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Theorem mulexpz 11158
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 10053 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 simpl 443 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  e.  CC )
3 simpl 443 . . . . . 6  |-  ( ( B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
42, 3anim12i 549 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  e.  CC  /\  B  e.  CC ) )
5 mulexp 11157 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
653expa 1151 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  N  e.  NN0 )  ->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N
)  x.  ( B ^ N ) ) )
74, 6sylan 457 . . . 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 734 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
9 simplrl 736 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  B  e.  CC )
108, 9mulcld 8871 . . . . . 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 8843 . . . . . . 7  |-  ( N  e.  RR  ->  N  e.  CC )
1211ad2antrl 708 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
13 nnnn0 9988 . . . . . . 7  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
1413ad2antll 709 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
15 expneg2 11128 . . . . . 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 1182 . . . . 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 11128 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
188, 12, 14, 17syl3anc 1182 . . . . . . 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 11128 . . . . . . . 8  |-  ( ( B  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( B ^ N )  =  ( 1  /  ( B ^ -u N ) ) )
209, 12, 14, 19syl3anc 1182 . . . . . . 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 5892 . . . . . 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 11157 . . . . . . . . . 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 1182 . . . . . . . . 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 5890 . . . . . . . 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 9886 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
2625oveq1i 5884 . . . . . . . 8  |-  ( ( 1  x.  1 )  /  ( ( A ^ -u N )  x.  ( B ^ -u N ) ) )  =  ( 1  / 
( ( A ^ -u N )  x.  ( B ^ -u N ) ) )
2724, 26syl6eqr 2346 . . . . . . 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 11137 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ -u N
)  e.  CC )
298, 14, 28syl2anc 642 . . . . . . . 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 735 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  =/=  0 )
31 nnz 10061 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
3231ad2antll 709 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
33 expne0i 11150 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u N  e.  ZZ )  ->  ( A ^ -u N )  =/=  0 )
348, 30, 32, 33syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ -u N )  =/=  0 )
35 expcl 11137 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  -u N  e.  NN0 )  ->  ( B ^ -u N
)  e.  CC )
369, 14, 35syl2anc 642 . . . . . . . 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 737 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  B  =/=  0 )
38 expne0i 11150 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  -u N  e.  ZZ )  ->  ( B ^ -u N )  =/=  0 )
399, 37, 32, 38syl3anc 1182 . . . . . . . 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 8811 . . . . . . . . 9  |-  1  e.  CC
41 divmuldiv 9476 . . . . . . . . 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 663 . . . . . . . 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 1183 . . . . . . 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 2331 . . . . . 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 2331 . . . . 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 2331 . . . 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 760 . . 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 461 . 2  |-  ( ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  /\  N  e.  ZZ )  ->  ( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
49483impa 1146 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 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758   -ucneg 9054    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120
This theorem is referenced by:  exprec  11159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-exp 11121
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