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Theorem abscxp2 20589
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 9096 . . . . . 6  |-  0  e.  RR
21a1i 11 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  0  e.  RR )
3 0le0 10086 . . . . . 6  |-  0  <_  0
43a1i 11 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  0  <_  0 )
5 simplr 733 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  B  e.  RR )
6 recxpcl 20571 . . . . 5  |-  ( ( 0  e.  RR  /\  0  <_  0  /\  B  e.  RR )  ->  (
0  ^ c  B
)  e.  RR )
72, 4, 5, 6syl3anc 1185 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( 0  ^ c  B )  e.  RR )
8 cxpge0 20579 . . . . 5  |-  ( ( 0  e.  RR  /\  0  <_  0  /\  B  e.  RR )  ->  0  <_  ( 0  ^ c  B ) )
92, 4, 5, 8syl3anc 1185 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  0  <_  ( 0  ^ c  B
) )
107, 9absidd 12230 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( abs `  ( 0  ^ c  B ) )  =  ( 0  ^ c  B ) )
11 simpr 449 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  A  = 
0 )
1211oveq1d 6099 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( A  ^ c  B )  =  ( 0  ^ c  B ) )
1312fveq2d 5735 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( abs `  ( A  ^ c  B ) )  =  ( abs `  (
0  ^ c  B
) ) )
1411abs00bd 12101 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( abs `  A )  =  0 )
1514oveq1d 6099 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( ( abs `  A )  ^ c  B )  =  ( 0  ^ c  B
) )
1610, 13, 153eqtr4d 2480 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =  0 )  ->  ( abs `  ( A  ^ c  B ) )  =  ( ( abs `  A
)  ^ c  B
) )
17 simplr 733 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  B  e.  RR )
1817recnd 9119 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  B  e.  CC )
19 logcl 20471 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
2019adantlr 697 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( log `  A )  e.  CC )
2118, 20mulcld 9113 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( B  x.  ( log `  A
) )  e.  CC )
22 absef 12803 . . . . 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 12037 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( Re `  ( B  x.  ( log `  A ) ) )  =  ( B  x.  ( Re `  ( log `  A ) ) ) )
25 relog 20496 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  =  ( log `  ( abs `  A ) ) )
2625adantlr 697 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( Re `  ( log `  A
) )  =  ( log `  ( abs `  A ) ) )
2726oveq2d 6100 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( B  x.  ( Re `  ( log `  A ) ) )  =  ( B  x.  ( log `  ( abs `  A ) ) ) )
2824, 27eqtrd 2470 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( Re `  ( B  x.  ( log `  A ) ) )  =  ( B  x.  ( log `  ( abs `  A ) ) ) )
2928fveq2d 5735 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( exp `  ( Re `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( B  x.  ( log `  ( abs `  A
) ) ) ) )
3023, 29eqtrd 2470 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  ( exp `  ( B  x.  ( log `  A ) ) ) )  =  ( exp `  ( B  x.  ( log `  ( abs `  A
) ) ) ) )
31 simpll 732 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  A  e.  CC )
32 simpr 449 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  A  =/=  0 )
33 cxpef 20561 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  B  e.  CC )  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
3431, 32, 18, 33syl3anc 1185 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( A  ^ c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
3534fveq2d 5735 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  ( A  ^ c  B ) )  =  ( abs `  ( exp `  ( B  x.  ( log `  A ) ) ) ) )
3631abscld 12243 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  RR )
3736recnd 9119 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  A )  e.  CC )
38 abs00 12099 . . . . . . 7  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
3938adantr 453 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  RR )  ->  ( ( abs `  A
)  =  0  <->  A  =  0 ) )
4039necon3bid 2638 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  RR )  ->  ( ( abs `  A
)  =/=  0  <->  A  =/=  0 ) )
4140biimpar 473 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  A )  =/=  0
)
42 cxpef 20561 . . . 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 1185 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( ( abs `  A )  ^ c  B )  =  ( exp `  ( B  x.  ( log `  ( abs `  A ) ) ) ) )
4430, 35, 433eqtr4d 2480 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( abs `  ( A  ^ c  B ) )  =  ( ( abs `  A
)  ^ c  B
) )
4516, 44pm2.61dane 2684 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995    x. cmul 9000    <_ cle 9126   Recre 11907   abscabs 12044   expce 12669   logclog 20457    ^ c ccxp 20458
This theorem is referenced by:  root1cj  20645  rlimcxp  20817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678  df-pi 12680  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-log 20459  df-cxp 20460
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