Theorem rpcxplt2 15646

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)

Assertion

Ref	Expression
rpcxplt2	|- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A < B <-> ( A ^c C ) < ( B ^c C ) ) )

Proof of Theorem rpcxplt2

Step	Hyp	Ref	Expression
1		simp3 1025	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. RR+ )
2	1	rpred 9931	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. RR )
3		simp1 1023	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> A e. RR+ )
4	3	relogcld 15609	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( log ` A ) e. RR )
5	2, 4	remulcld 8210	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( C x. ( log ` A ) ) e. RR )
6		simp2 1024	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> B e. RR+ )
7	6	relogcld 15609	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( log ` B ) e. RR )
8	2, 7	remulcld 8210	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( C x. ( log ` B ) ) e. RR )
9		eflt 15502	. . 3 |- ( ( ( C x. ( log ` A ) ) e. RR /\ ( C x. ( log ` B ) ) e. RR ) -> ( ( C x. ( log ` A ) ) < ( C x. ( log ` B ) ) <-> ( exp ` ( C x. ( log ` A ) ) ) < ( exp ` ( C x. ( log ` B ) ) ) ) )
10	5, 8, 9	syl2anc 411	. 2 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( C x. ( log ` A ) ) < ( C x. ( log ` B ) ) <-> ( exp ` ( C x. ( log ` A ) ) ) < ( exp ` ( C x. ( log ` B ) ) ) ) )
11		eflt 15502	. . . 4 |- ( ( ( log ` A ) e. RR /\ ( log ` B ) e. RR ) -> ( ( log ` A ) < ( log ` B ) <-> ( exp ` ( log ` A ) ) < ( exp ` ( log ` B ) ) ) )
12	4, 7, 11	syl2anc 411	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( log ` A ) < ( log ` B ) <-> ( exp ` ( log ` A ) ) < ( exp ` ( log ` B ) ) ) )
13	4, 7, 1	ltmul2d 9974	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( log ` A ) < ( log ` B ) <-> ( C x. ( log ` A ) ) < ( C x. ( log ` B ) ) ) )
14	3	reeflogd 15610	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( exp ` ( log ` A ) ) = A )
15	6	reeflogd 15610	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( exp ` ( log ` B ) ) = B )
16	14, 15	breq12d 4101	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( exp ` ( log ` A ) ) < ( exp ` ( log ` B ) ) <-> A < B ) )
17	12, 13, 16	3bitr3rd 219	. 2 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A < B <-> ( C x. ( log ` A ) ) < ( C x. ( log ` B ) ) ) )
18	1	rpcnd 9933	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. CC )
19		rpcxpef 15621	. . . 4 |- ( ( A e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
20	3, 18, 19	syl2anc 411	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
21		rpcxpef 15621	. . . 4 |- ( ( B e. RR+ /\ C e. CC ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
22	6, 18, 21	syl2anc 411	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
23	20, 22	breq12d 4101	. 2 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( A ^c C ) < ( B ^c C ) <-> ( exp ` ( C x. ( log ` A ) ) ) < ( exp ` ( C x. ( log ` B ) ) ) ) )
24	10, 17, 23	3bitr4d 220	1 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A < B <-> ( A ^c C ) < ( B ^c C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   x. cmul 8037   < clt 8214   RR+crp 9888   expce 12205   logclog 15583   ^c ccxp 15584

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152  ax-pre-suploc 8153  ax-addf 8154  ax-mulf 8155

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-map 6819  df-pm 6820  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-ioo 10127  df-ico 10129  df-icc 10130  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-exp 10802  df-fac 10989  df-bc 11011  df-ihash 11039  df-shft 11377  df-cj 11404  df-re 11405  df-im 11406  df-rsqrt 11560  df-abs 11561  df-clim 11841  df-sumdc 11916  df-ef 12211  df-e 12212  df-rest 13326  df-topgen 13345  df-psmet 14560  df-xmet 14561  df-met 14562  df-bl 14563  df-mopn 14564  df-top 14725  df-topon 14738  df-bases 14770  df-ntr 14823  df-cn 14915  df-cnp 14916  df-tx 14980  df-cncf 15298  df-limced 15383  df-dvap 15384  df-relog 15585  df-rpcxp 15586

This theorem is referenced by: (None)