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Theorem eflt 14067
Description: The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.)
Assertion
Ref Expression
eflt  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )

Proof of Theorem eflt
Dummy variables  d  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efltim 11699 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  ( exp `  A
)  <  ( exp `  B ) ) )
2 efcn 14060 . . . . 5  |-  exp  e.  ( CC -cn-> CC )
3 simplr 528 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  ->  B  e.  RR )
43recnd 7982 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  ->  B  e.  CC )
5 simpr 110 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  -> 
( exp `  A
)  <  ( exp `  B ) )
6 simpll 527 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  ->  A  e.  RR )
76reefcld 11670 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  -> 
( exp `  A
)  e.  RR )
83reefcld 11670 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  -> 
( exp `  B
)  e.  RR )
9 difrp 9688 . . . . . . 7  |-  ( ( ( exp `  A
)  e.  RR  /\  ( exp `  B )  e.  RR )  -> 
( ( exp `  A
)  <  ( exp `  B )  <->  ( ( exp `  B )  -  ( exp `  A ) )  e.  RR+ )
)
107, 8, 9syl2anc 411 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  -> 
( ( exp `  A
)  <  ( exp `  B )  <->  ( ( exp `  B )  -  ( exp `  A ) )  e.  RR+ )
)
115, 10mpbid 147 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  -> 
( ( exp `  B
)  -  ( exp `  A ) )  e.  RR+ )
12 cncfi 13936 . . . . 5  |-  ( ( exp  e.  ( CC
-cn-> CC )  /\  B  e.  CC  /\  ( ( exp `  B )  -  ( exp `  A
) )  e.  RR+ )  ->  E. d  e.  RR+  A. x  e.  CC  (
( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) )
132, 4, 11, 12mp3an2i 1342 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  ->  E. d  e.  RR+  A. x  e.  CC  ( ( abs `  ( x  -  B
) )  <  d  ->  ( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) )
146adantr 276 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A )  < 
( exp `  B
) )  /\  (
d  e.  RR+  /\  A. x  e.  CC  (
( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) ) )  ->  A  e.  RR )
153adantr 276 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A )  < 
( exp `  B
) )  /\  (
d  e.  RR+  /\  A. x  e.  CC  (
( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) ) )  ->  B  e.  RR )
16 simplr 528 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A )  < 
( exp `  B
) )  /\  (
d  e.  RR+  /\  A. x  e.  CC  (
( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) ) )  -> 
( exp `  A
)  <  ( exp `  B ) )
17 simprl 529 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A )  < 
( exp `  B
) )  /\  (
d  e.  RR+  /\  A. x  e.  CC  (
( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) ) )  -> 
d  e.  RR+ )
18 fvoveq1 5895 . . . . . . . 8  |-  ( x  =  A  ->  ( abs `  ( x  -  B ) )  =  ( abs `  ( A  -  B )
) )
1918breq1d 4012 . . . . . . 7  |-  ( x  =  A  ->  (
( abs `  (
x  -  B ) )  <  d  <->  ( abs `  ( A  -  B
) )  <  d
) )
2019imbrov2fvoveq 5897 . . . . . 6  |-  ( x  =  A  ->  (
( ( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) )  <->  ( ( abs `  ( A  -  B
) )  <  d  ->  ( abs `  (
( exp `  A
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) ) )
21 simprr 531 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A )  < 
( exp `  B
) )  /\  (
d  e.  RR+  /\  A. x  e.  CC  (
( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) ) )  ->  A. x  e.  CC  ( ( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) )
2214recnd 7982 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A )  < 
( exp `  B
) )  /\  (
d  e.  RR+  /\  A. x  e.  CC  (
( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) ) )  ->  A  e.  CC )
2320, 21, 22rspcdva 2846 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A )  < 
( exp `  B
) )  /\  (
d  e.  RR+  /\  A. x  e.  CC  (
( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) ) )  -> 
( ( abs `  ( A  -  B )
)  <  d  ->  ( abs `  ( ( exp `  A )  -  ( exp `  B
) ) )  < 
( ( exp `  B
)  -  ( exp `  A ) ) ) )
2414, 15, 16, 17, 23efltlemlt 14066 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A )  < 
( exp `  B
) )  /\  (
d  e.  RR+  /\  A. x  e.  CC  (
( abs `  (
x  -  B ) )  <  d  -> 
( abs `  (
( exp `  x
)  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) ) )  ->  A  <  B )
2513, 24rexlimddv 2599 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( exp `  A
)  <  ( exp `  B ) )  ->  A  <  B )
2625ex 115 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( exp `  A
)  <  ( exp `  B )  ->  A  <  B ) )
271, 26impbid 129 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( exp `  A )  <  ( exp `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4002   ` cfv 5215  (class class class)co 5872   CCcc 7806   RRcr 7807    < clt 7988    - cmin 8124   RR+crp 9649   abscabs 10999   expce 11643   -cn->ccncf 13928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-mulrcl 7907  ax-addcom 7908  ax-mulcom 7909  ax-addass 7910  ax-mulass 7911  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-1rid 7915  ax-0id 7916  ax-rnegex 7917  ax-precex 7918  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-apti 7923  ax-pre-ltadd 7924  ax-pre-mulgt0 7925  ax-pre-mulext 7926  ax-arch 7927  ax-caucvg 7928  ax-addf 7930  ax-mulf 7931
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-disj 3980  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-isom 5224  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-of 6080  df-1st 6138  df-2nd 6139  df-recs 6303  df-irdg 6368  df-frec 6389  df-1o 6414  df-oadd 6418  df-er 6532  df-map 6647  df-pm 6648  df-en 6738  df-dom 6739  df-fin 6740  df-sup 6980  df-inf 6981  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-reap 8528  df-ap 8535  df-div 8626  df-inn 8916  df-2 8974  df-3 8975  df-4 8976  df-n0 9173  df-z 9250  df-uz 9525  df-q 9616  df-rp 9650  df-xneg 9768  df-xadd 9769  df-ico 9890  df-fz 10005  df-fzo 10138  df-seqfrec 10441  df-exp 10515  df-fac 10699  df-bc 10721  df-ihash 10749  df-shft 10817  df-cj 10844  df-re 10845  df-im 10846  df-rsqrt 11000  df-abs 11001  df-clim 11280  df-sumdc 11355  df-ef 11649  df-rest 12678  df-topgen 12697  df-psmet 13316  df-xmet 13317  df-met 13318  df-bl 13319  df-mopn 13320  df-top 13367  df-topon 13380  df-bases 13412  df-ntr 13467  df-cn 13559  df-cnp 13560  df-tx 13624  df-cncf 13929  df-limced 13996  df-dvap 13997
This theorem is referenced by:  efle  14068  reefiso  14069  reapef  14070  logdivlti  14173  cxplt  14207  rpcxplt2  14210
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