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Theorem expmordi 26703
Description: Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Assertion
Ref Expression
expmordi  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B
)  /\  N  e.  NN )  ->  ( A ^ N )  < 
( B ^ N
) )

Proof of Theorem expmordi
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6030 . . . . . 6  |-  ( a  =  1  ->  ( A ^ a )  =  ( A ^ 1 ) )
2 oveq2 6030 . . . . . 6  |-  ( a  =  1  ->  ( B ^ a )  =  ( B ^ 1 ) )
31, 2breq12d 4168 . . . . 5  |-  ( a  =  1  ->  (
( A ^ a
)  <  ( B ^ a )  <->  ( A ^ 1 )  < 
( B ^ 1 ) ) )
43imbi2d 308 . . . 4  |-  ( a  =  1  ->  (
( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^
a )  <  ( B ^ a ) )  <-> 
( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^
1 )  <  ( B ^ 1 ) ) ) )
5 oveq2 6030 . . . . . 6  |-  ( a  =  b  ->  ( A ^ a )  =  ( A ^ b
) )
6 oveq2 6030 . . . . . 6  |-  ( a  =  b  ->  ( B ^ a )  =  ( B ^ b
) )
75, 6breq12d 4168 . . . . 5  |-  ( a  =  b  ->  (
( A ^ a
)  <  ( B ^ a )  <->  ( A ^ b )  < 
( B ^ b
) ) )
87imbi2d 308 . . . 4  |-  ( a  =  b  ->  (
( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^
a )  <  ( B ^ a ) )  <-> 
( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^
b )  <  ( B ^ b ) ) ) )
9 oveq2 6030 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( A ^ a )  =  ( A ^ (
b  +  1 ) ) )
10 oveq2 6030 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( B ^ a )  =  ( B ^ (
b  +  1 ) ) )
119, 10breq12d 4168 . . . . 5  |-  ( a  =  ( b  +  1 )  ->  (
( A ^ a
)  <  ( B ^ a )  <->  ( A ^ ( b  +  1 ) )  < 
( B ^ (
b  +  1 ) ) ) )
1211imbi2d 308 . . . 4  |-  ( a  =  ( b  +  1 )  ->  (
( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^
a )  <  ( B ^ a ) )  <-> 
( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^
( b  +  1 ) )  <  ( B ^ ( b  +  1 ) ) ) ) )
13 oveq2 6030 . . . . . 6  |-  ( a  =  N  ->  ( A ^ a )  =  ( A ^ N
) )
14 oveq2 6030 . . . . . 6  |-  ( a  =  N  ->  ( B ^ a )  =  ( B ^ N
) )
1513, 14breq12d 4168 . . . . 5  |-  ( a  =  N  ->  (
( A ^ a
)  <  ( B ^ a )  <->  ( A ^ N )  <  ( B ^ N ) ) )
1615imbi2d 308 . . . 4  |-  ( a  =  N  ->  (
( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^
a )  <  ( B ^ a ) )  <-> 
( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^ N )  <  ( B ^ N ) ) ) )
17 recn 9015 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
18 recn 9015 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
19 exp1 11316 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
20 exp1 11316 . . . . . . . 8  |-  ( B  e.  CC  ->  ( B ^ 1 )  =  B )
2119, 20breqan12d 4170 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
1 )  <  ( B ^ 1 )  <->  A  <  B ) )
2217, 18, 21syl2an 464 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ^
1 )  <  ( B ^ 1 )  <->  A  <  B ) )
2322biimpar 472 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <  B
)  ->  ( A ^ 1 )  < 
( B ^ 1 ) )
2423adantrl 697 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B
) )  ->  ( A ^ 1 )  < 
( B ^ 1 ) )
25 simp2ll 1024 . . . . . . . . . 10  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  A  e.  RR )
26 nnnn0 10162 . . . . . . . . . . 11  |-  ( b  e.  NN  ->  b  e.  NN0 )
27263ad2ant1 978 . . . . . . . . . 10  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  b  e.  NN0 )
2825, 27reexpcld 11469 . . . . . . . . 9  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( A ^
b )  e.  RR )
29 simp2lr 1025 . . . . . . . . . 10  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  B  e.  RR )
3029, 27reexpcld 11469 . . . . . . . . 9  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( B ^
b )  e.  RR )
3128, 30jca 519 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( ( A ^ b )  e.  RR  /\  ( B ^ b )  e.  RR ) )
32 simp2rl 1026 . . . . . . . . . 10  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  0  <_  A
)
3325, 27, 32expge0d 11470 . . . . . . . . 9  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  0  <_  ( A ^ b ) )
34 simp3 959 . . . . . . . . 9  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( A ^
b )  <  ( B ^ b ) )
3533, 34jca 519 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( 0  <_ 
( A ^ b
)  /\  ( A ^ b )  < 
( B ^ b
) ) )
36 simp2l 983 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( A  e.  RR  /\  B  e.  RR ) )
37 simp2r 984 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( 0  <_  A  /\  A  <  B
) )
38 ltmul12a 9800 . . . . . . . 8  |-  ( ( ( ( ( A ^ b )  e.  RR  /\  ( B ^ b )  e.  RR )  /\  (
0  <_  ( A ^ b )  /\  ( A ^ b )  <  ( B ^
b ) ) )  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) ) )  ->  ( ( A ^ b )  x.  A )  <  (
( B ^ b
)  x.  B ) )
3931, 35, 36, 37, 38syl22anc 1185 . . . . . . 7  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( ( A ^ b )  x.  A )  <  (
( B ^ b
)  x.  B ) )
4025recnd 9049 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  A  e.  CC )
4140, 27expp1d 11453 . . . . . . 7  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( A ^
( b  +  1 ) )  =  ( ( A ^ b
)  x.  A ) )
4229recnd 9049 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  B  e.  CC )
4342, 27expp1d 11453 . . . . . . 7  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( B ^
( b  +  1 ) )  =  ( ( B ^ b
)  x.  B ) )
4439, 41, 433brtr4d 4185 . . . . . 6  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( A ^
( b  +  1 ) )  <  ( B ^ ( b  +  1 ) ) )
45443exp 1152 . . . . 5  |-  ( b  e.  NN  ->  (
( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( ( A ^ b )  < 
( B ^ b
)  ->  ( A ^ ( b  +  1 ) )  < 
( B ^ (
b  +  1 ) ) ) ) )
4645a2d 24 . . . 4  |-  ( b  e.  NN  ->  (
( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^
b )  <  ( B ^ b ) )  ->  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( A ^ ( b  +  1 ) )  < 
( B ^ (
b  +  1 ) ) ) ) )
474, 8, 12, 16, 24, 46nnind 9952 . . 3  |-  ( N  e.  NN  ->  (
( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^ N )  <  ( B ^ N ) ) )
4847impcom 420 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  N  e.  NN )  ->  ( A ^ N )  <  ( B ^ N ) )
49483impa 1148 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B
)  /\  N  e.  NN )  ->  ( A ^ N )  < 
( B ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4155  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930    < clt 9055    <_ cle 9056   NNcn 9934   NN0cn0 10155   ^cexp 11311
This theorem is referenced by:  rpexpmord  26704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-n0 10156  df-z 10217  df-uz 10423  df-seq 11253  df-exp 11312
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