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Theorem rpexpmord 26356
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 5952 . . 3  |-  ( a  =  b  ->  (
a ^ N )  =  ( b ^ N ) )
2 oveq1 5952 . . 3  |-  ( a  =  A  ->  (
a ^ N )  =  ( A ^ N ) )
3 oveq1 5952 . . 3  |-  ( a  =  B  ->  (
a ^ N )  =  ( B ^ N ) )
4 rpssre 10456 . . 3  |-  RR+  C_  RR
5 rpre 10452 . . . 4  |-  ( a  e.  RR+  ->  a  e.  RR )
6 nnnn0 10064 . . . 4  |-  ( N  e.  NN  ->  N  e.  NN0 )
7 reexpcl 11213 . . . 4  |-  ( ( a  e.  RR  /\  N  e.  NN0 )  -> 
( a ^ N
)  e.  RR )
85, 6, 7syl2anr 464 . . 3  |-  ( ( N  e.  NN  /\  a  e.  RR+ )  -> 
( a ^ N
)  e.  RR )
9 simplrl 736 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  e.  RR+ )
109rpred 10482 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  e.  RR )
11 simplrr 737 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  b  e.  RR+ )
1211rpred 10482 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  b  e.  RR )
139rpge0d 10486 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  0  <_  a )
14 simpr 447 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  a  <  b )
15 simpll 730 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  N  e.  NN )
16 expmordi 26355 . . . . 5  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( 0  <_ 
a  /\  a  <  b )  /\  N  e.  NN )  ->  (
a ^ N )  <  ( b ^ N ) )
1710, 12, 13, 14, 15, 16syl221anc 1193 . . . 4  |-  ( ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  /\  a  <  b )  ->  ( a ^ N )  <  (
b ^ N ) )
1817ex 423 . . 3  |-  ( ( N  e.  NN  /\  ( a  e.  RR+  /\  b  e.  RR+ )
)  ->  ( a  <  b  ->  ( a ^ N )  <  (
b ^ N ) ) )
191, 2, 3, 4, 8, 18ltord1 9389 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  RR+  /\  B  e.  RR+ ) )  -> 
( A  <  B  <->  ( A ^ N )  <  ( B ^ N ) ) )
20193impb 1147 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 176    /\ wa 358    /\ w3a 934    e. wcel 1710   class class class wbr 4104  (class class class)co 5945   RRcr 8826   0cc0 8827    < clt 8957    <_ cle 8958   NNcn 9836   NN0cn0 10057   RR+crp 10446   ^cexp 11197
This theorem is referenced by:  jm3.1lem1  26433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-seq 11139  df-exp 11198
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