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Theorem mulexpzap 9269
Description: Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
Assertion
Ref Expression
mulexpzap  |-  ( ( ( 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 mulexpzap
StepHypRef Expression
1 elznn0nn 8257 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 simpl 102 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0 )  ->  A  e.  CC )
3 simpl 102 . . . . . 6  |-  ( ( B  e.  CC  /\  B #  0 )  ->  B  e.  CC )
42, 3anim12i 321 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( A  e.  CC  /\  B  e.  CC ) )
5 mulexp 9268 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
653expa 1104 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  N  e.  NN0 )  ->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N
)  x.  ( B ^ N ) ) )
74, 6sylan 267 . . . 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 485 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
9 simplrl 487 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  B  e.  CC )
108, 9mulcld 7045 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A  x.  B )  e.  CC )
11 simpllr 486 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A #  0 )
12 simplrr 488 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  B #  0 )
138, 9, 11, 12mulap0d 7637 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A  x.  B ) #  0 )
14 recn 7012 . . . . . . 7  |-  ( N  e.  RR  ->  N  e.  CC )
1514ad2antrl 459 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
16 nnnn0 8186 . . . . . . 7  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
1716ad2antll 460 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
18 expineg2 9238 . . . . . 6  |-  ( ( ( ( A  x.  B )  e.  CC  /\  ( A  x.  B
) #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  ->  (
( A  x.  B
) ^ N )  =  ( 1  / 
( ( A  x.  B ) ^ -u N
) ) )
1910, 13, 15, 17, 18syl22anc 1136 . . . . 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
) ) )
20 expineg2 9238 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
218, 11, 15, 17, 20syl22anc 1136 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
22 expineg2 9238 . . . . . . . 8  |-  ( ( ( B  e.  CC  /\  B #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  ->  ( B ^ N )  =  ( 1  /  ( B ^ -u N ) ) )
239, 12, 15, 17, 22syl22anc 1136 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( B ^ N )  =  ( 1  /  ( B ^ -u N ) ) )
2421, 23oveq12d 5530 . . . . . 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 ) ) ) )
25 mulexp 9268 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  -u N  e.  NN0 )  ->  (
( A  x.  B
) ^ -u N
)  =  ( ( A ^ -u N
)  x.  ( B ^ -u N ) ) )
268, 9, 17, 25syl3anc 1135 . . . . . . . . 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 ) ) )
2726oveq2d 5528 . . . . . . . 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 ) ) ) )
28 1t1e1 8065 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
2928oveq1i 5522 . . . . . . . 8  |-  ( ( 1  x.  1 )  /  ( ( A ^ -u N )  x.  ( B ^ -u N ) ) )  =  ( 1  / 
( ( A ^ -u N )  x.  ( B ^ -u N ) ) )
3027, 29syl6eqr 2090 . . . . . . 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 ) ) ) )
31 expcl 9247 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ -u N
)  e.  CC )
328, 17, 31syl2anc 391 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ -u N )  e.  CC )
33 nnz 8262 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
3433ad2antll 460 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
35 expap0i 9261 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  -u N  e.  ZZ )  ->  ( A ^ -u N ) #  0 )
368, 11, 34, 35syl3anc 1135 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ -u N ) #  0 )
37 expcl 9247 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  -u N  e.  NN0 )  ->  ( B ^ -u N
)  e.  CC )
389, 17, 37syl2anc 391 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( B ^ -u N )  e.  CC )
39 expap0i 9261 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B #  0  /\  -u N  e.  ZZ )  ->  ( B ^ -u N ) #  0 )
409, 12, 34, 39syl3anc 1135 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( B ^ -u N ) #  0 )
41 ax-1cn 6975 . . . . . . . . 9  |-  1  e.  CC
42 divmuldivap 7686 . . . . . . . . 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 ) ) ) )
4341, 41, 42mpanl12 412 . . . . . . . 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 ) ) ) )
4432, 36, 38, 40, 43syl22anc 1136 . . . . . . 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 ) ) ) )
4530, 44eqtr4d 2075 . . . . . 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 ) ) ) )
4624, 45eqtr4d 2075 . . . . 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
) ) )
4719, 46eqtr4d 2075 . . . 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
) ) )
487, 47jaodan 710 . . 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 ) ) )
491, 48sylan2b 271 . 2  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  N  e.  ZZ )  ->  ( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
50493impa 1099 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    /\ wa 97    \/ wo 629    /\ w3a 885    = wceq 1243    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   CCcc 6885   RRcr 6886   0cc0 6887   1c1 6888    x. cmul 6892   -ucneg 7181   # cap 7570    / cdiv 7649   NNcn 7912   NN0cn0 8179   ZZcz 8243   ^cexp 9228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6973  ax-resscn 6974  ax-1cn 6975  ax-1re 6976  ax-icn 6977  ax-addcl 6978  ax-addrcl 6979  ax-mulcl 6980  ax-mulrcl 6981  ax-addcom 6982  ax-mulcom 6983  ax-addass 6984  ax-mulass 6985  ax-distr 6986  ax-i2m1 6987  ax-1rid 6989  ax-0id 6990  ax-rnegex 6991  ax-precex 6992  ax-cnre 6993  ax-pre-ltirr 6994  ax-pre-ltwlin 6995  ax-pre-lttrn 6996  ax-pre-apti 6997  ax-pre-ltadd 6998  ax-pre-mulgt0 6999  ax-pre-mulext 7000
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-iltp 6566  df-enr 6809  df-nr 6810  df-ltr 6813  df-0r 6814  df-1r 6815  df-0 6894  df-1 6895  df-r 6897  df-lt 6900  df-pnf 7060  df-mnf 7061  df-xr 7062  df-ltxr 7063  df-le 7064  df-sub 7182  df-neg 7183  df-reap 7564  df-ap 7571  df-div 7650  df-inn 7913  df-n0 8180  df-z 8244  df-uz 8472  df-iseq 9186  df-iexp 9229
This theorem is referenced by:  exprecap  9270
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