Theorem cxplt 13887

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)

Assertion

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

Proof of Theorem cxplt

Step	Hyp	Ref	Expression
1		simprl 529	. . . 4 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> B e. RR )
2		rplogcl 13851	. . . . . 6 |- ( ( A e. RR /\ 1 < A ) -> ( log ` A ) e. RR+ )
3	2	adantr 276	. . . . 5 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) e. RR+ )
4	3	rpred 9665	. . . 4 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) e. RR )
5	1, 4	remulcld 7962	. . 3 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B x. ( log ` A ) ) e. RR )
6		simprr 531	. . . 4 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> C e. RR )
7	6, 4	remulcld 7962	. . 3 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( C x. ( log ` A ) ) e. RR )
8		eflt 13747	. . 3 |- ( ( ( B x. ( log ` A ) ) e. RR /\ ( C x. ( log ` A ) ) e. RR ) -> ( ( B x. ( log ` A ) ) < ( C x. ( log ` A ) ) <-> ( exp ` ( B x. ( log ` A ) ) ) < ( exp ` ( C x. ( log ` A ) ) ) ) )
9	5, 7, 8	syl2anc 411	. 2 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( B x. ( log ` A ) ) < ( C x. ( log ` A ) ) <-> ( exp ` ( B x. ( log ` A ) ) ) < ( exp ` ( C x. ( log ` A ) ) ) ) )
10	1, 6, 3	ltmul1d 9707	. 2 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( B x. ( log ` A ) ) < ( C x. ( log ` A ) ) ) )
11		simpll 527	. . . . 5 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> A e. RR )
12		0red 7933	. . . . . 6 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 0 e. RR )
13		1red 7947	. . . . . 6 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 1 e. RR )
14		0lt1 8058	. . . . . . 7 |- 0 < 1
15	14	a1i 9	. . . . . 6 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 0 < 1 )
16		simplr 528	. . . . . 6 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 1 < A )
17	12, 13, 11, 15, 16	lttrd 8057	. . . . 5 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 0 < A )
18	11, 17	elrpd 9662	. . . 4 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> A e. RR+ )
19	1	recnd 7960	. . . 4 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> B e. CC )
20		rpcxpef 13866	. . . 4 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
21	18, 19, 20	syl2anc 411	. . 3 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
22	6	recnd 7960	. . . 4 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> C e. CC )
23		rpcxpef 13866	. . . 4 |- ( ( A e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
24	18, 22, 23	syl2anc 411	. . 3 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
25	21, 24	breq12d 4011	. 2 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( A ^c B ) < ( A ^c C ) <-> ( exp ` ( B x. ( log ` A ) ) ) < ( exp ` ( C x. ( log ` A ) ) ) ) )
26	9, 10, 25	3bitr4d 220	1 |- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c B ) < ( A ^c C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   CCcc 7784   RRcr 7785   0cc0 7786   1c1 7787   x. cmul 7791   < clt 7966   RR+crp 9622   expce 11616   logclog 13828   ^c ccxp 13829

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906  ax-pre-suploc 7907  ax-addf 7908  ax-mulf 7909

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-disj 3976  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-of 6073  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-frec 6382  df-1o 6407  df-oadd 6411  df-er 6525  df-map 6640  df-pm 6641  df-en 6731  df-dom 6732  df-fin 6733  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-xneg 9741  df-xadd 9742  df-ioo 9861  df-ico 9863  df-icc 9864  df-fz 9978  df-fzo 10111  df-seqfrec 10414  df-exp 10488  df-fac 10672  df-bc 10694  df-ihash 10722  df-shft 10790  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-clim 11253  df-sumdc 11328  df-ef 11622  df-e 11623  df-rest 12610  df-topgen 12629  df-psmet 13038  df-xmet 13039  df-met 13040  df-bl 13041  df-mopn 13042  df-top 13047  df-topon 13060  df-bases 13092  df-ntr 13147  df-cn 13239  df-cnp 13240  df-tx 13304  df-cncf 13609  df-limced 13676  df-dvap 13677  df-relog 13830  df-rpcxp 13831

This theorem is referenced by:  cxple  13888  cxplt3  13891  cxpltd  13899