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Theorem abscxp2 20572
Description: Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
abscxp2  |-  ( ( A  e.  CC  /\  B  e.  RR )  ->  ( abs `  ( A  ^ c  B ) )  =  ( ( abs `  A )  ^ c  B ) )

Proof of Theorem abscxp2
StepHypRef Expression
1 0re 9080 . . . . . 6  |-  0  e.  RR
21a1i 11 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  0  e.  RR )
3 0le0 10070 . . . . . 6  |-  0  <_  0
43a1i 11 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  0  <_  0 )
5 simplr 732 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  B  e.  RR )
6 recxpcl 20554 . . . . 5  |-  ( ( 0  e.  RR  /\  0  <_  0  /\  B  e.  RR )  ->  (
0  ^ c  B
)  e.  RR )
72, 4, 5, 6syl3anc 1184 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( 0  ^ c  B )  e.  RR )
8 cxpge0 20562 . . . . 5  |-  ( ( 0  e.  RR  /\  0  <_  0  /\  B  e.  RR )  ->  0  <_  ( 0  ^ c  B ) )
92, 4, 5, 8syl3anc 1184 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  0  <_  ( 0  ^ c  B
) )
107, 9absidd 12213 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( abs `  ( 0  ^ c  B ) )  =  ( 0  ^ c  B ) )
11 simpr 448 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  A  = 
0 )
1211oveq1d 6087 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( A  ^ c  B )  =  ( 0  ^ c  B ) )
1312fveq2d 5723 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( abs `  ( A  ^ c  B ) )  =  ( abs `  (
0  ^ c  B
) ) )
1411abs00bd 12084 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( abs `  A )  =  0 )
1514oveq1d 6087 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( ( abs `  A )  ^ c  B )  =  ( 0  ^ c  B
) )
1610, 13, 153eqtr4d 2477 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( abs `  ( A  ^ c  B ) )  =  ( ( abs `  A
)  ^ c  B
) )
17 simplr 732 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  B  e.  RR )
1817recnd 9103 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  B  e.  CC )
19 logcl 20454 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
2019adantlr 696 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( log `  A )  e.  CC )
2118, 20mulcld 9097 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( B  x.  ( log `  A
) )  e.  CC )
22 absef 12786 . . . . 5  |-  ( ( B  x.  ( log `  A ) )  e.  CC  ->  ( abs `  ( exp `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( Re `  ( B  x.  ( log `  A ) ) ) ) )
2321, 22syl 16 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  ( exp `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( Re `  ( B  x.  ( log `  A ) ) ) ) )
2417, 20remul2d 12020 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( Re `  ( B  x.  ( log `  A ) ) )  =  ( B  x.  ( Re `  ( log `  A ) ) ) )
25 relog 20479 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  =  ( log `  ( abs `  A ) ) )
2625adantlr 696 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( Re `  ( log `  A
) )  =  ( log `  ( abs `  A ) ) )
2726oveq2d 6088 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( B  x.  ( Re `  ( log `  A ) ) )  =  ( B  x.  ( log `  ( abs `  A ) ) ) )
2824, 27eqtrd 2467 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( Re `  ( B  x.  ( log `  A ) ) )  =  ( B  x.  ( log `  ( abs `  A ) ) ) )
2928fveq2d 5723 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( exp `  ( Re `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( B  x.  ( log `  ( abs `  A
) ) ) ) )
3023, 29eqtrd 2467 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  ( exp `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( B  x.  ( log `  ( abs `  A
) ) ) ) )
31 simpll 731 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  A  e.  CC )
32 simpr 448 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  A  =/=  0 )
33 cxpef 20544 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
3431, 32, 18, 33syl3anc 1184 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
3534fveq2d 5723 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  ( A  ^ c  B ) )  =  ( abs `  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
3631abscld 12226 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  RR )
3736recnd 9103 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  CC )
38 abs00 12082 . . . . . . 7  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
3938adantr 452 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  RR )  ->  ( ( abs `  A
)  =  0  <->  A  =  0 ) )
4039necon3bid 2633 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  RR )  ->  ( ( abs `  A
)  =/=  0  <->  A  =/=  0 ) )
4140biimpar 472 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  A )  =/=  0
)
42 cxpef 20544 . . . 4  |-  ( ( ( abs `  A
)  e.  CC  /\  ( abs `  A )  =/=  0  /\  B  e.  CC )  ->  (
( abs `  A
)  ^ c  B
)  =  ( exp `  ( B  x.  ( log `  ( abs `  A
) ) ) ) )
4337, 41, 18, 42syl3anc 1184 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( ( abs `  A )  ^ c  B )  =  ( exp `  ( B  x.  ( log `  ( abs `  A ) ) ) ) )
4430, 35, 433eqtr4d 2477 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  ( A  ^ c  B ) )  =  ( ( abs `  A
)  ^ c  B
) )
4516, 44pm2.61dane 2676 1  |-  ( ( A  e.  CC  /\  B  e.  RR )  ->  ( abs `  ( A  ^ c  B ) )  =  ( ( abs `  A )  ^ c  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978   0cc0 8979    x. cmul 8984    <_ cle 9110   Recre 11890   abscabs 12027   expce 12652   logclog 20440    ^ c ccxp 20441
This theorem is referenced by:  root1cj  20628  rlimcxp  20800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ioc 10910  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-mod 11239  df-seq 11312  df-exp 11371  df-fac 11555  df-bc 11582  df-hash 11607  df-shft 11870  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-limsup 12253  df-clim 12270  df-rlim 12271  df-sum 12468  df-ef 12658  df-sin 12660  df-cos 12661  df-pi 12663  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742  df-log 20442  df-cxp 20443
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