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Theorem rpexpmord 31049
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 6203 . . 3  |-  ( a  =  b  ->  (
a ^ N )  =  ( b ^ N ) )
2 oveq1 6203 . . 3  |-  ( a  =  A  ->  (
a ^ N )  =  ( A ^ N ) )
3 oveq1 6203 . . 3  |-  ( a  =  B  ->  (
a ^ N )  =  ( B ^ N ) )
4 rpssre 11149 . . 3  |-  RR+  C_  RR
5 rpre 11145 . . . 4  |-  ( a  e.  RR+  ->  a  e.  RR )
6 nnnn0 10719 . . . 4  |-  ( N  e.  NN  ->  N  e.  NN0 )
7 reexpcl 12086 . . . 4  |-  ( ( a  e.  RR  /\  N  e.  NN0 )  -> 
( a ^ N
)  e.  RR )
85, 6, 7syl2anr 476 . . 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 11177 . . . . 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 11177 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  b  e.  RR )
139rpge0d 11181 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  0  <_  a )
14 simpr 459 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  <  b )
15 simpll 751 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  N  e.  NN )
16 expmordi 31048 . . . . 5  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( 0  <_ 
a  /\  a  <  b )  /\  N  e.  NN )  ->  (
a ^ N )  <  ( b ^ N ) )
1710, 12, 13, 14, 15, 16syl221anc 1237 . . . 4  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  ( a ^ N )  <  (
b ^ N ) )
1817ex 432 . . 3  |-  ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  ->  ( a  <  b  ->  ( a ^ N )  <  (
b ^ N ) ) )
191, 2, 3, 4, 8, 18ltord1 9996 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  RR+  /\  B  e.  RR+ ) )  -> 
( A  <  B  <->  ( A ^ N )  <  ( B ^ N ) ) )
20193impb 1190 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 367    /\ w3a 971    e. wcel 1826   class class class wbr 4367  (class class class)co 6196   RRcr 9402   0cc0 9403    < clt 9539    <_ cle 9540   NNcn 10452   NN0cn0 10712   RR+crp 11139   ^cexp 12069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-seq 12011  df-exp 12070
This theorem is referenced by:  jm3.1lem1  31125
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