ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expadd Unicode version

Theorem expadd 10286
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 5748 . . . . . . 7  |-  ( j  =  0  ->  ( M  +  j )  =  ( M  + 
0 ) )
21oveq2d 5756 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  0 ) ) )
3 oveq2 5748 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
43oveq2d 5756 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) )
52, 4eqeq12d 2130 . . . . 5  |-  ( j  =  0  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  + 
0 ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) ) )
65imbi2d 229 . . . 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 5748 . . . . . . 7  |-  ( j  =  k  ->  ( M  +  j )  =  ( M  +  k ) )
87oveq2d 5756 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  k )
) )
9 oveq2 5748 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
109oveq2d 5756 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ k
) ) )
118, 10eqeq12d 2130 . . . . 5  |-  ( j  =  k  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k ) ) ) )
1211imbi2d 229 . . . 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 5748 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  +  j )  =  ( M  +  ( k  +  1 ) ) )
1413oveq2d 5756 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
15 oveq2 5748 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1615oveq2d 5756 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) ) )
1714, 16eqeq12d 2130 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M )  x.  ( A ^ ( k  +  1 ) ) ) ) )
1817imbi2d 229 . . . 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 5748 . . . . . . 7  |-  ( j  =  N  ->  ( M  +  j )  =  ( M  +  N ) )
2019oveq2d 5756 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  N )
) )
21 oveq2 5748 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
2221oveq2d 5756 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ N
) ) )
2320, 22eqeq12d 2130 . . . . 5  |-  ( j  =  N  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
2423imbi2d 229 . . . 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 8941 . . . . . . . . 9  |-  ( M  e.  NN0  ->  M  e.  CC )
2625addid1d 7875 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( M  +  0 )  =  M )
2726adantl 273 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
2827oveq2d 5756 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( A ^ M ) )
29 expcl 10262 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
3029mulid1d 7747 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  1 )  =  ( A ^ M ) )
3128, 30eqtr4d 2151 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  1 ) )
32 exp0 10248 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3332adantr 272 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ 0 )  =  1 )
3433oveq2d 5756 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  ( A ^ 0 ) )  =  ( ( A ^ M )  x.  1 ) )
3531, 34eqtr4d 2151 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  ( A ^
0 ) ) )
36 oveq1 5747 . . . . . . 7  |-  ( ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k
) )  ->  (
( A ^ ( M  +  k )
)  x.  A )  =  ( ( ( A ^ M )  x.  ( A ^
k ) )  x.  A ) )
37 nn0cn 8941 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 7677 . . . . . . . . . . . . 13  |-  1  e.  CC
39 addass 7714 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4038, 39mp3an3 1287 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4125, 37, 40syl2an 285 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4241adantll 465 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4342oveq2d 5756 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
44 simpll 501 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  A  e.  CC )
45 nn0addcl 8966 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  +  k )  e.  NN0 )
4645adantll 465 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  +  k )  e.  NN0 )
47 expp1 10251 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( M  +  k
)  e.  NN0 )  ->  ( A ^ (
( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k ) )  x.  A ) )
4844, 46, 47syl2anc 406 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
4943, 48eqtr3d 2150 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  +  ( k  +  1 ) ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
50 expp1 10251 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5150adantlr 466 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
5251oveq2d 5756 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) )  =  ( ( A ^ M
)  x.  ( ( A ^ k )  x.  A ) ) )
5329adantr 272 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^ M )  e.  CC )
54 expcl 10262 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
5554adantlr 466 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
k )  e.  CC )
5653, 55, 44mulassd 7753 . . . . . . . . 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 2151 . . . . . . . 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 2130 . . . . . . 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 155 . . . . . 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 115 . . . . 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 26 . . . 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 9119 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
6362expdcom 1401 . 2  |-  ( A  e.  CC  ->  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) ) )
64633imp 1158 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 103    /\ w3a 945    = wceq 1314    e. wcel 1463  (class class class)co 5740   CCcc 7582   0cc0 7584   1c1 7585    + caddc 7587    x. cmul 7589   NN0cn0 8931   ^cexp 10243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-seqfrec 10170  df-exp 10244
This theorem is referenced by:  expaddzaplem  10287  expaddzap  10288  expmul  10289  i4  10346  expaddd  10377  ef01bndlem  11373
  Copyright terms: Public domain W3C validator