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Theorem rpexpmord 29214
Description: Mantissa ordering relationship for exponentiation of positive reals. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Assertion
Ref Expression
rpexpmord  |-  ( ( N  e.  NN  /\  A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <  B  <->  ( A ^ N )  <  ( B ^ N ) ) )

Proof of Theorem rpexpmord
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6097 . . 3  |-  ( a  =  b  ->  (
a ^ N )  =  ( b ^ N ) )
2 oveq1 6097 . . 3  |-  ( a  =  A  ->  (
a ^ N )  =  ( A ^ N ) )
3 oveq1 6097 . . 3  |-  ( a  =  B  ->  (
a ^ N )  =  ( B ^ N ) )
4 rpssre 10997 . . 3  |-  RR+  C_  RR
5 rpre 10993 . . . 4  |-  ( a  e.  RR+  ->  a  e.  RR )
6 nnnn0 10582 . . . 4  |-  ( N  e.  NN  ->  N  e.  NN0 )
7 reexpcl 11878 . . . 4  |-  ( ( a  e.  RR  /\  N  e.  NN0 )  -> 
( a ^ N
)  e.  RR )
85, 6, 7syl2anr 475 . . 3  |-  ( ( N  e.  NN  /\  a  e.  RR+ )  -> 
( a ^ N
)  e.  RR )
9 simplrl 754 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  e.  RR+ )
109rpred 11023 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  e.  RR )
11 simplrr 755 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  b  e.  RR+ )
1211rpred 11023 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  b  e.  RR )
139rpge0d 11027 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  0  <_  a )
14 simpr 458 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  <  b )
15 simpll 748 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  N  e.  NN )
16 expmordi 29213 . . . . 5  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( 0  <_ 
a  /\  a  <  b )  /\  N  e.  NN )  ->  (
a ^ N )  <  ( b ^ N ) )
1710, 12, 13, 14, 15, 16syl221anc 1224 . . . 4  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  ( a ^ N )  <  (
b ^ N ) )
1817ex 434 . . 3  |-  ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  ->  ( a  <  b  ->  ( a ^ N )  <  (
b ^ N ) ) )
191, 2, 3, 4, 8, 18ltord1 9862 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  RR+  /\  B  e.  RR+ ) )  -> 
( A  <  B  <->  ( A ^ N )  <  ( B ^ N ) ) )
20193impb 1178 1  |-  ( ( N  e.  NN  /\  A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <  B  <->  ( A ^ N )  <  ( B ^ N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    e. wcel 1761   class class class wbr 4289  (class class class)co 6090   RRcr 9277   0cc0 9278    < clt 9414    <_ cle 9415   NNcn 10318   NN0cn0 10575   RR+crp 10987   ^cexp 11861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-seq 11803  df-exp 11862
This theorem is referenced by:  jm3.1lem1  29291
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