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Theorem expadd 12190
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 6278 . . . . . . 7  |-  ( j  =  0  ->  ( M  +  j )  =  ( M  + 
0 ) )
21oveq2d 6286 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  0 ) ) )
3 oveq2 6278 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
43oveq2d 6286 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) )
52, 4eqeq12d 2476 . . . . 5  |-  ( j  =  0  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  + 
0 ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) ) )
65imbi2d 314 . . . 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 6278 . . . . . . 7  |-  ( j  =  k  ->  ( M  +  j )  =  ( M  +  k ) )
87oveq2d 6286 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  k )
) )
9 oveq2 6278 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
109oveq2d 6286 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ k
) ) )
118, 10eqeq12d 2476 . . . . 5  |-  ( j  =  k  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k ) ) ) )
1211imbi2d 314 . . . 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 6278 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  +  j )  =  ( M  +  ( k  +  1 ) ) )
1413oveq2d 6286 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
15 oveq2 6278 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1615oveq2d 6286 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) ) )
1714, 16eqeq12d 2476 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M )  x.  ( A ^ ( k  +  1 ) ) ) ) )
1817imbi2d 314 . . . 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 6278 . . . . . . 7  |-  ( j  =  N  ->  ( M  +  j )  =  ( M  +  N ) )
2019oveq2d 6286 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  N )
) )
21 oveq2 6278 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
2221oveq2d 6286 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ N
) ) )
2320, 22eqeq12d 2476 . . . . 5  |-  ( j  =  N  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
2423imbi2d 314 . . . 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 10801 . . . . . . . . 9  |-  ( M  e.  NN0  ->  M  e.  CC )
2625addid1d 9769 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( M  +  0 )  =  M )
2726adantl 464 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
2827oveq2d 6286 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( A ^ M ) )
29 expcl 12166 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
3029mulid1d 9602 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  1 )  =  ( A ^ M ) )
3128, 30eqtr4d 2498 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  1 ) )
32 exp0 12152 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3332adantr 463 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ 0 )  =  1 )
3433oveq2d 6286 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  ( A ^ 0 ) )  =  ( ( A ^ M )  x.  1 ) )
3531, 34eqtr4d 2498 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  ( A ^
0 ) ) )
36 oveq1 6277 . . . . . . 7  |-  ( ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k
) )  ->  (
( A ^ ( M  +  k )
)  x.  A )  =  ( ( ( A ^ M )  x.  ( A ^
k ) )  x.  A ) )
37 nn0cn 10801 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 9539 . . . . . . . . . . . . 13  |-  1  e.  CC
39 addass 9568 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4038, 39mp3an3 1311 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4125, 37, 40syl2an 475 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4241adantll 711 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4342oveq2d 6286 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
44 simpll 751 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  A  e.  CC )
45 nn0addcl 10827 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  +  k )  e.  NN0 )
4645adantll 711 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  +  k )  e.  NN0 )
47 expp1 12155 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( M  +  k
)  e.  NN0 )  ->  ( A ^ (
( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k ) )  x.  A ) )
4844, 46, 47syl2anc 659 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
4943, 48eqtr3d 2497 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  +  ( k  +  1 ) ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
50 expp1 12155 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5150adantlr 712 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
5251oveq2d 6286 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) )  =  ( ( A ^ M
)  x.  ( ( A ^ k )  x.  A ) ) )
5329adantr 463 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^ M )  e.  CC )
54 expcl 12166 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
5554adantlr 712 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
k )  e.  CC )
5653, 55, 44mulassd 9608 . . . . . . . . 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 2498 . . . . . . . 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 2476 . . . . . . 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 221 . . . . . 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 433 . . . . 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 10955 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
6362expdcom 437 . 2  |-  ( A  e.  CC  ->  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) ) )
64633imp 1188 1  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   NN0cn0 10791   ^cexp 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-seq 12090  df-exp 12149
This theorem is referenced by:  expaddzlem  12191  expaddz  12192  expmul  12193  i4  12252  expaddd  12294  faclbnd4lem1  12353  ef01bndlem  14001  modxai  14638  numexp2x  14649  expmhm  18680  quart1lem  23383  log2ublem2  23475  bposlem8  23764  fallrisefac  29388  fsumcube  30050
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