre 13016 | . . . 4 ⊢ ℝ+ ⊆ ℝ | |
6 | ax-resscn 11197 | . . . 4 ⊢ ℝ ⊆ ℂ | |
7 | 5, 6 | sstri 3986 | . . 3 ⊢ ℝ+ ⊆ ℂ |
8 | rpmulcl 13032 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑥 · 𝑦) ∈ ℝ+) | |
9 | 1rp 13013 | . . 3 ⊢ 1 ∈ ℝ+ | |
10 | rpreccl 13035 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+) | |
11 | 10 | adantr 479 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℝ+) |
12 | 7, 8, 9, 11 | expcl2lem 14074 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
13 | 1, 3, 4, 12 | syl3anc 1368 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 / cdiv 11903 ℤcz 12591 ℝ+crp 13009 ↑cexp 14062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 |
This theorem is referenced by: expgt0 14096 ltexp2a 14166 expcan 14169 ltexp2 14170 leexp2a 14172 ltexp2r 14173 expnlbnd2 14232 rpexpcld 14245 expcnv 15846 effsumlt 16091 ef01bndlem 16164 rpnnen2lem11 16204 iscmet3lem3 25262 iscmet3lem1 25263 iscmet3lem2 25264 iscmet3 25265 minveclem3 25401 pjthlem1 25409 aaliou3lem1 26322 aaliou3lem2 26323 aaliou3lem3 26324 aaliou3lem8 26325 aaliou3lem5 26327 aaliou3lem6 26328 aaliou3lem7 26329 aaliou3lem9 26330 tanregt0 26518 asinlem3 26848 cxp2limlem 26953 ftalem5 27054 basellem3 27060 basellem4 27061 basellem8 27065 chebbnd1lem3 27449 dchrisum0lem1a 27464 dchrisum0lem1b 27493 dchrisum0lem1 27494 dchrisum0lem2a 27495 dchrisum0lem2 27496 dchrisum0lem3 27497 pntlemd 27572 pntlema 27574 pntlemb 27575 pntlemh 27577 pntlemr 27580 pntlemi 27582 pntlemf 27583 pntlemo 27585 pntlem3 27587 pntleml 27589 ostth2lem1 27596 ostth3 27616 minvecolem3 30758 pjhthlem1 31273 dpexpp1 32716 dya2icoseg 34028 faclimlem3 35470 geomcau 37363 dignnld 47862 |
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