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Theorem eflt 12720
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 1331 . 2  |-  T.
2 fveq2 5730 . . 3  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
3 fveq2 5730 . . 3  |-  ( x  =  A  ->  ( exp `  x )  =  ( exp `  A
) )
4 fveq2 5730 . . 3  |-  ( x  =  B  ->  ( exp `  x )  =  ( exp `  B
) )
5 ssid 3369 . . 3  |-  RR  C_  RR
6 reefcl 12691 . . . 4  |-  ( x  e.  RR  ->  ( exp `  x )  e.  RR )
76adantl 454 . . 3  |-  ( (  T.  /\  x  e.  RR )  ->  ( exp `  x )  e.  RR )
8 simp2 959 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  RR )
9 simp1 958 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  RR )
108, 9resubcld 9467 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR )
11 posdif 9523 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  0  <  ( y  -  x ) ) )
1211biimp3a 1284 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( y  -  x
) )
1310, 12elrpd 10648 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR+ )
14 efgt1 12719 . . . . . . . 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 12692 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  e.  RR )
1710reefcld 12692 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( y  -  x ) )  e.  RR )
18 efgt0 12706 . . . . . . . . 9  |-  ( x  e.  RR  ->  0  <  ( exp `  x
) )
199, 18syl 16 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( exp `  x
) )
20 ltmulgt11 9872 . . . . . . . 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 1185 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
1  <  ( exp `  ( y  -  x
) )  <->  ( exp `  x )  <  (
( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) ) )
2215, 21mpbid 203 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  < 
( ( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) )
239recnd 9116 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  CC )
2410recnd 9116 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  CC )
25 efadd 12698 . . . . . . . 8  |-  ( ( x  e.  CC  /\  ( y  -  x
)  e.  CC )  ->  ( exp `  (
x  +  ( y  -  x ) ) )  =  ( ( exp `  x )  x.  ( exp `  (
y  -  x ) ) ) )
2623, 24, 25syl2anc 644 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( ( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) )
278recnd 9116 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  CC )
2823, 27pncan3d 9416 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
x  +  ( y  -  x ) )  =  y )
2928fveq2d 5734 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( exp `  y
) )
3026, 29eqtr3d 2472 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
( exp `  x
)  x.  ( exp `  ( y  -  x
) ) )  =  ( exp `  y
) )
3122, 30breqtrd 4238 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  < 
( exp `  y
) )
32313expia 1156 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
3332adantl 454 . . 3  |-  ( (  T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
342, 3, 4, 5, 7, 33ltord1 9555 . 2  |-  ( (  T.  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )
351, 34mpan 653 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    T. wtru 1326    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    < clt 9122    - cmin 9293   RR+crp 10614   expce 12666
This theorem is referenced by:  efle  12721  reefiso  20366  logdivlti  20517  divlogrlim  20528  cxplt  20587  birthday  20795  cxploglim  20818  emgt0  20847  bposlem6  21075  bposlem9  21078  pntpbnd1a  21281  pntibndlem2  21287  pntlemb  21293  ostth2lem3  21331  ostth2  21333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ico 10924  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672
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