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Theorem expadd 9271
Description: Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
Assertion
Ref Expression
expadd  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )

Proof of Theorem expadd
Dummy variables  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5520 . . . . . . 7  |-  ( j  =  0  ->  ( M  +  j )  =  ( M  + 
0 ) )
21oveq2d 5528 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  0 ) ) )
3 oveq2 5520 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
43oveq2d 5528 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) )
52, 4eqeq12d 2054 . . . . 5  |-  ( j  =  0  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  + 
0 ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) ) )
65imbi2d 219 . . . 4  |-  ( j  =  0  ->  (
( ( A  e.  CC  /\  M  e. 
NN0 )  ->  ( A ^ ( M  +  j ) )  =  ( ( A ^ M )  x.  ( A ^ j ) ) )  <->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  + 
0 ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) ) ) )
7 oveq2 5520 . . . . . . 7  |-  ( j  =  k  ->  ( M  +  j )  =  ( M  +  k ) )
87oveq2d 5528 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  k )
) )
9 oveq2 5520 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
109oveq2d 5528 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ k
) ) )
118, 10eqeq12d 2054 . . . . 5  |-  ( j  =  k  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k ) ) ) )
1211imbi2d 219 . . . 4  |-  ( j  =  k  ->  (
( ( A  e.  CC  /\  M  e. 
NN0 )  ->  ( A ^ ( M  +  j ) )  =  ( ( A ^ M )  x.  ( A ^ j ) ) )  <->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k ) ) ) ) )
13 oveq2 5520 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  +  j )  =  ( M  +  ( k  +  1 ) ) )
1413oveq2d 5528 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
15 oveq2 5520 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1615oveq2d 5528 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) ) )
1714, 16eqeq12d 2054 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M )  x.  ( A ^ ( k  +  1 ) ) ) ) )
1817imbi2d 219 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( A  e.  CC  /\  M  e. 
NN0 )  ->  ( A ^ ( M  +  j ) )  =  ( ( A ^ M )  x.  ( A ^ j ) ) )  <->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M )  x.  ( A ^ ( k  +  1 ) ) ) ) ) )
19 oveq2 5520 . . . . . . 7  |-  ( j  =  N  ->  ( M  +  j )  =  ( M  +  N ) )
2019oveq2d 5528 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  N )
) )
21 oveq2 5520 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
2221oveq2d 5528 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ N
) ) )
2320, 22eqeq12d 2054 . . . . 5  |-  ( j  =  N  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
2423imbi2d 219 . . . 4  |-  ( j  =  N  ->  (
( ( A  e.  CC  /\  M  e. 
NN0 )  ->  ( A ^ ( M  +  j ) )  =  ( ( A ^ M )  x.  ( A ^ j ) ) )  <->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) ) )
25 nn0cn 8189 . . . . . . . . 9  |-  ( M  e.  NN0  ->  M  e.  CC )
2625addid1d 7160 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( M  +  0 )  =  M )
2726adantl 262 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
2827oveq2d 5528 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( A ^ M ) )
29 expcl 9247 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
3029mulid1d 7042 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  1 )  =  ( A ^ M ) )
3128, 30eqtr4d 2075 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  1 ) )
32 exp0 9233 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3332adantr 261 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ 0 )  =  1 )
3433oveq2d 5528 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  ( A ^ 0 ) )  =  ( ( A ^ M )  x.  1 ) )
3531, 34eqtr4d 2075 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  ( A ^
0 ) ) )
36 oveq1 5519 . . . . . . 7  |-  ( ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k
) )  ->  (
( A ^ ( M  +  k )
)  x.  A )  =  ( ( ( A ^ M )  x.  ( A ^
k ) )  x.  A ) )
37 nn0cn 8189 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 6975 . . . . . . . . . . . . 13  |-  1  e.  CC
39 addass 7009 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4038, 39mp3an3 1221 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4125, 37, 40syl2an 273 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4241adantll 445 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4342oveq2d 5528 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
44 simpll 481 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  A  e.  CC )
45 nn0addcl 8215 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  +  k )  e.  NN0 )
4645adantll 445 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  +  k )  e.  NN0 )
47 expp1 9236 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( M  +  k
)  e.  NN0 )  ->  ( A ^ (
( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k ) )  x.  A ) )
4844, 46, 47syl2anc 391 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
4943, 48eqtr3d 2074 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  +  ( k  +  1 ) ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
50 expp1 9236 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5150adantlr 446 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
5251oveq2d 5528 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) )  =  ( ( A ^ M
)  x.  ( ( A ^ k )  x.  A ) ) )
5329adantr 261 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^ M )  e.  CC )
54 expcl 9247 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
5554adantlr 446 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
k )  e.  CC )
5653, 55, 44mulassd 7048 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( ( A ^ M )  x.  ( A ^
k ) )  x.  A )  =  ( ( A ^ M
)  x.  ( ( A ^ k )  x.  A ) ) )
5752, 56eqtr4d 2075 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) )  =  ( ( ( A ^ M )  x.  ( A ^ k ) )  x.  A ) )
5849, 57eqeq12d 2054 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ ( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M )  x.  ( A ^ ( k  +  1 ) ) )  <-> 
( ( A ^
( M  +  k ) )  x.  A
)  =  ( ( ( A ^ M
)  x.  ( A ^ k ) )  x.  A ) ) )
5936, 58syl5ibr 145 . . . . . 6  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k ) )  ->  ( A ^
( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M
)  x.  ( A ^ ( k  +  1 ) ) ) ) )
6059expcom 109 . . . . 5  |-  ( k  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^
( M  +  k ) )  =  ( ( A ^ M
)  x.  ( A ^ k ) )  ->  ( A ^
( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M
)  x.  ( A ^ ( k  +  1 ) ) ) ) ) )
6160a2d 23 . . . 4  |-  ( k  e.  NN0  ->  ( ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  +  k )
)  =  ( ( A ^ M )  x.  ( A ^
k ) ) )  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M )  x.  ( A ^ ( k  +  1 ) ) ) ) ) )
626, 12, 18, 24, 35, 61nn0ind 8350 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
6362expdcom 1331 . 2  |-  ( A  e.  CC  ->  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) ) )
64633imp 1098 1  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    /\ w3a 885    = wceq 1243    e. wcel 1393  (class class class)co 5512   CCcc 6885   0cc0 6887   1c1 6888    + caddc 6890    x. cmul 6892   NN0cn0 8179   ^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:  expaddzaplem  9272  expaddzap  9273  expmul  9274  i4  9329  expaddd  9357
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