Theorem cxplt3 15777

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)

Assertion

Ref	Expression
cxplt3	|- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c C ) < ( A ^c B ) ) )

Proof of Theorem cxplt3

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> A e. RR+ )
2		simprl 531	. . . . 5 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. RR )
3	2	recnd 8301	. . . 4 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. CC )
4		cxprec 15767	. . . 4 |- ( ( A e. RR+ /\ B e. CC ) -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
5	1, 3, 4	syl2anc 411	. . 3 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
6		simprr 533	. . . . 5 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. RR )
7	6	recnd 8301	. . . 4 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. CC )
8		cxprec 15767	. . . 4 |- ( ( A e. RR+ /\ C e. CC ) -> ( ( 1 / A ) ^c C ) = ( 1 / ( A ^c C ) ) )
9	1, 7, 8	syl2anc 411	. . 3 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( 1 / A ) ^c C ) = ( 1 / ( A ^c C ) ) )
10	5, 9	breq12d 4121	. 2 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( ( 1 / A ) ^c B ) < ( ( 1 / A ) ^c C ) <-> ( 1 / ( A ^c B ) ) < ( 1 / ( A ^c C ) ) ) )
11	1	rprecred 10040	. . 3 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( 1 / A ) e. RR )
12		simplr 529	. . . 4 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> A < 1 )
13	1	reclt1d 10042	. . . 4 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( A < 1 <-> 1 < ( 1 / A ) ) )
14	12, 13	mpbid 147	. . 3 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> 1 < ( 1 / A ) )
15		cxplt 15773	. . 3 |- ( ( ( ( 1 / A ) e. RR /\ 1 < ( 1 / A ) ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( ( 1 / A ) ^c B ) < ( ( 1 / A ) ^c C ) ) )
16	11, 14, 2, 6, 15	syl22anc 1275	. 2 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( ( 1 / A ) ^c B ) < ( ( 1 / A ) ^c C ) ) )
17		rpcxpcl 15760	. . . 4 |- ( ( A e. RR+ /\ C e. RR ) -> ( A ^c C ) e. RR+ )
18	17	ad2ant2rl 511	. . 3 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c C ) e. RR+ )
19		rpcxpcl 15760	. . . 4 |- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR+ )
20	19	ad2ant2r 509	. . 3 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c B ) e. RR+ )
21	18, 20	ltrecd 10047	. 2 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( A ^c C ) < ( A ^c B ) <-> ( 1 / ( A ^c B ) ) < ( 1 / ( A ^c C ) ) ) )
22	10, 16, 21	3bitr4d 220	1 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c C ) < ( A ^c B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   class class class wbr 4108  (class class class)co 6049   CCcc 8124   RRcr 8125   1c1 8127   < clt 8307   / cdiv 8945   RR+crp 9985   ^c ccxp 15714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246  ax-pre-suploc 8247  ax-addf 8248  ax-mulf 8249

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-disj 4085  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-of 6265  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-oadd 6650  df-er 6766  df-map 6883  df-pm 6884  df-en 6975  df-dom 6976  df-fin 6977  df-sup 7274  df-inf 7275  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-xneg 10104  df-xadd 10105  df-ioo 10224  df-ico 10226  df-icc 10227  df-fz 10342  df-fzo 10476  df-seqfrec 10809  df-exp 10900  df-fac 11087  df-bc 11109  df-ihash 11137  df-shft 11496  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-clim 11960  df-sumdc 12035  df-ef 12330  df-e 12331  df-rest 13446  df-topgen 13465  df-psmet 14683  df-xmet 14684  df-met 14685  df-bl 14686  df-mopn 14687  df-top 14855  df-topon 14868  df-bases 14900  df-ntr 14953  df-cn 15045  df-cnp 15046  df-tx 15110  df-cncf 15428  df-limced 15513  df-dvap 15514  df-relog 15715  df-rpcxp 15716

This theorem is referenced by:  cxple3  15778  cxplt3d  15792