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Mirrors > Home > MPE Home > Th. List > recxpcl | Structured version Visualization version GIF version |
Description: Real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
recxpcl | ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) → (𝐴↑𝑐𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 11235 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | recn 11235 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
3 | cxpval 26648 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) | |
4 | 1, 2, 3 | syl2an 594 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) |
5 | 4 | 3adant2 1128 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) |
6 | 1re 11251 | . . . . 5 ⊢ 1 ∈ ℝ | |
7 | 0re 11253 | . . . . 5 ⊢ 0 ∈ ℝ | |
8 | 6, 7 | ifcli 4577 | . . . 4 ⊢ if(𝐵 = 0, 1, 0) ∈ ℝ |
9 | 8 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → if(𝐵 = 0, 1, 0) ∈ ℝ) |
10 | df-ne 2930 | . . . 4 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0) | |
11 | simpl3 1190 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℝ) | |
12 | simpl1 1188 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℝ) | |
13 | simpl2 1189 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 0 ≤ 𝐴) | |
14 | simpr 483 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0) | |
15 | 12, 13, 14 | ne0gt0d 11388 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 0 < 𝐴) |
16 | 12, 15 | elrpd 13053 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℝ+) |
17 | 16 | relogcld 26607 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℝ) |
18 | 11, 17 | remulcld 11281 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (𝐵 · (log‘𝐴)) ∈ ℝ) |
19 | 18 | reefcld 16073 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ 0) → (exp‘(𝐵 · (log‘𝐴))) ∈ ℝ) |
20 | 10, 19 | sylan2br 593 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) ∧ ¬ 𝐴 = 0) → (exp‘(𝐵 · (log‘𝐴))) ∈ ℝ) |
21 | 9, 20 | ifclda 4565 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) → if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) ∈ ℝ) |
22 | 5, 21 | eqeltrd 2825 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ) → (𝐴↑𝑐𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ifcif 4530 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ℂcc 11143 ℝcr 11144 0cc0 11145 1c1 11146 · cmul 11150 ≤ cle 11286 expce 16046 logclog 26538 ↑𝑐ccxp 26539 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 |
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-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 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-se 5634 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-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-fi 9441 df-sup 9472 df-inf 9473 df-oi 9540 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-mod 13876 df-seq 14008 df-exp 14068 df-fac 14274 df-bc 14303 df-hash 14331 df-shft 15055 df-cj 15087 df-re 15088 df-im 15089 df-sqrt 15223 df-abs 15224 df-limsup 15456 df-clim 15473 df-rlim 15474 df-sum 15674 df-ef 16052 df-sin 16054 df-cos 16055 df-pi 16057 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-starv 17256 df-sca 17257 df-vsca 17258 df-ip 17259 df-tset 17260 df-ple 17261 df-ds 17263 df-unif 17264 df-hom 17265 df-cco 17266 df-rest 17412 df-topn 17413 df-0g 17431 df-gsum 17432 df-topgen 17433 df-pt 17434 df-prds 17437 df-xrs 17492 df-qtop 17497 df-imas 17498 df-xps 17500 df-mre 17574 df-mrc 17575 df-acs 17577 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18749 df-mulg 19037 df-cntz 19285 df-cmn 19754 df-psmet 21293 df-xmet 21294 df-met 21295 df-bl 21296 df-mopn 21297 df-fbas 21298 df-fg 21299 df-cnfld 21302 df-top 22845 df-topon 22862 df-topsp 22884 df-bases 22898 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-lp 23089 df-perf 23090 df-cn 23180 df-cnp 23181 df-haus 23268 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-xms 24275 df-ms 24276 df-tms 24277 df-cncf 24847 df-limc 25844 df-dv 25845 df-log 26540 df-cxp 26541 |
This theorem is referenced by: rpcxpcl 26660 abscxp2 26677 cxple 26679 cxplt2 26682 recxpcld 26707 cxpaddlelem 26736 |
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