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Theorem mulexpz 12900
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 11391 . . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2 simpl 473 . . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> A e. CC )
3 simpl 473 . . . . . 6 |- ( ( B e. CC /\ B =/= 0 ) -> B e. CC )
42, 3anim12i 590 . . . . 5 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( A e. CC /\ B e. CC ) )
5 mulexp 12899 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
653expa 1265 . . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
74, 6sylan 488 . . . 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 798 . . . . . . 7 |- ( ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
9 simplrl 800 . . . . . . 7 |- ( ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> B e. CC )
108, 9mulcld 10060 . . . . . 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 10026 . . . . . . 7 |- ( N e. RR -> N e. CC )
1211ad2antrl 764 . . . . . 6 |- ( ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
13 nnnn0 11299 . . . . . . 7 |- ( -u N e. NN -> -u N e. NN0 )
1413ad2antll 765 . . . . . 6 |- ( ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
15 expneg2 12869 . . . . . 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 1326 . . . . 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 12869 . . . . . . . 8 |- ( ( A e. CC /\ N e. CC /\ -u N e. NN0 ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
188, 12, 14, 17syl3anc 1326 . . . . . . 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 12869 . . . . . . . 8 |- ( ( B e. CC /\ N e. CC /\ -u N e. NN0 ) -> ( B ^ N ) = ( 1 / ( B ^ -u N ) ) )
209, 12, 14, 19syl3anc 1326 . . . . . . 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 6668 . . . . . 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 12899 . . . . . . . . . 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 1326 . . . . . . . . 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 6666 . . . . . . . 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 11175 . . . . . . . . 9 |- ( 1 x. 1 ) = 1
2625oveq1i 6660 . . . . . . . 8 |- ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) = ( 1 / ( ( A ^ -u N ) x. ( B ^ -u N ) ) )
2724, 26syl6eqr 2674 . . . . . . 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 12878 . . . . . . . . 9 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
298, 14, 28syl2anc 693 . . . . . . . 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 799 . . . . . . . . 9 |- ( ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A =/= 0 )
31 nnz 11399 . . . . . . . . . 10 |- ( -u N e. NN -> -u N e. ZZ )
3231ad2antll 765 . . . . . . . . 9 |- ( ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
33 expne0i 12892 . . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) =/= 0 )
348, 30, 32, 33syl3anc 1326 . . . . . . . 8 |- ( ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) =/= 0 )
35 expcl 12878 . . . . . . . . 9 |- ( ( B e. CC /\ -u N e. NN0 ) -> ( B ^ -u N ) e. CC )
369, 14, 35syl2anc 693 . . . . . . . 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 801 . . . . . . . . 9 |- ( ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> B =/= 0 )
38 expne0i 12892 . . . . . . . . 9 |- ( ( B e. CC /\ B =/= 0 /\ -u N e. ZZ ) -> ( B ^ -u N ) =/= 0 )
399, 37, 32, 38syl3anc 1326 . . . . . . . 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 9994 . . . . . . . . 9 |- 1 e. CC
41 divmuldiv 10725 . . . . . . . . 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 718 . . . . . . . 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 1327 . . . . . . 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 2659 . . . . . 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 2659 . . . . 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 2659 . . . 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 826 . . 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 492 . 2 |- ( ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) /\ N e. ZZ ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
49483impa 1259 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 setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ^cexp 12860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861
This theorem is referenced by:  exprec  12901  knoppndvlem14  32516  knoppndvlem17  32519
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