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Theorem eflt 13514
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 1374 . 2  |- T.
2 fveq2 5794 . . 3  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
3 fveq2 5794 . . 3  |-  ( x  =  A  ->  ( exp `  x )  =  ( exp `  A
) )
4 fveq2 5794 . . 3  |-  ( x  =  B  ->  ( exp `  x )  =  ( exp `  B
) )
5 ssid 3478 . . 3  |-  RR  C_  RR
6 reefcl 13485 . . . 4  |-  ( x  e.  RR  ->  ( exp `  x )  e.  RR )
76adantl 466 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  ( exp `  x )  e.  RR )
8 simp2 989 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  RR )
9 simp1 988 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  RR )
108, 9resubcld 9882 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR )
11 posdif 9938 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  0  <  ( y  -  x ) ) )
1211biimp3a 1319 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( y  -  x
) )
1310, 12elrpd 11131 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR+ )
14 efgt1 13513 . . . . . . . 8  |-  ( ( y  -  x )  e.  RR+  ->  1  < 
( exp `  (
y  -  x ) ) )
1513, 14syl 16 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  1  <  ( exp `  (
y  -  x ) ) )
169reefcld 13486 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  e.  RR )
1710reefcld 13486 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( y  -  x ) )  e.  RR )
18 efgt0 13500 . . . . . . . . 9  |-  ( x  e.  RR  ->  0  <  ( exp `  x
) )
199, 18syl 16 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( exp `  x
) )
20 ltmulgt11 10295 . . . . . . . 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 1219 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
1  <  ( exp `  ( y  -  x
) )  <->  ( exp `  x )  <  (
( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) ) )
2215, 21mpbid 210 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  < 
( ( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) )
239recnd 9518 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  CC )
2410recnd 9518 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  CC )
25 efadd 13492 . . . . . . . 8  |-  ( ( x  e.  CC  /\  ( y  -  x
)  e.  CC )  ->  ( exp `  (
x  +  ( y  -  x ) ) )  =  ( ( exp `  x )  x.  ( exp `  (
y  -  x ) ) ) )
2623, 24, 25syl2anc 661 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( ( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) )
278recnd 9518 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  CC )
2823, 27pncan3d 9828 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
x  +  ( y  -  x ) )  =  y )
2928fveq2d 5798 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( exp `  y
) )
3026, 29eqtr3d 2495 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
( exp `  x
)  x.  ( exp `  ( y  -  x
) ) )  =  ( exp `  y
) )
3122, 30breqtrd 4419 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  < 
( exp `  y
) )
32313expia 1190 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
3332adantl 466 . . 3  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
342, 3, 4, 5, 7, 33ltord1 9972 . 2  |-  ( ( T.  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )
351, 34mpan 670 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   T. wtru 1371    e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   CCcc 9386   RRcr 9387   0cc0 9388   1c1 9389    + caddc 9391    x. cmul 9393    < clt 9524    - cmin 9701   RR+crp 11097   expce 13460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-ico 11412  df-fz 11550  df-fzo 11661  df-fl 11754  df-seq 11919  df-exp 11978  df-fac 12164  df-bc 12191  df-hash 12216  df-shft 12669  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-limsup 13062  df-clim 13079  df-rlim 13080  df-sum 13277  df-ef 13466
This theorem is referenced by:  efle  13515  reefiso  22041  logdivlti  22197  divlogrlim  22208  cxplt  22267  birthday  22476  cxploglim  22499  emgt0  22528  bposlem6  22756  bposlem9  22759  pntpbnd1a  22962  pntibndlem2  22968  pntlemb  22974  ostth2lem3  23012  ostth2  23014
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