Theorem cxplt3 15240

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 529	. . . . 5 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. RR )
3	2	recnd 8072	. . . 4 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. CC )
4		cxprec 15230	. . . 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 531	. . . . 5 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. RR )
7	6	recnd 8072	. . . 4 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. CC )
8		cxprec 15230	. . . 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 4047	. 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 9800	. . 3 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( 1 / A ) e. RR )
12		simplr 528	. . . 4 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> A < 1 )
13	1	reclt1d 9802	. . . 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 15236	. . 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 1250	. 2 |- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( ( 1 / A ) ^c B ) < ( ( 1 / A ) ^c C ) ) )
17		rpcxpcl 15223	. . . 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 15223	. . . 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 9807	. 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 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   RRcr 7895   1c1 7897   < clt 8078   / cdiv 8716   RR+crp 9745   ^c ccxp 15177

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016  ax-pre-suploc 8017  ax-addf 8018  ax-mulf 8019

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-disj 4012  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-map 6718  df-pm 6719  df-en 6809  df-dom 6810  df-fin 6811  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-xneg 9864  df-xadd 9865  df-ioo 9984  df-ico 9986  df-icc 9987  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-fac 10835  df-bc 10857  df-ihash 10885  df-shft 10997  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-sumdc 11536  df-ef 11830  df-e 11831  df-rest 12943  df-topgen 12962  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178  df-mopn 14179  df-top 14318  df-topon 14331  df-bases 14363  df-ntr 14416  df-cn 14508  df-cnp 14509  df-tx 14573  df-cncf 14891  df-limced 14976  df-dvap 14977  df-relog 15178  df-rpcxp 15179

This theorem is referenced by:  cxple3  15241  cxplt3d  15255