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Theorem expmordi 27001
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 6081 . . . . . 6  |-  ( a  =  1  ->  ( A ^ a )  =  ( A ^ 1 ) )
2 oveq2 6081 . . . . . 6  |-  ( a  =  1  ->  ( B ^ a )  =  ( B ^ 1 ) )
31, 2breq12d 4217 . . . . 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 6081 . . . . . 6  |-  ( a  =  b  ->  ( A ^ a )  =  ( A ^ b
) )
6 oveq2 6081 . . . . . 6  |-  ( a  =  b  ->  ( B ^ a )  =  ( B ^ b
) )
75, 6breq12d 4217 . . . . 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 6081 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( A ^ a )  =  ( A ^ (
b  +  1 ) ) )
10 oveq2 6081 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( B ^ a )  =  ( B ^ (
b  +  1 ) ) )
119, 10breq12d 4217 . . . . 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 6081 . . . . . 6  |-  ( a  =  N  ->  ( A ^ a )  =  ( A ^ N
) )
14 oveq2 6081 . . . . . 6  |-  ( a  =  N  ->  ( B ^ a )  =  ( B ^ N
) )
1513, 14breq12d 4217 . . . . 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 9072 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
18 recn 9072 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
19 exp1 11379 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
20 exp1 11379 . . . . . . . 8  |-  ( B  e.  CC  ->  ( B ^ 1 )  =  B )
2119, 20breqan12d 4219 . . . . . . 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 10220 . . . . . . . . . . 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 11532 . . . . . . . . 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 11532 . . . . . . . . 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 11533 . . . . . . . . 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 9858 . . . . . . . 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 9106 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  A  e.  CC )
4140, 27expp1d 11516 . . . . . . 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 9106 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  B  e.  CC )
4342, 27expp1d 11516 . . . . . . 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 4234 . . . . . 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 10010 . . 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 1652    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113   NNcn 9992   NN0cn0 10213   ^cexp 11374
This theorem is referenced by:  rpexpmord  27002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316  df-exp 11375
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