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Theorem expaddzap 8933
Description: Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
Assertion
Ref Expression
expaddzap  CC #  0  M  ZZ  N  ZZ  ^ M  +  N  ^ M  x.  ^ N

Proof of Theorem expaddzap
StepHypRef Expression
1 elznn0nn 8015 . . 3  N  ZZ  N  NN0  N  RR  -u N  NN
2 elznn0nn 8015 . . . 4  M  ZZ  M  NN0  M  RR  -u M  NN
3 expadd 8931 . . . . . . . 8  CC  M  NN0  N  NN0  ^ M  +  N  ^ M  x.  ^ N
433expia 1105 . . . . . . 7  CC  M  NN0  N  NN0  ^ M  +  N  ^ M  x.  ^ N
54adantlr 446 . . . . . 6  CC #  0  M  NN0  N  NN0  ^ M  +  N  ^ M  x.  ^ N
6 expaddzaplem 8932 . . . . . . 7  CC #  0  M  RR  -u M  NN  N  NN0  ^ M  +  N  ^ M  x.  ^ N
763expia 1105 . . . . . 6  CC #  0  M  RR  -u M  NN  N  NN0  ^ M  +  N  ^ M  x.  ^ N
85, 7jaodan 709 . . . . 5  CC #  0  M  NN0  M  RR  -u M  NN  N  NN0  ^ M  +  N  ^ M  x.  ^ N
9 expaddzaplem 8932 . . . . . . . . 9  CC #  0  N  RR  -u N  NN  M  NN0  ^ N  +  M  ^ N  x.  ^ M
10 simp3 905 . . . . . . . . . . . 12  CC #  0  N  RR  -u N  NN  M  NN0  M  NN0
1110nn0cnd 7993 . . . . . . . . . . 11  CC #  0  N  RR  -u N  NN  M  NN0  M  CC
12 simp2l 929 . . . . . . . . . . . 12  CC #  0  N  RR  -u N  NN  M  NN0  N  RR
1312recnd 6831 . . . . . . . . . . 11  CC #  0  N  RR  -u N  NN  M  NN0  N  CC
1411, 13addcomd 6941 . . . . . . . . . 10  CC #  0  N  RR  -u N  NN  M  NN0  M  +  N  N  +  M
1514oveq2d 5471 . . . . . . . . 9  CC #  0  N  RR  -u N  NN  M  NN0  ^ M  +  N  ^ N  +  M
16 simp1l 927 . . . . . . . . . . 11  CC #  0  N  RR  -u N  NN  M  NN0  CC
17 expcl 8907 . . . . . . . . . . 11  CC  M  NN0  ^ M  CC
1816, 10, 17syl2anc 391 . . . . . . . . . 10  CC #  0  N  RR  -u N  NN  M  NN0  ^ M  CC
19 simp1r 928 . . . . . . . . . . 11  CC #  0  N  RR  -u N  NN  M  NN0 #  0
2013negnegd 7089 . . . . . . . . . . . 12  CC #  0  N  RR  -u N  NN  M  NN0  -u -u N  N
21 simp2r 930 . . . . . . . . . . . . . 14  CC #  0  N  RR  -u N  NN  M  NN0  -u N  NN
2221nnnn0d 7991 . . . . . . . . . . . . 13  CC #  0  N  RR  -u N  NN  M  NN0  -u N  NN0
23 nn0negz 8035 . . . . . . . . . . . . 13  -u N  NN0  -u -u N  ZZ
2422, 23syl 14 . . . . . . . . . . . 12  CC #  0  N  RR  -u N  NN  M  NN0  -u -u N  ZZ
2520, 24eqeltrrd 2112 . . . . . . . . . . 11  CC #  0  N  RR  -u N  NN  M  NN0  N  ZZ
26 expclzap 8914 . . . . . . . . . . 11  CC #  0  N  ZZ  ^ N  CC
2716, 19, 25, 26syl3anc 1134 . . . . . . . . . 10  CC #  0  N  RR  -u N  NN  M  NN0  ^ N  CC
2818, 27mulcomd 6826 . . . . . . . . 9  CC #  0  N  RR  -u N  NN  M  NN0  ^ M  x.  ^ N  ^ N  x.  ^ M
299, 15, 283eqtr4d 2079 . . . . . . . 8  CC #  0  N  RR  -u N  NN  M  NN0  ^ M  +  N  ^ M  x.  ^ N
30293expia 1105 . . . . . . 7  CC #  0  N  RR  -u N  NN  M  NN0  ^ M  +  N  ^ M  x.  ^ N
3130impancom 247 . . . . . 6  CC #  0  M  NN0  N  RR  -u N  NN  ^ M  +  N  ^ M  x.  ^ N
32 simp2l 929 . . . . . . . . . . . . . . 15  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN 
M  RR
3332recnd 6831 . . . . . . . . . . . . . 14  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN 
M  CC
34 simp3l 931 . . . . . . . . . . . . . . 15  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN 
N  RR
3534recnd 6831 . . . . . . . . . . . . . 14  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN 
N  CC
3633, 35negdid 7111 . . . . . . . . . . . . 13  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  +  N  -u M  +  -u N
3736oveq2d 5471 . . . . . . . . . . . 12  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M  +  N  ^ -u M  +  -u N
38 simp1l 927 . . . . . . . . . . . . 13  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  CC
39 simp2r 930 . . . . . . . . . . . . . 14  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  NN
4039nnnn0d 7991 . . . . . . . . . . . . 13  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  NN0
41 simp3r 932 . . . . . . . . . . . . . 14  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u N  NN
4241nnnn0d 7991 . . . . . . . . . . . . 13  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u N  NN0
43 expadd 8931 . . . . . . . . . . . . 13  CC  -u M  NN0  -u N  NN0  ^ -u M  +  -u N  ^ -u M  x.  ^ -u N
4438, 40, 42, 43syl3anc 1134 . . . . . . . . . . . 12  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M  +  -u N  ^ -u M  x.  ^ -u N
4537, 44eqtrd 2069 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M  +  N  ^ -u M  x.  ^ -u N
4645oveq2d 5471 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  ^ -u M  +  N  1  ^ -u M  x.  ^ -u N
47 1t1e1 7825 . . . . . . . . . . 11  1  x.  1  1
4847oveq1i 5465 . . . . . . . . . 10  1  x.  1  ^ -u M  x.  ^ -u N  1  ^ -u M  x.  ^ -u N
4946, 48syl6eqr 2087 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  ^ -u M  +  N  1  x.  1  ^ -u M  x.  ^ -u N
50 expcl 8907 . . . . . . . . . . 11  CC  -u M  NN0  ^ -u M  CC
5138, 40, 50syl2anc 391 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M  CC
52 simp1r 928 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN #  0
5340nn0zd 8114 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  ZZ
54 expap0i 8921 . . . . . . . . . . 11  CC #  0  -u M  ZZ  ^ -u M #  0
5538, 52, 53, 54syl3anc 1134 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u M #  0
56 expcl 8907 . . . . . . . . . . 11  CC  -u N  NN0  ^ -u N  CC
5738, 42, 56syl2anc 391 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u N  CC
5842nn0zd 8114 . . . . . . . . . . 11  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u N  ZZ
59 expap0i 8921 . . . . . . . . . . 11  CC #  0  -u N  ZZ  ^ -u N #  0
6038, 52, 58, 59syl3anc 1134 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ -u N #  0
61 ax-1cn 6756 . . . . . . . . . . 11  1  CC
62 divmuldivap 7450 . . . . . . . . . . 11  1  CC  1  CC  ^ -u M  CC  ^ -u M #  0  ^ -u N  CC  ^ -u N #  0  1  ^ -u M  x. 
1  ^ -u N  1  x.  1  ^ -u M  x.  ^ -u N
6361, 61, 62mpanl12 412 . . . . . . . . . 10  ^ -u M  CC  ^ -u M #  0  ^ -u N  CC  ^ -u N #  0  1  ^ -u M  x.  1  ^ -u N  1  x.  1  ^ -u M  x.  ^ -u N
6451, 55, 57, 60, 63syl22anc 1135 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  ^ -u M  x.  1  ^ -u N  1  x.  1  ^ -u M  x.  ^ -u N
6549, 64eqtr4d 2072 . . . . . . . 8  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  1  ^ -u M  +  N  1  ^ -u M  x.  1  ^ -u N
6633, 35addcld 6824 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  M  +  N  CC
6740, 42nn0addcld 7995 . . . . . . . . . 10  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  +  -u N  NN0
6836, 67eqeltrd 2111 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  -u M  +  N  NN0
69 expineg2 8898 . . . . . . . . 9  CC #  0  M  +  N  CC  -u M  +  N  NN0  ^ M  +  N  1  ^ -u M  +  N
7038, 52, 66, 68, 69syl22anc 1135 . . . . . . . 8  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  +  N  1  ^ -u M  +  N
71 expineg2 8898 . . . . . . . . . 10  CC #  0  M  CC  -u M  NN0  ^ M  1  ^ -u M
7238, 52, 33, 40, 71syl22anc 1135 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  1  ^ -u M
73 expineg2 8898 . . . . . . . . . 10  CC #  0  N  CC  -u N  NN0  ^ N  1  ^ -u N
7438, 52, 35, 42, 73syl22anc 1135 . . . . . . . . 9  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ N  1  ^ -u N
7572, 74oveq12d 5473 . . . . . . . 8  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  x.  ^ N  1  ^ -u M  x. 
1  ^ -u N
7665, 70, 753eqtr4d 2079 . . . . . . 7  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  +  N  ^ M  x.  ^ N
77763expia 1105 . . . . . 6  CC #  0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  +  N  ^ M  x.  ^ N
7831, 77jaodan 709 . . . . 5  CC #  0  M  NN0  M  RR  -u M  NN  N  RR  -u N  NN  ^ M  +  N  ^ M  x.  ^ N
798, 78jaod 636 . . . 4  CC #  0  M  NN0  M  RR  -u M  NN  N  NN0  N  RR  -u N  NN  ^ M  +  N  ^ M  x.  ^ N
802, 79sylan2b 271 . . 3  CC #  0  M  ZZ  N 
NN0  N  RR  -u N  NN  ^ M  +  N  ^ M  x.  ^ N
811, 80syl5bi 141 . 2  CC #  0  M  ZZ  N  ZZ  ^ M  +  N  ^ M  x.  ^ N
8281impr 361 1  CC #  0  M  ZZ  N  ZZ  ^ M  +  N  ^ M  x.  ^ 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 6689   RRcr 6690   0cc0 6691   1c1 6692    + caddc 6694    x. cmul 6696   -ucneg 6960   # cap 7345   cdiv 7413   NNcn 7675   NN0cn0 7937   ZZcz 8001   ^cexp 8888
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-bnd 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 6754  ax-resscn 6755  ax-1cn 6756  ax-1re 6757  ax-icn 6758  ax-addcl 6759  ax-addrcl 6760  ax-mulcl 6761  ax-mulrcl 6762  ax-addcom 6763  ax-mulcom 6764  ax-addass 6765  ax-mulass 6766  ax-distr 6767  ax-i2m1 6768  ax-1rid 6770  ax-0id 6771  ax-rnegex 6772  ax-precex 6773  ax-cnre 6774  ax-pre-ltirr 6775  ax-pre-ltwlin 6776  ax-pre-lttrn 6777  ax-pre-apti 6778  ax-pre-ltadd 6779  ax-pre-mulgt0 6780  ax-pre-mulext 6781
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 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-iltp 6452  df-enr 6634  df-nr 6635  df-ltr 6638  df-0r 6639  df-1r 6640  df-0 6698  df-1 6699  df-r 6701  df-lt 6704  df-pnf 6839  df-mnf 6840  df-xr 6841  df-ltxr 6842  df-le 6843  df-sub 6961  df-neg 6962  df-reap 7339  df-ap 7346  df-div 7414  df-inn 7676  df-n0 7938  df-z 8002  df-uz 8230  df-iseq 8873  df-iexp 8889
This theorem is referenced by:  expsubap  8936  expp1zap  8937
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