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Theorem mulexpz 11020
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 9916 . . 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 11019 . . . . . 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 8735 . . . . . 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 8707 . . . . . . 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 9851 . . . . . . 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 10990 . . . . . 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 10990 . . . . . . . 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 10990 . . . . . . . 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 5728 . . . . . 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 11019 . . . . . . . . . 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 5726 . . . . . . . 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 9749 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
2625oveq1i 5720 . . . . . . . 8  |-  ( ( 1  x.  1 )  /  ( ( A ^ -u N )  x.  ( B ^ -u N ) ) )  =  ( 1  / 
( ( A ^ -u N )  x.  ( B ^ -u N ) ) )
2724, 26syl6eqr 2303 . . . . . . 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 10999 . . . . . . . . 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 9924 . . . . . . . . . 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 11012 . . . . . . . . 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 10999 . . . . . . . . 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 11012 . . . . . . . . 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 8675 . . . . . . . . 9  |-  1  e.  CC
41 divmuldiv 9340 . . . . . . . . 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 2288 . . . . . 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 2288 . . . . 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 2288 . . . 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 2412  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    x. cmul 8622   -ucneg 8918    / cdiv 9303   NNcn 9626   NN0cn0 9844   ZZcz 9903   ^cexp 10982
This theorem is referenced by:  exprec  11021
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-n0 9845  df-z 9904  df-uz 10110  df-seq 10925  df-exp 10983
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