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Theorem exprec 11349
Description: Nonnegative integer exponentiation of a reciprocal. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
exprec  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( 1  /  A
) ^ N )  =  ( 1  / 
( A ^ N
) ) )

Proof of Theorem exprec
StepHypRef Expression
1 reccl 9618 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
213adant3 977 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
1  /  A )  e.  CC )
3 recne0 9624 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =/=  0 )
433adant3 977 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
1  /  A )  =/=  0 )
5 simp3 959 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  N  e.  ZZ )
6 expclz 11334 . . . 4  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0  /\  N  e.  ZZ )  ->  ( ( 1  /  A ) ^ N )  e.  CC )
72, 4, 5, 6syl3anc 1184 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( 1  /  A
) ^ N )  e.  CC )
8 expclz 11334 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
9 expne0i 11340 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  =/=  0 )
107, 8, 9divcan3d 9728 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( ( A ^ N )  x.  (
( 1  /  A
) ^ N ) )  /  ( A ^ N ) )  =  ( ( 1  /  A ) ^ N ) )
11 simp1 957 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  A  e.  CC )
12 simp2 958 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  A  =/=  0 )
1311, 12recidd 9718 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A  x.  ( 1  /  A ) )  =  1 )
1413oveq1d 6036 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( A  x.  (
1  /  A ) ) ^ N )  =  ( 1 ^ N ) )
15 mulexpz 11348 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( ( 1  /  A )  e.  CC  /\  ( 1  /  A )  =/=  0 )  /\  N  e.  ZZ )  ->  (
( A  x.  (
1  /  A ) ) ^ N )  =  ( ( A ^ N )  x.  ( ( 1  /  A ) ^ N
) ) )
1611, 12, 2, 4, 5, 15syl221anc 1195 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( A  x.  (
1  /  A ) ) ^ N )  =  ( ( A ^ N )  x.  ( ( 1  /  A ) ^ N
) ) )
1714, 16eqtr3d 2422 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
1 ^ N )  =  ( ( A ^ N )  x.  ( ( 1  /  A ) ^ N
) ) )
18 1exp 11337 . . . . 5  |-  ( N  e.  ZZ  ->  (
1 ^ N )  =  1 )
195, 18syl 16 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
1 ^ N )  =  1 )
2017, 19eqtr3d 2422 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( A ^ N
)  x.  ( ( 1  /  A ) ^ N ) )  =  1 )
2120oveq1d 6036 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( ( A ^ N )  x.  (
( 1  /  A
) ^ N ) )  /  ( A ^ N ) )  =  ( 1  / 
( A ^ N
) ) )
2210, 21eqtr3d 2422 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( 1  /  A
) ^ N )  =  ( 1  / 
( A ^ N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551  (class class class)co 6021   CCcc 8922   0cc0 8924   1c1 8925    x. cmul 8929    / cdiv 9610   ZZcz 10215   ^cexp 11310
This theorem is referenced by:  expmulz  11354  expdiv  11358  ltexp2r  11364  sqrecd  11455  exprecd  11459  expcnv  12571  geo2lim  12580
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-seq 11252  df-exp 11311
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