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Theorem expdiv 11422
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 9686 . . . . 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 6088 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( A  /  B ) ^ N )  =  ( ( A  x.  (
1  /  B ) ) ^ N ) )
5 reccl 9677 . . 3  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  CC )
6 mulexp 11411 . . 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 10296 . . . . . 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 11413 . . . . 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 6089 . . 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 11391 . . . . 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 11391 . . . . . 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 11404 . . . . 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 9785 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  NN0 )  ->  ( ( A ^ N )  / 
( B ^ N
) )  =  ( ( A ^ N
)  x.  ( 1  /  ( B ^ N ) ) ) )
2314, 22eqtr4d 2470 . 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 2471 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 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987    / cdiv 9669   NN0cn0 10213   ZZcz 10274   ^cexp 11374
This theorem is referenced by:  expdivd  11529  stoweidlem7  27713  onetansqsecsq  28431  cotsqcscsq  28432
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-rmo 2705  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-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316  df-exp 11375
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