Theorem rpcxplt2 13380

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 988	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. RR+ )
2	1	rpred 9623	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. RR )
3		simp1 986	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> A e. RR+ )
4	3	relogcld 13344	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( log ` A ) e. RR )
5	2, 4	remulcld 7920	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( C x. ( log ` A ) ) e. RR )
6		simp2 987	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> B e. RR+ )
7	6	relogcld 13344	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( log ` B ) e. RR )
8	2, 7	remulcld 7920	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( C x. ( log ` B ) ) e. RR )
9		eflt 13237	. . 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 409	. 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 13237	. . . 4 |- ( ( ( log ` A ) e. RR /\ ( log ` B ) e. RR ) -> ( ( log ` A ) < ( log ` B ) <-> ( exp ` ( log ` A ) ) < ( exp ` ( log ` B ) ) ) )
12	4, 7, 11	syl2anc 409	. . 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 9666	. . 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 13345	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( exp ` ( log ` A ) ) = A )
15	6	reeflogd 13345	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( exp ` ( log ` B ) ) = B )
16	14, 15	breq12d 3989	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( exp ` ( log ` A ) ) < ( exp ` ( log ` B ) ) <-> A < B ) )
17	12, 13, 16	3bitr3rd 218	. 2 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A < B <-> ( C x. ( log ` A ) ) < ( C x. ( log ` B ) ) ) )
18	1	rpcnd 9625	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. CC )
19		rpcxpef 13356	. . . 4 |- ( ( A e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
20	3, 18, 19	syl2anc 409	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
21		rpcxpef 13356	. . . 4 |- ( ( B e. RR+ /\ C e. CC ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
22	6, 18, 21	syl2anc 409	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
23	20, 22	breq12d 3989	. 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 219	1 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A < B <-> ( A ^c C ) < ( B ^c C ) ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 967   = wceq 1342   e. wcel 2135   class class class wbr 3976   ` cfv 5182  (class class class)co 5836   CCcc 7742   RRcr 7743   x. cmul 7749   < clt 7924   RR+crp 9580   expce 11569   logclog 13318   ^c ccxp 13319

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864  ax-pre-suploc 7865  ax-addf 7866  ax-mulf 7867

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-disj 3954  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-isom 5191  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-of 6044  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-frec 6350  df-1o 6375  df-oadd 6379  df-er 6492  df-map 6607  df-pm 6608  df-en 6698  df-dom 6699  df-fin 6700  df-sup 6940  df-inf 6941  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-q 9549  df-rp 9581  df-xneg 9699  df-xadd 9700  df-ioo 9819  df-ico 9821  df-icc 9822  df-fz 9936  df-fzo 10068  df-seqfrec 10371  df-exp 10445  df-fac 10628  df-bc 10650  df-ihash 10678  df-shft 10743  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927  df-clim 11206  df-sumdc 11281  df-ef 11575  df-e 11576  df-rest 12494  df-topgen 12513  df-psmet 12528  df-xmet 12529  df-met 12530  df-bl 12531  df-mopn 12532  df-top 12537  df-topon 12550  df-bases 12582  df-ntr 12637  df-cn 12729  df-cnp 12730  df-tx 12794  df-cncf 13099  df-limced 13166  df-dvap 13167  df-relog 13320  df-rpcxp 13321

This theorem is referenced by: (None)