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Theorem eflt 13383
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 1366 . 2  |- T.
2 fveq2 5679 . . 3  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
3 fveq2 5679 . . 3  |-  ( x  =  A  ->  ( exp `  x )  =  ( exp `  A
) )
4 fveq2 5679 . . 3  |-  ( x  =  B  ->  ( exp `  x )  =  ( exp `  B
) )
5 ssid 3363 . . 3  |-  RR  C_  RR
6 reefcl 13354 . . . 4  |-  ( x  e.  RR  ->  ( exp `  x )  e.  RR )
76adantl 463 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  ( exp `  x )  e.  RR )
8 simp2 982 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  RR )
9 simp1 981 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  RR )
108, 9resubcld 9763 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR )
11 posdif 9819 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  0  <  ( y  -  x ) ) )
1211biimp3a 1311 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( y  -  x
) )
1310, 12elrpd 11012 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  RR+ )
14 efgt1 13382 . . . . . . . 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 13355 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  e.  RR )
1710reefcld 13355 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( y  -  x ) )  e.  RR )
18 efgt0 13369 . . . . . . . . 9  |-  ( x  e.  RR  ->  0  <  ( exp `  x
) )
199, 18syl 16 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  0  <  ( exp `  x
) )
20 ltmulgt11 10176 . . . . . . . 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 1211 . . . . . . 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 9399 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  x  e.  CC )
2410recnd 9399 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
y  -  x )  e.  CC )
25 efadd 13361 . . . . . . . 8  |-  ( ( x  e.  CC  /\  ( y  -  x
)  e.  CC )  ->  ( exp `  (
x  +  ( y  -  x ) ) )  =  ( ( exp `  x )  x.  ( exp `  (
y  -  x ) ) ) )
2623, 24, 25syl2anc 654 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( ( exp `  x
)  x.  ( exp `  ( y  -  x
) ) ) )
278recnd 9399 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  y  e.  CC )
2823, 27pncan3d 9709 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
x  +  ( y  -  x ) )  =  y )
2928fveq2d 5683 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  ( x  +  ( y  -  x
) ) )  =  ( exp `  y
) )
3026, 29eqtr3d 2467 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  (
( exp `  x
)  x.  ( exp `  ( y  -  x
) ) )  =  ( exp `  y
) )
3122, 30breqtrd 4304 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <  y )  ->  ( exp `  x )  < 
( exp `  y
) )
32313expia 1182 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
3332adantl 463 . . 3  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  <  y  ->  ( exp `  x
)  <  ( exp `  y ) ) )
342, 3, 4, 5, 7, 33ltord1 9853 . 2  |-  ( ( T.  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )
351, 34mpan 663 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 958    = wceq 1362   T. wtru 1363    e. wcel 1755   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270    + caddc 9272    x. cmul 9274    < clt 9405    - cmin 9582   RR+crp 10978   expce 13329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-n0 10567  df-z 10634  df-uz 10849  df-rp 10979  df-ico 11293  df-fz 11424  df-fzo 11532  df-fl 11625  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-sum 13147  df-ef 13335
This theorem is referenced by:  efle  13384  reefiso  21797  logdivlti  21953  divlogrlim  21964  cxplt  22023  birthday  22232  cxploglim  22255  emgt0  22284  bposlem6  22512  bposlem9  22515  pntpbnd1a  22718  pntibndlem2  22724  pntlemb  22730  ostth2lem3  22768  ostth2  22770
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