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Theorem mulexpz 11073
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 9969 . . 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 11072 . . . . . 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 8788 . . . . . 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 8760 . . . . . . 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 9904 . . . . . . 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 11043 . . . . . 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 11043 . . . . . . . 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 11043 . . . . . . . 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 5775 . . . . . 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 11072 . . . . . . . . . 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 5773 . . . . . . . 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 9802 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
2625oveq1i 5767 . . . . . . . 8  |-  ( ( 1  x.  1 )  /  ( ( A ^ -u N )  x.  ( B ^ -u N ) ) )  =  ( 1  / 
( ( A ^ -u N )  x.  ( B ^ -u N ) ) )
2724, 26syl6eqr 2306 . . . . . . 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 11052 . . . . . . . . 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 9977 . . . . . . . . . 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 11065 . . . . . . . . 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 11052 . . . . . . . . 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 11065 . . . . . . . . 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 8728 . . . . . . . . 9  |-  1  e.  CC
41 divmuldiv 9393 . . . . . . . . 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 2291 . . . . . 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 2291 . . . . 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 2291 . . . 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 1619    e. wcel 1621    =/= wne 2419  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    x. cmul 8675   -ucneg 8971    / cdiv 9356   NNcn 9679   NN0cn0 9897   ZZcz 9956   ^cexp 11035
This theorem is referenced by:  exprec  11074
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
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 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-n0 9898  df-z 9957  df-uz 10163  df-seq 10978  df-exp 11036
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