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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 485 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ ℝ+) | |
2 | rpne0 12406 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
3 | 2 | adantr 483 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝐴 ≠ 0) |
4 | simpr 487 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
5 | rpssre 12397 | . . . 4 ⊢ ℝ+ ⊆ ℝ | |
6 | ax-resscn 10594 | . . . 4 ⊢ ℝ ⊆ ℂ | |
7 | 5, 6 | sstri 3976 | . . 3 ⊢ ℝ+ ⊆ ℂ |
8 | rpmulcl 12413 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑥 · 𝑦) ∈ ℝ+) | |
9 | 1rp 12394 | . . 3 ⊢ 1 ∈ ℝ+ | |
10 | rpreccl 12416 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+) | |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℝ+) |
12 | 7, 8, 9, 11 | expcl2lem 13442 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
13 | 1, 3, 4, 12 | syl3anc 1367 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ≠ wne 3016 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 / cdiv 11297 ℤcz 11982 ℝ+crp 12390 ↑cexp 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 |
This theorem is referenced by: expgt0 13463 ltexp2a 13531 expcan 13534 ltexp2 13535 leexp2a 13537 ltexp2r 13538 expnlbnd2 13596 rpexpcld 13609 expcnv 15219 effsumlt 15464 ef01bndlem 15537 rpnnen2lem11 15577 iscmet3lem3 23893 iscmet3lem1 23894 iscmet3lem2 23895 iscmet3 23896 minveclem3 24032 pjthlem1 24040 aaliou3lem1 24931 aaliou3lem2 24932 aaliou3lem3 24933 aaliou3lem8 24934 aaliou3lem5 24936 aaliou3lem6 24937 aaliou3lem7 24938 aaliou3lem9 24939 tanregt0 25123 asinlem3 25449 cxp2limlem 25553 ftalem5 25654 basellem3 25660 basellem4 25661 basellem8 25665 chebbnd1lem3 26047 dchrisum0lem1a 26062 dchrisum0lem1b 26091 dchrisum0lem1 26092 dchrisum0lem2a 26093 dchrisum0lem2 26094 dchrisum0lem3 26095 pntlemd 26170 pntlema 26172 pntlemb 26173 pntlemh 26175 pntlemr 26178 pntlemi 26180 pntlemf 26181 pntlemo 26183 pntlem3 26185 pntleml 26187 ostth2lem1 26194 ostth3 26214 minvecolem3 28653 pjhthlem1 29168 dpexpp1 30584 dya2icoseg 31535 faclimlem3 32977 geomcau 35049 dignnld 44712 |
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