Theorem logltb 15597

Description: The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Assertion

Ref	Expression
logltb	|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A < B <-> ( log ` A ) < ( log ` B ) ) )

Proof of Theorem logltb

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relogiso 15596	. . . . 5 |- ( log |` RR+ ) Isom < , < ( RR+ , RR )
2		df-isom 5335	. . . . 5 |- ( ( log |` RR+ ) Isom < , < ( RR+ , RR ) <-> ( ( log |` RR+ ) : RR+ -1-1-onto-> RR /\ A. x e. RR+ A. y e. RR+ ( x < y <-> ( ( log |` RR+ ) ` x ) < ( ( log |` RR+ ) ` y ) ) ) )
3	1, 2	mpbi 145	. . . 4 |- ( ( log |` RR+ ) : RR+ -1-1-onto-> RR /\ A. x e. RR+ A. y e. RR+ ( x < y <-> ( ( log |` RR+ ) ` x ) < ( ( log |` RR+ ) ` y ) ) )
4	3	simpri 113	. . 3 |- A. x e. RR+ A. y e. RR+ ( x < y <-> ( ( log |` RR+ ) ` x ) < ( ( log |` RR+ ) ` y ) )
5		breq1 4091	. . . . 5 |- ( x = A -> ( x < y <-> A < y ) )
6		fveq2 5639	. . . . . 6 |- ( x = A -> ( ( log |` RR+ ) ` x ) = ( ( log |` RR+ ) ` A ) )
7	6	breq1d 4098	. . . . 5 |- ( x = A -> ( ( ( log |` RR+ ) ` x ) < ( ( log |` RR+ ) ` y ) <-> ( ( log |` RR+ ) ` A ) < ( ( log |` RR+ ) ` y ) ) )
8	5, 7	bibi12d 235	. . . 4 |- ( x = A -> ( ( x < y <-> ( ( log |` RR+ ) ` x ) < ( ( log |` RR+ ) ` y ) ) <-> ( A < y <-> ( ( log |` RR+ ) ` A ) < ( ( log |` RR+ ) ` y ) ) ) )
9		breq2 4092	. . . . 5 |- ( y = B -> ( A < y <-> A < B ) )
10		fveq2 5639	. . . . . 6 |- ( y = B -> ( ( log |` RR+ ) ` y ) = ( ( log |` RR+ ) ` B ) )
11	10	breq2d 4100	. . . . 5 |- ( y = B -> ( ( ( log |` RR+ ) ` A ) < ( ( log |` RR+ ) ` y ) <-> ( ( log |` RR+ ) ` A ) < ( ( log |` RR+ ) ` B ) ) )
12	9, 11	bibi12d 235	. . . 4 |- ( y = B -> ( ( A < y <-> ( ( log |` RR+ ) ` A ) < ( ( log |` RR+ ) ` y ) ) <-> ( A < B <-> ( ( log |` RR+ ) ` A ) < ( ( log |` RR+ ) ` B ) ) ) )
13	8, 12	rspc2v 2923	. . 3 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A. x e. RR+ A. y e. RR+ ( x < y <-> ( ( log |` RR+ ) ` x ) < ( ( log |` RR+ ) ` y ) ) -> ( A < B <-> ( ( log |` RR+ ) ` A ) < ( ( log |` RR+ ) ` B ) ) ) )
14	4, 13	mpi 15	. 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A < B <-> ( ( log |` RR+ ) ` A ) < ( ( log |` RR+ ) ` B ) ) )
15		fvres 5663	. . 3 |- ( A e. RR+ -> ( ( log |` RR+ ) ` A ) = ( log ` A ) )
16		fvres 5663	. . 3 |- ( B e. RR+ -> ( ( log |` RR+ ) ` B ) = ( log ` B ) )
17	15, 16	breqan12d 4104	. 2 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( ( ( log |` RR+ ) ` A ) < ( ( log |` RR+ ) ` B ) <-> ( log ` A ) < ( log ` B ) ) )
18	14, 17	bitrd 188	1 |- ( ( A e. RR+ /\ B e. RR+ ) -> ( A < B <-> ( log ` A ) < ( log ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   class class class wbr 4088   |` cres 4727   -1-1-onto->wf1o 5325   ` cfv 5326   Isom wiso 5327   RRcr 8030   < clt 8213   RR+crp 9887   logclog 15579

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151  ax-pre-suploc 8152  ax-addf 8153  ax-mulf 8154

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-of 6234  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-map 6818  df-pm 6819  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-xneg 10006  df-xadd 10007  df-ioo 10126  df-ico 10128  df-icc 10129  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-fac 10987  df-bc 11009  df-ihash 11037  df-shft 11375  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914  df-ef 12208  df-e 12209  df-rest 13323  df-topgen 13342  df-psmet 14556  df-xmet 14557  df-met 14558  df-bl 14559  df-mopn 14560  df-top 14721  df-topon 14734  df-bases 14766  df-ntr 14819  df-cn 14911  df-cnp 14912  df-tx 14976  df-cncf 15294  df-limced 15379  df-dvap 15380  df-relog 15581

This theorem is referenced by:  logleb  15598  logrpap0b  15599  rplogcl  15602  loggt0b  15614  loglt1b  15616  logblt  15685