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Theorem cxplt 13037
Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
Assertion
Ref Expression
cxplt  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( B  <  C  <->  ( A  ^c  B )  <  ( A  ^c  C ) ) )

Proof of Theorem cxplt
StepHypRef Expression
1 simprl 521 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  B  e.  RR )
2 rplogcl 13001 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
( log `  A
)  e.  RR+ )
32adantr 274 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( log `  A
)  e.  RR+ )
43rpred 9511 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( log `  A
)  e.  RR )
51, 4remulcld 7818 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( B  x.  ( log `  A ) )  e.  RR )
6 simprr 522 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  C  e.  RR )
76, 4remulcld 7818 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( C  x.  ( log `  A ) )  e.  RR )
8 eflt 12897 . . 3  |-  ( ( ( B  x.  ( log `  A ) )  e.  RR  /\  ( C  x.  ( log `  A ) )  e.  RR )  ->  (
( B  x.  ( log `  A ) )  <  ( C  x.  ( log `  A ) )  <->  ( exp `  ( B  x.  ( log `  A ) ) )  <  ( exp `  ( C  x.  ( log `  A ) ) ) ) )
95, 7, 8syl2anc 409 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( ( B  x.  ( log `  A ) )  <  ( C  x.  ( log `  A
) )  <->  ( exp `  ( B  x.  ( log `  A ) ) )  <  ( exp `  ( C  x.  ( log `  A ) ) ) ) )
101, 6, 3ltmul1d 9553 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( B  <  C  <->  ( B  x.  ( log `  A ) )  < 
( C  x.  ( log `  A ) ) ) )
11 simpll 519 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  A  e.  RR )
12 0red 7789 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
0  e.  RR )
13 1red 7803 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
1  e.  RR )
14 0lt1 7911 . . . . . . 7  |-  0  <  1
1514a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
0  <  1 )
16 simplr 520 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
1  <  A )
1712, 13, 11, 15, 16lttrd 7910 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
0  <  A )
1811, 17elrpd 9508 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  A  e.  RR+ )
191recnd 7816 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  B  e.  CC )
20 rpcxpef 13016 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  CC )  ->  ( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A ) ) ) )
2118, 19, 20syl2anc 409 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( A  ^c  B )  =  ( exp `  ( B  x.  ( log `  A
) ) ) )
226recnd 7816 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  C  e.  CC )
23 rpcxpef 13016 . . . 4  |-  ( ( A  e.  RR+  /\  C  e.  CC )  ->  ( A  ^c  C )  =  ( exp `  ( C  x.  ( log `  A ) ) ) )
2418, 22, 23syl2anc 409 . . 3  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( A  ^c  C )  =  ( exp `  ( C  x.  ( log `  A
) ) ) )
2521, 24breq12d 3948 . 2  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( ( A  ^c  B )  <  ( A  ^c  C )  <-> 
( exp `  ( B  x.  ( log `  A ) ) )  <  ( exp `  ( C  x.  ( log `  A ) ) ) ) )
269, 10, 253bitr4d 219 1  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( B  e.  RR  /\  C  e.  RR ) )  -> 
( B  <  C  <->  ( A  ^c  B )  <  ( A  ^c  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   class class class wbr 3935   ` cfv 5129  (class class class)co 5780   CCcc 7640   RRcr 7641   0cc0 7642   1c1 7643    x. cmul 7647    < clt 7822   RR+crp 9468   expce 11378   logclog 12978    ^c ccxp 12979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-iinf 4508  ax-cnex 7733  ax-resscn 7734  ax-1cn 7735  ax-1re 7736  ax-icn 7737  ax-addcl 7738  ax-addrcl 7739  ax-mulcl 7740  ax-mulrcl 7741  ax-addcom 7742  ax-mulcom 7743  ax-addass 7744  ax-mulass 7745  ax-distr 7746  ax-i2m1 7747  ax-0lt1 7748  ax-1rid 7749  ax-0id 7750  ax-rnegex 7751  ax-precex 7752  ax-cnre 7753  ax-pre-ltirr 7754  ax-pre-ltwlin 7755  ax-pre-lttrn 7756  ax-pre-apti 7757  ax-pre-ltadd 7758  ax-pre-mulgt0 7759  ax-pre-mulext 7760  ax-arch 7761  ax-caucvg 7762  ax-pre-suploc 7763  ax-addf 7764  ax-mulf 7765
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-if 3478  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-disj 3913  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-id 4221  df-po 4224  df-iso 4225  df-iord 4294  df-on 4296  df-ilim 4297  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-isom 5138  df-riota 5736  df-ov 5783  df-oprab 5784  df-mpo 5785  df-of 5988  df-1st 6044  df-2nd 6045  df-recs 6208  df-irdg 6273  df-frec 6294  df-1o 6319  df-oadd 6323  df-er 6435  df-map 6550  df-pm 6551  df-en 6641  df-dom 6642  df-fin 6643  df-sup 6877  df-inf 6878  df-pnf 7824  df-mnf 7825  df-xr 7826  df-ltxr 7827  df-le 7828  df-sub 7957  df-neg 7958  df-reap 8359  df-ap 8366  df-div 8455  df-inn 8743  df-2 8801  df-3 8802  df-4 8803  df-n0 9000  df-z 9077  df-uz 9349  df-q 9437  df-rp 9469  df-xneg 9587  df-xadd 9588  df-ioo 9703  df-ico 9705  df-icc 9706  df-fz 9820  df-fzo 9949  df-seqfrec 10248  df-exp 10322  df-fac 10502  df-bc 10524  df-ihash 10552  df-shft 10617  df-cj 10644  df-re 10645  df-im 10646  df-rsqrt 10800  df-abs 10801  df-clim 11078  df-sumdc 11153  df-ef 11384  df-e 11385  df-rest 12154  df-topgen 12173  df-psmet 12188  df-xmet 12189  df-met 12190  df-bl 12191  df-mopn 12192  df-top 12197  df-topon 12210  df-bases 12242  df-ntr 12297  df-cn 12389  df-cnp 12390  df-tx 12454  df-cncf 12759  df-limced 12826  df-dvap 12827  df-relog 12980  df-rpcxp 12981
This theorem is referenced by:  cxple  13038  cxplt3  13041  cxpltd  13049
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