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Theorem rpexpmord 27024
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 6091 . . 3  |-  ( a  =  b  ->  (
a ^ N )  =  ( b ^ N ) )
2 oveq1 6091 . . 3  |-  ( a  =  A  ->  (
a ^ N )  =  ( A ^ N ) )
3 oveq1 6091 . . 3  |-  ( a  =  B  ->  (
a ^ N )  =  ( B ^ N ) )
4 rpssre 10627 . . 3  |-  RR+  C_  RR
5 rpre 10623 . . . 4  |-  ( a  e.  RR+  ->  a  e.  RR )
6 nnnn0 10233 . . . 4  |-  ( N  e.  NN  ->  N  e.  NN0 )
7 reexpcl 11403 . . . 4  |-  ( ( a  e.  RR  /\  N  e.  NN0 )  -> 
( a ^ N
)  e.  RR )
85, 6, 7syl2anr 466 . . 3  |-  ( ( N  e.  NN  /\  a  e.  RR+ )  -> 
( a ^ N
)  e.  RR )
9 simplrl 738 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  e.  RR+ )
109rpred 10653 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  e.  RR )
11 simplrr 739 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  b  e.  RR+ )
1211rpred 10653 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  b  e.  RR )
139rpge0d 10657 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  0  <_  a )
14 simpr 449 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  <  b )
15 simpll 732 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  N  e.  NN )
16 expmordi 27023 . . . . 5  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( 0  <_ 
a  /\  a  <  b )  /\  N  e.  NN )  ->  (
a ^ N )  <  ( b ^ N ) )
1710, 12, 13, 14, 15, 16syl221anc 1196 . . . 4  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  ( a ^ N )  <  (
b ^ N ) )
1817ex 425 . . 3  |-  ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  ->  ( a  <  b  ->  ( a ^ N )  <  (
b ^ N ) ) )
191, 2, 3, 4, 8, 18ltord1 9558 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  RR+  /\  B  e.  RR+ ) )  -> 
( A  <  B  <->  ( A ^ N )  <  ( B ^ N ) ) )
20193impb 1150 1  |-  ( ( N  e.  NN  /\  A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <  B  <->  ( A ^ N )  <  ( B ^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726   class class class wbr 4215  (class class class)co 6084   RRcr 8994   0cc0 8995    < clt 9125    <_ cle 9126   NNcn 10005   NN0cn0 10226   RR+crp 10617   ^cexp 11387
This theorem is referenced by:  jm3.1lem1  27101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 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-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-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-seq 11329  df-exp 11388
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