MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eflt Structured version   Visualization version   Unicode version

Theorem eflt 14171
Description: The exponential function on the reals is strictly monotonic. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 17-Jul-2014.)
Assertion
Ref Expression
eflt  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )

Proof of Theorem eflt
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tru 1448 . 2  |- T.
2 fveq2 5865 . . 3  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
3 fveq2 5865 . . 3  |-  ( x  =  A  ->  ( exp `  x )  =  ( exp `  A
) )
4 fveq2 5865 . . 3  |-  ( x  =  B  ->  ( exp `  x )  =  ( exp `  B
) )
5 ssid 3451 . . 3  |-  RR  C_  RR
6 reefcl 14141 . . . 4  |-  ( x  e.  RR  ->  ( exp `  x )  e.  RR )
76adantl 468 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  ( exp `  x )  e.  RR )
8 simp2 1009 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  RR )
9 simp1 1008 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  RR )
108, 9resubcld 10047 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR )
11 posdif 10107 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  0  <  ( y  -  x ) ) )
1211biimp3a 1369 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( y  -  x
) )
1310, 12elrpd 11338 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR+ )
14 efgt1 14170 . . . . . . . 8  |-  ( ( y  -  x )  e.  RR+  ->  1  < 
( exp `  (
y  -  x ) ) )
1513, 14syl 17 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  1  <  ( exp `  (
y  -  x ) ) )
169reefcld 14142 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  e.  RR )
1710reefcld 14142 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( y  -  x ) )  e.  RR )
18 efgt0 14157 . . . . . . . . 9  |-  ( x  e.  RR  ->  0  <  ( exp `  x
) )
199, 18syl 17 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( exp `  x
) )
20 ltmulgt11 10465 . . . . . . . 8  |-  ( ( ( exp `  x
)  e.  RR  /\  ( exp `  ( y  -  x ) )  e.  RR  /\  0  <  ( exp `  x
) )  ->  (
1  <  ( exp `  ( y  -  x
) )  <->  ( exp `  x )  <  (
( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) ) )
2116, 17, 19, 20syl3anc 1268 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
1  <  ( exp `  ( y  -  x
) )  <->  ( exp `  x )  <  (
( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) ) )
2215, 21mpbid 214 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  < 
( ( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) )
239recnd 9669 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  CC )
2410recnd 9669 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  CC )
25 efadd 14148 . . . . . . . 8  |-  ( ( x  e.  CC  /\  ( y  -  x
)  e.  CC )  ->  ( exp `  (
x  +  ( y  -  x ) ) )  =  ( ( exp `  x )  x.  ( exp `  (
y  -  x ) ) ) )
2623, 24, 25syl2anc 667 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( ( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) )
278recnd 9669 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  CC )
2823, 27pncan3d 9989 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
x  +  ( y  -  x ) )  =  y )
2928fveq2d 5869 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( exp `  y
) )
3026, 29eqtr3d 2487 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
( exp `  x
)  x.  ( exp `  ( y  -  x
) ) )  =  ( exp `  y
) )
3122, 30breqtrd 4427 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  < 
( exp `  y
) )
32313expia 1210 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
3332adantl 468 . . 3  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
342, 3, 4, 5, 7, 33ltord1 10140 . 2  |-  ( ( T.  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )
351, 34mpan 676 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444   T. wtru 1445    e. wcel 1887   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    - cmin 9860   RR+crp 11302   expce 14114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121
This theorem is referenced by:  efle  14172  reefiso  23403  logdivlti  23569  divlogrlim  23580  cxplt  23639  birthday  23880  cxploglim  23903  emgt0  23932  bposlem6  24217  bposlem9  24220  pntpbnd1a  24423  pntibndlem2  24429  pntlemb  24435  ostth2lem3  24473  ostth2  24475
  Copyright terms: Public domain W3C validator