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Theorem expmulzap 8955
Description: Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
Assertion
Ref Expression
expmulzap  CC #  0  M  ZZ  N  ZZ  ^ M  x.  N  ^ M ^ N

Proof of Theorem expmulzap
StepHypRef Expression
1 elznn0nn 8035 . . 3  N  ZZ  N  NN0  N  RR  -u N  NN
2 elznn0nn 8035 . . . 4  M  ZZ  M  NN0  M  RR  -u M  NN
3 expmul 8954 . . . . . . . 8  CC  M  NN0  N  NN0  ^ M  x.  N  ^ M ^ N
433expia 1105 . . . . . . 7  CC  M  NN0  N  NN0  ^ M  x.  N  ^ M ^ N
54adantlr 446 . . . . . 6  CC #  0  M  NN0  N  NN0  ^ M  x.  N  ^ M ^ N
6 simp2l 929 . . . . . . . . . . . . . 14  CC #  0  M  RR  -u M  NN  N  NN0  M  RR
76recnd 6851 . . . . . . . . . . . . 13  CC #  0  M  RR  -u M  NN  N  NN0  M  CC
8 simp3 905 . . . . . . . . . . . . . 14  CC #  0  M  RR  -u M  NN  N  NN0  N  NN0
98nn0cnd 8013 . . . . . . . . . . . . 13  CC #  0  M  RR  -u M  NN  N  NN0  N  CC
107, 9mulneg1d 7204 . . . . . . . . . . . 12  CC #  0  M  RR  -u M  NN  N  NN0  -u M  x.  N  -u M  x.  N
1110oveq2d 5471 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  NN0  ^ -u M  x.  N  ^ -u M  x.  N
12 simp1l 927 . . . . . . . . . . . 12  CC #  0  M  RR  -u M  NN  N  NN0  CC
13 simp2r 930 . . . . . . . . . . . . 13  CC #  0  M  RR  -u M  NN  N  NN0  -u M  NN
1413nnnn0d 8011 . . . . . . . . . . . 12  CC #  0  M  RR  -u M  NN  N  NN0  -u M  NN0
15 expmul 8954 . . . . . . . . . . . 12  CC  -u M  NN0  N  NN0  ^ -u M  x.  N  ^ -u M ^ N
1612, 14, 8, 15syl3anc 1134 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  NN0  ^ -u M  x.  N  ^ -u M ^ N
1711, 16eqtr3d 2071 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  NN0  ^ -u M  x.  N  ^ -u M ^ N
1817oveq2d 5471 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  NN0  1  ^ -u M  x.  N 
1  ^ -u M ^ N
19 expcl 8927 . . . . . . . . . . 11  CC  -u M  NN0  ^ -u M  CC
2012, 14, 19syl2anc 391 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  NN0  ^ -u M  CC
21 simp1r 928 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  NN0 #  0
2213nnzd 8135 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  NN0  -u M  ZZ
23 expap0i 8941 . . . . . . . . . . 11  CC #  0  -u M  ZZ  ^ -u M #  0
2412, 21, 22, 23syl3anc 1134 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  NN0  ^ -u M #  0
258nn0zd 8134 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  NN0  N  ZZ
26 exprecap 8950 . . . . . . . . . 10  ^ -u M  CC  ^ -u M #  0  N  ZZ  1  ^ -u M ^ N  1  ^ -u M ^ N
2720, 24, 25, 26syl3anc 1134 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  NN0  1  ^ -u M ^ N  1  ^ -u M ^ N
2818, 27eqtr4d 2072 . . . . . . . 8  CC #  0  M  RR  -u M  NN  N  NN0  1  ^ -u M  x.  N  1  ^ -u M ^ N
297, 9mulcld 6845 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  NN0  M  x.  N  CC
3014, 8nn0mulcld 8016 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  NN0  -u M  x.  N  NN0
3110, 30eqeltrrd 2112 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  NN0  -u M  x.  N  NN0
32 expineg2 8918 . . . . . . . . 9  CC #  0  M  x.  N  CC  -u M  x.  N  NN0  ^ M  x.  N  1  ^ -u M  x.  N
3312, 21, 29, 31, 32syl22anc 1135 . . . . . . . 8  CC #  0  M  RR  -u M  NN  N  NN0  ^ M  x.  N 
1  ^ -u M  x.  N
34 expineg2 8918 . . . . . . . . . 10  CC #  0  M  CC  -u M  NN0  ^ M  1  ^ -u M
3512, 21, 7, 14, 34syl22anc 1135 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  NN0  ^ M 
1  ^ -u M
3635oveq1d 5470 . . . . . . . 8  CC #  0  M  RR  -u M  NN  N  NN0  ^ M ^ N  1  ^ -u M ^ N
3728, 33, 363eqtr4d 2079 . . . . . . 7  CC #  0  M  RR  -u M  NN  N  NN0  ^ M  x.  N  ^ M ^ N
38373expia 1105 . . . . . 6  CC #  0  M  RR  -u M  NN  N  NN0  ^ M  x.  N  ^ M ^ N
395, 38jaodan 709 . . . . 5  CC #  0  M  NN0  M  RR  -u M  NN  N  NN0  ^ M  x.  N  ^ M ^ N
40 simp2 904 . . . . . . . . . . . . 13  CC #  0  M  NN0  N  RR  -u N  NN 
M  NN0
4140nn0cnd 8013 . . . . . . . . . . . 12  CC #  0  M  NN0  N  RR  -u N  NN 
M  CC
42 simp3l 931 . . . . . . . . . . . . 13  CC #  0  M  NN0  N  RR  -u N  NN 
N  RR
4342recnd 6851 . . . . . . . . . . . 12  CC #  0  M  NN0  N  RR  -u N  NN 
N  CC
4441, 43mulneg2d 7205 . . . . . . . . . . 11  CC #  0  M  NN0  N  RR  -u N  NN  M  x.  -u N  -u M  x.  N
4544oveq2d 5471 . . . . . . . . . 10  CC #  0  M  NN0  N  RR  -u N  NN  ^ M  x.  -u N  ^ -u M  x.  N
46 simp1l 927 . . . . . . . . . . 11  CC #  0  M  NN0  N  RR  -u N  NN  CC
47 simp3r 932 . . . . . . . . . . . 12  CC #  0  M  NN0  N  RR  -u N  NN  -u N  NN
4847nnnn0d 8011 . . . . . . . . . . 11  CC #  0  M  NN0  N  RR  -u N  NN  -u N  NN0
49 expmul 8954 . . . . . . . . . . 11  CC  M  NN0  -u N  NN0  ^ M  x.  -u N  ^ M ^ -u N
5046, 40, 48, 49syl3anc 1134 . . . . . . . . . 10  CC #  0  M  NN0  N  RR  -u N  NN  ^ M  x.  -u N  ^ M ^ -u N
5145, 50eqtr3d 2071 . . . . . . . . 9  CC #  0  M  NN0  N  RR  -u N  NN  ^ -u M  x.  N  ^ M ^ -u N
5251oveq2d 5471 . . . . . . . 8  CC #  0  M  NN0  N  RR  -u N  NN  1  ^ -u M  x.  N  1  ^ M ^ -u N
53 simp1r 928 . . . . . . . . 9  CC #  0  M  NN0  N  RR  -u N  NN #  0
5441, 43mulcld 6845 . . . . . . . . 9  CC #  0  M  NN0  N  RR  -u N  NN  M  x.  N  CC
5540, 48nn0mulcld 8016 . . . . . . . . . 10  CC #  0  M  NN0  N  RR  -u N  NN  M  x.  -u N  NN0
5644, 55eqeltrrd 2112 . . . . . . . . 9  CC #  0  M  NN0  N  RR  -u N  NN  -u M  x.  N  NN0
5746, 53, 54, 56, 32syl22anc 1135 . . . . . . . 8  CC #  0  M  NN0  N  RR  -u N  NN  ^ M  x.  N  1  ^ -u M  x.  N
58 expcl 8927 . . . . . . . . . 10  CC  M  NN0  ^ M  CC
5946, 40, 58syl2anc 391 . . . . . . . . 9  CC #  0  M  NN0  N  RR  -u N  NN  ^ M  CC
6040nn0zd 8134 . . . . . . . . . 10  CC #  0  M  NN0  N  RR  -u N  NN 
M  ZZ
61 expap0i 8941 . . . . . . . . . 10  CC #  0  M  ZZ  ^ M #  0
6246, 53, 60, 61syl3anc 1134 . . . . . . . . 9  CC #  0  M  NN0  N  RR  -u N  NN  ^ M #  0
63 expineg2 8918 . . . . . . . . 9  ^ M  CC  ^ M #  0  N  CC  -u N  NN0  ^ M ^ N  1  ^ M ^ -u N
6459, 62, 43, 48, 63syl22anc 1135 . . . . . . . 8  CC #  0  M  NN0  N  RR  -u N  NN  ^ M ^ N  1  ^ M ^ -u N
6552, 57, 643eqtr4d 2079 . . . . . . 7  CC #  0  M  NN0  N  RR  -u N  NN  ^ M  x.  N  ^ M ^ N
66653expia 1105 . . . . . 6  CC #  0  M  NN0  N  RR  -u N  NN  ^ M  x.  N  ^ M ^ N
67 simp1l 927 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  CC
68 simp1r 928 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN #  0
69 simp2l 929 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN 
M  RR
7069recnd 6851 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN 
M  CC
71 simp2r 930 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  NN
7271nnnn0d 8011 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  NN0
7367, 68, 70, 72, 34syl22anc 1135 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  1  ^ -u M
7473oveq1d 5470 . . . . . . . 8  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M ^ N  1  ^ -u M ^ N
7567, 72, 19syl2anc 391 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M  CC
7671nnzd 8135 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  ZZ
7767, 68, 76, 23syl3anc 1134 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M #  0
7875, 77recclapd 7539 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  ^ -u M  CC
7975, 77recap0d 7540 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  ^ -u M #  0
80 simp3l 931 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN 
N  RR
8180recnd 6851 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN 
N  CC
82 simp3r 932 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u N  NN
8382nnnn0d 8011 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u N  NN0
84 expineg2 8918 . . . . . . . . 9  1  ^ -u M  CC  1  ^ -u M #  0  N  CC  -u N  NN0  1  ^ -u M ^ N  1  1  ^ -u M ^ -u N
8578, 79, 81, 83, 84syl22anc 1135 . . . . . . . 8  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  ^ -u M ^ N  1  1  ^ -u M ^ -u N
8682nnzd 8135 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u N  ZZ
87 exprecap 8950 . . . . . . . . . . 11  ^ -u M  CC  ^ -u M #  0  -u N  ZZ  1  ^ -u M ^ -u N  1  ^ -u M ^ -u N
8875, 77, 86, 87syl3anc 1134 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  ^ -u M ^ -u N  1  ^ -u M ^ -u N
8988oveq2d 5471 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  1  ^ -u M ^ -u N 
1  1  ^ -u M ^ -u N
90 expcl 8927 . . . . . . . . . . 11  ^ -u M  CC  -u N  NN0  ^ -u M ^ -u N  CC
9175, 83, 90syl2anc 391 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M ^ -u N  CC
92 expap0i 8941 . . . . . . . . . . 11  ^ -u M  CC  ^ -u M #  0  -u N  ZZ  ^ -u M ^ -u N #  0
9375, 77, 86, 92syl3anc 1134 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M ^ -u N #  0
9491, 93recrecapd 7543 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1 
1  ^ -u M ^ -u N  ^ -u M ^ -u N
95 expmul 8954 . . . . . . . . . . 11  CC  -u M  NN0  -u N  NN0  ^ -u M  x.  -u N  ^ -u M ^ -u N
9667, 72, 83, 95syl3anc 1134 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M  x.  -u N  ^ -u M ^ -u N
9770, 81mul2negd 7206 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  x.  -u N  M  x.  N
9897oveq2d 5471 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M  x.  -u N  ^ M  x.  N
9996, 98eqtr3d 2071 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M ^ -u N  ^ M  x.  N
10089, 94, 993eqtrd 2073 . . . . . . . 8  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  1  ^ -u M ^ -u N  ^ M  x.  N
10174, 85, 1003eqtrrd 2074 . . . . . . 7  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  x.  N  ^ M ^ N
1021013expia 1105 . . . . . 6  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  x.  N  ^ M ^ N
10366, 102jaodan 709 . . . . 5  CC #  0  M  NN0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  x.  N  ^ M ^ N
10439, 103jaod 636 . . . 4  CC #  0  M  NN0  M  RR  -u M  NN  N  NN0  N  RR  -u N  NN  ^ M  x.  N  ^ M ^ N
1052, 104sylan2b 271 . . 3  CC #  0  M  ZZ  N 
NN0  N  RR  -u N  NN  ^ M  x.  N  ^ M ^ N
1061, 105syl5bi 141 . 2  CC #  0  M  ZZ  N  ZZ  ^ M  x.  N  ^ M ^ N
107106impr 361 1  CC #  0  M  ZZ  N  ZZ  ^ M  x.  N  ^ M ^ N
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wo 628   w3a 884   wceq 1242   wcel 1390   class class class wbr 3755  (class class class)co 5455   CCcc 6709   RRcr 6710   0cc0 6711   1c1 6712    x. cmul 6716   -ucneg 6980   # cap 7365   cdiv 7433   NNcn 7695   NN0cn0 7957   ZZcz 8021   ^cexp 8908
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulrcl 6782  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-precex 6793  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799  ax-pre-mulgt0 6800  ax-pre-mulext 6801
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-if 3326  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-frec 5918  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-reap 7359  df-ap 7366  df-div 7434  df-inn 7696  df-n0 7958  df-z 8022  df-uz 8250  df-iseq 8893  df-iexp 8909
This theorem is referenced by: (None)
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