Theorem rpcxple2 15943

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)

Assertion

Ref	Expression
rpcxple2	|- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )

Proof of Theorem rpcxple2

Step	Hyp	Ref	Expression
1		simp3 1030	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. RR+ )
2	1	rpred 10076	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. RR )
3		simp1 1028	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> A e. RR+ )
4	3	relogcld 15906	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( log ` A ) e. RR )
5	2, 4	remulcld 8346	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( C x. ( log ` A ) ) e. RR )
6		simp2 1029	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> B e. RR+ )
7	6	relogcld 15906	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( log ` B ) e. RR )
8	2, 7	remulcld 8346	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( C x. ( log ` B ) ) e. RR )
9		efle 15800	. . 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 415	. 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		efle 15800	. . . 4 |- ( ( ( log ` A ) e. RR /\ ( log ` B ) e. RR ) -> ( ( log ` A ) <_ ( log ` B ) <-> ( exp ` ( log ` A ) ) <_ ( exp ` ( log ` B ) ) ) )
12	4, 7, 11	syl2anc 415	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( log ` A ) <_ ( log ` B ) <-> ( exp ` ( log ` A ) ) <_ ( exp ` ( log ` B ) ) ) )
13	4, 7, 1	lemul2d 10121	. . 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 15907	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( exp ` ( log ` A ) ) = A )
15	6	reeflogd 15907	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( exp ` ( log ` B ) ) = B )
16	14, 15	breq12d 4138	. . 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 10078	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. CC )
19		rpcxpef 15919	. . . 4 |- ( ( A e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
20	3, 18, 19	syl2anc 415	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
21		rpcxpef 15919	. . . 4 |- ( ( B e. RR+ /\ C e. CC ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
22	6, 18, 21	syl2anc 415	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
23	20, 22	breq12d 4138	. 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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   x. cmul 8174   <_ cle 8351   RR+crp 10033   expce 12387   logclog 15880   ^c ccxp 15881

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289  ax-pre-suploc 8290  ax-addf 8291  ax-mulf 8292

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-disj 4102  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-map 6914  df-pm 6915  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-xneg 10153  df-xadd 10154  df-ioo 10273  df-ico 10275  df-icc 10276  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-fac 11142  df-bc 11164  df-ihash 11193  df-shft 11558  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098  df-ef 12393  df-e 12394  df-rest 13572  df-topgen 13591  df-psmet 14852  df-xmet 14853  df-met 14854  df-bl 14855  df-mopn 14856  df-top 15022  df-topon 15035  df-bases 15067  df-ntr 15120  df-cn 15212  df-cnp 15213  df-tx 15277  df-cncf 15595  df-limced 15680  df-dvap 15681  df-relog 15882  df-rpcxp 15883

This theorem is referenced by:  rpabscxpbnd  15965