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

Theorem expdiv 11357
Description: Nonnegative integer exponentiation of a quotient. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expdiv  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( A  /  B ) ^ N )  =  ( ( A ^ N
)  /  ( B ^ N ) ) )

Proof of Theorem expdiv
StepHypRef Expression
1 divrec 9626 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )
213expb 1154 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
323adant3 977 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
43oveq1d 6035 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( A  /  B ) ^ N )  =  ( ( A  x.  (
1  /  B ) ) ^ N ) )
5 reccl 9617 . . 3  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  CC )
6 mulexp 11346 . . 3  |-  ( ( A  e.  CC  /\  ( 1  /  B
)  e.  CC  /\  N  e.  NN0 )  -> 
( ( A  x.  ( 1  /  B
) ) ^ N
)  =  ( ( A ^ N )  x.  ( ( 1  /  B ) ^ N ) ) )
75, 6syl3an2 1218 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( A  x.  ( 1  /  B ) ) ^ N )  =  ( ( A ^ N
)  x.  ( ( 1  /  B ) ^ N ) ) )
8 simp2l 983 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  B  e.  CC )
9 simp2r 984 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  B  =/=  0
)
10 nn0z 10236 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  ZZ )
11103ad2ant3 980 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
12 exprec 11348 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  N  e.  ZZ )  ->  (
( 1  /  B
) ^ N )  =  ( 1  / 
( B ^ N
) ) )
138, 9, 11, 12syl3anc 1184 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( 1  /  B ) ^ N )  =  ( 1  /  ( B ^ N ) ) )
1413oveq2d 6036 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( A ^ N )  x.  ( ( 1  /  B ) ^ N
) )  =  ( ( A ^ N
)  x.  ( 1  /  ( B ^ N ) ) ) )
15 expcl 11326 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
16153adant2 976 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
17 expcl 11326 . . . . . 6  |-  ( ( B  e.  CC  /\  N  e.  NN0 )  -> 
( B ^ N
)  e.  CC )
1817adantlr 696 . . . . 5  |-  ( ( ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( B ^ N )  e.  CC )
19183adant1 975 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( B ^ N )  e.  CC )
20 expne0i 11339 . . . . 5  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  N  e.  ZZ )  ->  ( B ^ N )  =/=  0 )
218, 9, 11, 20syl3anc 1184 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( B ^ N )  =/=  0
)
2216, 19, 21divrecd 9725 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( A ^ N )  / 
( B ^ N
) )  =  ( ( A ^ N
)  x.  ( 1  /  ( B ^ N ) ) ) )
2314, 22eqtr4d 2422 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( A ^ N )  x.  ( ( 1  /  B ) ^ N
) )  =  ( ( A ^ N
)  /  ( B ^ N ) ) )
244, 7, 233eqtrd 2423 1  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( A  /  B ) ^ N )  =  ( ( A ^ N
)  /  ( B ^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550  (class class class)co 6020   CCcc 8921   0cc0 8923   1c1 8924    x. cmul 8928    / cdiv 9609   NN0cn0 10153   ZZcz 10214   ^cexp 11309
This theorem is referenced by:  expdivd  11464  stoweidlem7  27424  onetansqsecsq  27850  cotsqcscsq  27851
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-seq 11251  df-exp 11310
  Copyright terms: Public domain W3C validator