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Theorem eflt 13953
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 1409 . 2  |- T.
2 fveq2 5805 . . 3  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
3 fveq2 5805 . . 3  |-  ( x  =  A  ->  ( exp `  x )  =  ( exp `  A
) )
4 fveq2 5805 . . 3  |-  ( x  =  B  ->  ( exp `  x )  =  ( exp `  B
) )
5 ssid 3460 . . 3  |-  RR  C_  RR
6 reefcl 13923 . . . 4  |-  ( x  e.  RR  ->  ( exp `  x )  e.  RR )
76adantl 464 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  ( exp `  x )  e.  RR )
8 simp2 998 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  RR )
9 simp1 997 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  RR )
108, 9resubcld 9948 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR )
11 posdif 10006 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  0  <  ( y  -  x ) ) )
1211biimp3a 1330 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( y  -  x
) )
1310, 12elrpd 11219 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR+ )
14 efgt1 13952 . . . . . . . 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 13924 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  e.  RR )
1710reefcld 13924 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( y  -  x ) )  e.  RR )
18 efgt0 13939 . . . . . . . . 9  |-  ( x  e.  RR  ->  0  <  ( exp `  x
) )
199, 18syl 17 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( exp `  x
) )
20 ltmulgt11 10363 . . . . . . . 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 1230 . . . . . . 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 9572 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  CC )
2410recnd 9572 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  CC )
25 efadd 13930 . . . . . . . 8  |-  ( ( x  e.  CC  /\  ( y  -  x
)  e.  CC )  ->  ( exp `  (
x  +  ( y  -  x ) ) )  =  ( ( exp `  x )  x.  ( exp `  (
y  -  x ) ) ) )
2623, 24, 25syl2anc 659 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( ( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) )
278recnd 9572 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  CC )
2823, 27pncan3d 9890 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
x  +  ( y  -  x ) )  =  y )
2928fveq2d 5809 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( exp `  y
) )
3026, 29eqtr3d 2445 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
( exp `  x
)  x.  ( exp `  ( y  -  x
) ) )  =  ( exp `  y
) )
3122, 30breqtrd 4418 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  < 
( exp `  y
) )
32313expia 1199 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
3332adantl 464 . . 3  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
342, 3, 4, 5, 7, 33ltord1 10039 . 2  |-  ( ( T.  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )
351, 34mpan 668 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 367    /\ w3a 974    = wceq 1405   T. wtru 1406    e. wcel 1842   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   CCcc 9440   RRcr 9441   0cc0 9442   1c1 9443    + caddc 9445    x. cmul 9447    < clt 9578    - cmin 9761   RR+crp 11183   expce 13898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520  ax-addf 9521  ax-mulf 9522
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-ico 11506  df-fz 11644  df-fzo 11768  df-fl 11879  df-seq 12062  df-exp 12121  df-fac 12308  df-bc 12335  df-hash 12360  df-shft 12956  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-limsup 13350  df-clim 13367  df-rlim 13368  df-sum 13565  df-ef 13904
This theorem is referenced by:  efle  13954  reefiso  23027  logdivlti  23191  divlogrlim  23202  cxplt  23261  birthday  23502  cxploglim  23525  emgt0  23554  bposlem6  23837  bposlem9  23840  pntpbnd1a  24043  pntibndlem2  24049  pntlemb  24055  ostth2lem3  24093  ostth2  24095
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