MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltexp2 Structured version   Unicode version

Theorem ltexp2 12193
Description: Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
Assertion
Ref Expression
ltexp2  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  < 
N  <->  ( A ^ M )  <  ( A ^ N ) ) )

Proof of Theorem ltexp2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6285 . . . . . 6  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
2 oveq2 6285 . . . . . 6  |-  ( x  =  M  ->  ( A ^ x )  =  ( A ^ M
) )
3 oveq2 6285 . . . . . 6  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
4 zssre 10872 . . . . . 6  |-  ZZ  C_  RR
5 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR )
6 0red 9595 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
7 1red 9609 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
8 0lt1 10076 . . . . . . . . . . 11  |-  0  <  1
98a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  1 )
10 simpr 461 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  <  A )
116, 7, 5, 9, 10lttrd 9741 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
125, 11elrpd 11258 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
13 rpexpcl 12159 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  RR+ )
1412, 13sylan 471 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR+ )
1514rpred 11260 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR )
16 simpll 753 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  A  e.  RR )
17 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
18 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
y  e.  ZZ )
19 simplr 754 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
1  <  A )
20 ltexp2a 12191 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  ( 1  <  A  /\  x  <  y ) )  ->  ( A ^ x )  < 
( A ^ y
) )
2120expr 615 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  1  <  A )  ->  ( x  < 
y  ->  ( A ^ x )  < 
( A ^ y
) ) )
2216, 17, 18, 19, 21syl31anc 1230 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  <  y  ->  ( A ^ x
)  <  ( A ^ y ) ) )
231, 2, 3, 4, 15, 22ltord1 10080 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) )
2423ancom2s 800 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( N  e.  ZZ  /\  M  e.  ZZ ) )  -> 
( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) )
2524exp43 612 . . 3  |-  ( A  e.  RR  ->  (
1  <  A  ->  ( N  e.  ZZ  ->  ( M  e.  ZZ  ->  ( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) ) ) ) )
2625com24 87 . 2  |-  ( A  e.  RR  ->  ( M  e.  ZZ  ->  ( N  e.  ZZ  ->  ( 1  <  A  -> 
( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) ) ) ) )
27263imp1 1208 1  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  < 
N  <->  ( A ^ M )  <  ( A ^ N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    e. wcel 1802   class class class wbr 4433  (class class class)co 6277   RRcr 9489   0cc0 9490   1c1 9491    < clt 9626   ZZcz 10865   RR+crp 11224   ^cexp 12140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-seq 12082  df-exp 12141
This theorem is referenced by:  leexp2  12194  ltexp2r  12196  ltexp2d  12313
  Copyright terms: Public domain W3C validator