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Theorem rpexpmord 30818
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 6302 . . 3  |-  ( a  =  b  ->  (
a ^ N )  =  ( b ^ N ) )
2 oveq1 6302 . . 3  |-  ( a  =  A  ->  (
a ^ N )  =  ( A ^ N ) )
3 oveq1 6302 . . 3  |-  ( a  =  B  ->  (
a ^ N )  =  ( B ^ N ) )
4 rpssre 11242 . . 3  |-  RR+  C_  RR
5 rpre 11238 . . . 4  |-  ( a  e.  RR+  ->  a  e.  RR )
6 nnnn0 10814 . . . 4  |-  ( N  e.  NN  ->  N  e.  NN0 )
7 reexpcl 12163 . . . 4  |-  ( ( a  e.  RR  /\  N  e.  NN0 )  -> 
( a ^ N
)  e.  RR )
85, 6, 7syl2anr 478 . . 3  |-  ( ( N  e.  NN  /\  a  e.  RR+ )  -> 
( a ^ N
)  e.  RR )
9 simplrl 759 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  e.  RR+ )
109rpred 11268 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  e.  RR )
11 simplrr 760 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  b  e.  RR+ )
1211rpred 11268 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  b  e.  RR )
139rpge0d 11272 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  0  <_  a )
14 simpr 461 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  <  b )
15 simpll 753 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  N  e.  NN )
16 expmordi 30817 . . . . 5  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( 0  <_ 
a  /\  a  <  b )  /\  N  e.  NN )  ->  (
a ^ N )  <  ( b ^ N ) )
1710, 12, 13, 14, 15, 16syl221anc 1239 . . . 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 10091 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  RR+  /\  B  e.  RR+ ) )  -> 
( A  <  B  <->  ( A ^ N )  <  ( B ^ N ) ) )
20193impb 1192 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 973    e. wcel 1767   class class class wbr 4453  (class class class)co 6295   RRcr 9503   0cc0 9504    < clt 9640    <_ cle 9641   NNcn 10548   NN0cn0 10807   RR+crp 11232   ^cexp 12146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-seq 12088  df-exp 12147
This theorem is referenced by:  jm3.1lem1  30893
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