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Theorem mulexpz 11136
Description: Integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
mulexpz  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  ZZ )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )

Proof of Theorem mulexpz
StepHypRef Expression
1 elznn0nn 10032 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 simpl 445 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  e.  CC )
3 simpl 445 . . . . . 6  |-  ( ( B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
42, 3anim12i 551 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  e.  CC  /\  B  e.  CC ) )
5 mulexp 11135 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
653expa 1156 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  N  e.  NN0 )  ->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N
)  x.  ( B ^ N ) ) )
74, 6sylan 459 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  N  e.  NN0 )  -> 
( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
8 simplll 737 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
9 simplrl 739 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  B  e.  CC )
108, 9mulcld 8850 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A  x.  B )  e.  CC )
11 recn 8822 . . . . . . 7  |-  ( N  e.  RR  ->  N  e.  CC )
1211ad2antrl 711 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
13 nnnn0 9967 . . . . . . 7  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
1413ad2antll 712 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
15 expneg2 11106 . . . . . 6  |-  ( ( ( A  x.  B
)  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  (
( A  x.  B
) ^ N )  =  ( 1  / 
( ( A  x.  B ) ^ -u N
) ) )
1610, 12, 14, 15syl3anc 1187 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( A  x.  B
) ^ N )  =  ( 1  / 
( ( A  x.  B ) ^ -u N
) ) )
17 expneg2 11106 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
188, 12, 14, 17syl3anc 1187 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
19 expneg2 11106 . . . . . . . 8  |-  ( ( B  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( B ^ N )  =  ( 1  /  ( B ^ -u N ) ) )
209, 12, 14, 19syl3anc 1187 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( B ^ N )  =  ( 1  /  ( B ^ -u N ) ) )
2118, 20oveq12d 5837 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( A ^ N
)  x.  ( B ^ N ) )  =  ( ( 1  /  ( A ^ -u N ) )  x.  ( 1  /  ( B ^ -u N ) ) ) )
22 mulexp 11135 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  -u N  e.  NN0 )  ->  (
( A  x.  B
) ^ -u N
)  =  ( ( A ^ -u N
)  x.  ( B ^ -u N ) ) )
238, 9, 14, 22syl3anc 1187 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( A  x.  B
) ^ -u N
)  =  ( ( A ^ -u N
)  x.  ( B ^ -u N ) ) )
2423oveq2d 5835 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
1  /  ( ( A  x.  B ) ^ -u N ) )  =  ( 1  /  ( ( A ^ -u N )  x.  ( B ^ -u N ) ) ) )
25 1t1e1 9865 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
2625oveq1i 5829 . . . . . . . 8  |-  ( ( 1  x.  1 )  /  ( ( A ^ -u N )  x.  ( B ^ -u N ) ) )  =  ( 1  / 
( ( A ^ -u N )  x.  ( B ^ -u N ) ) )
2724, 26syl6eqr 2334 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
1  /  ( ( A  x.  B ) ^ -u N ) )  =  ( ( 1  x.  1 )  /  ( ( A ^ -u N )  x.  ( B ^ -u N ) ) ) )
28 expcl 11115 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ -u N
)  e.  CC )
298, 14, 28syl2anc 645 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ -u N )  e.  CC )
30 simpllr 738 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  =/=  0 )
31 nnz 10040 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
3231ad2antll 712 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
33 expne0i 11128 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u N  e.  ZZ )  ->  ( A ^ -u N )  =/=  0 )
348, 30, 32, 33syl3anc 1187 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ -u N )  =/=  0 )
35 expcl 11115 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  -u N  e.  NN0 )  ->  ( B ^ -u N
)  e.  CC )
369, 14, 35syl2anc 645 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( B ^ -u N )  e.  CC )
37 simplrr 740 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  B  =/=  0 )
38 expne0i 11128 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  -u N  e.  ZZ )  ->  ( B ^ -u N )  =/=  0 )
399, 37, 32, 38syl3anc 1187 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( B ^ -u N )  =/=  0 )
40 ax-1cn 8790 . . . . . . . . 9  |-  1  e.  CC
41 divmuldiv 9455 . . . . . . . . 9  |-  ( ( ( 1  e.  CC  /\  1  e.  CC )  /\  ( ( ( A ^ -u N
)  e.  CC  /\  ( A ^ -u N
)  =/=  0 )  /\  ( ( B ^ -u N )  e.  CC  /\  ( B ^ -u N )  =/=  0 ) ) )  ->  ( (
1  /  ( A ^ -u N ) )  x.  ( 1  /  ( B ^ -u N ) ) )  =  ( ( 1  x.  1 )  / 
( ( A ^ -u N )  x.  ( B ^ -u N ) ) ) )
4240, 40, 41mpanl12 666 . . . . . . . 8  |-  ( ( ( ( A ^ -u N )  e.  CC  /\  ( A ^ -u N
)  =/=  0 )  /\  ( ( B ^ -u N )  e.  CC  /\  ( B ^ -u N )  =/=  0 ) )  ->  ( ( 1  /  ( A ^ -u N ) )  x.  ( 1  /  ( B ^ -u N ) ) )  =  ( ( 1  x.  1 )  /  ( ( A ^ -u N
)  x.  ( B ^ -u N ) ) ) )
4329, 34, 36, 39, 42syl22anc 1188 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( 1  /  ( A ^ -u N ) )  x.  ( 1  /  ( B ^ -u N ) ) )  =  ( ( 1  x.  1 )  / 
( ( A ^ -u N )  x.  ( B ^ -u N ) ) ) )
4427, 43eqtr4d 2319 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
1  /  ( ( A  x.  B ) ^ -u N ) )  =  ( ( 1  /  ( A ^ -u N ) )  x.  ( 1  /  ( B ^ -u N ) ) ) )
4521, 44eqtr4d 2319 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( A ^ N
)  x.  ( B ^ N ) )  =  ( 1  / 
( ( A  x.  B ) ^ -u N
) ) )
4616, 45eqtr4d 2319 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
477, 46jaodan 763 . . 3  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )  -> 
( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
481, 47sylan2b 463 . 2  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  N  e.  ZZ )  ->  ( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
49483impa 1151 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  ZZ )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    =/= wne 2447  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733    x. cmul 8737   -ucneg 9033    / cdiv 9418   NNcn 9741   NN0cn0 9960   ZZcz 10019   ^cexp 11098
This theorem is referenced by:  exprec  11137
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-n0 9961  df-z 10020  df-uz 10226  df-seq 11041  df-exp 11099
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