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Theorem expadd 11909
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 6102 . . . . . . 7  |-  ( j  =  0  ->  ( M  +  j )  =  ( M  + 
0 ) )
21oveq2d 6110 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  0 ) ) )
3 oveq2 6102 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
43oveq2d 6110 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) )
52, 4eqeq12d 2457 . . . . 5  |-  ( j  =  0  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  + 
0 ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) ) )
65imbi2d 316 . . . 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 6102 . . . . . . 7  |-  ( j  =  k  ->  ( M  +  j )  =  ( M  +  k ) )
87oveq2d 6110 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  k )
) )
9 oveq2 6102 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
109oveq2d 6110 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ k
) ) )
118, 10eqeq12d 2457 . . . . 5  |-  ( j  =  k  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k ) ) ) )
1211imbi2d 316 . . . 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 6102 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  +  j )  =  ( M  +  ( k  +  1 ) ) )
1413oveq2d 6110 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
15 oveq2 6102 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1615oveq2d 6110 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) ) )
1714, 16eqeq12d 2457 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M )  x.  ( A ^ ( k  +  1 ) ) ) ) )
1817imbi2d 316 . . . 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 6102 . . . . . . 7  |-  ( j  =  N  ->  ( M  +  j )  =  ( M  +  N ) )
2019oveq2d 6110 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  N )
) )
21 oveq2 6102 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
2221oveq2d 6110 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ N
) ) )
2320, 22eqeq12d 2457 . . . . 5  |-  ( j  =  N  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
2423imbi2d 316 . . . 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 10592 . . . . . . . . 9  |-  ( M  e.  NN0  ->  M  e.  CC )
2625addid1d 9572 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( M  +  0 )  =  M )
2726adantl 466 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
2827oveq2d 6110 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( A ^ M ) )
29 expcl 11886 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
3029mulid1d 9406 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  1 )  =  ( A ^ M ) )
3128, 30eqtr4d 2478 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  1 ) )
32 exp0 11872 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3332adantr 465 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ 0 )  =  1 )
3433oveq2d 6110 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  ( A ^ 0 ) )  =  ( ( A ^ M )  x.  1 ) )
3531, 34eqtr4d 2478 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  ( A ^
0 ) ) )
36 oveq1 6101 . . . . . . 7  |-  ( ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k
) )  ->  (
( A ^ ( M  +  k )
)  x.  A )  =  ( ( ( A ^ M )  x.  ( A ^
k ) )  x.  A ) )
37 nn0cn 10592 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 9343 . . . . . . . . . . . . 13  |-  1  e.  CC
39 addass 9372 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4038, 39mp3an3 1303 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4125, 37, 40syl2an 477 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4241adantll 713 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4342oveq2d 6110 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
44 simpll 753 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  A  e.  CC )
45 nn0addcl 10618 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  +  k )  e.  NN0 )
4645adantll 713 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  +  k )  e.  NN0 )
47 expp1 11875 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( M  +  k
)  e.  NN0 )  ->  ( A ^ (
( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k ) )  x.  A ) )
4844, 46, 47syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
4943, 48eqtr3d 2477 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  +  ( k  +  1 ) ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
50 expp1 11875 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5150adantlr 714 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
5251oveq2d 6110 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) )  =  ( ( A ^ M
)  x.  ( ( A ^ k )  x.  A ) ) )
5329adantr 465 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^ M )  e.  CC )
54 expcl 11886 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
5554adantlr 714 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
k )  e.  CC )
5653, 55, 44mulassd 9412 . . . . . . . . 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 2478 . . . . . . . 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 2457 . . . . . . 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 435 . . . . 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 10741 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
6362expdcom 439 . 2  |-  ( A  e.  CC  ->  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) ) )
64633imp 1181 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756  (class class class)co 6094   CCcc 9283   0cc0 9285   1c1 9286    + caddc 9288    x. cmul 9290   NN0cn0 10582   ^cexp 11868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-n0 10583  df-z 10650  df-uz 10865  df-seq 11810  df-exp 11869
This theorem is referenced by:  expaddzlem  11910  expaddz  11911  expmul  11912  i4  11971  expaddd  12013  faclbnd4lem1  12072  ef01bndlem  13471  modxai  14100  numexp2x  14111  expmhm  17883  quart1lem  22253  log2ublem2  22345  bposlem8  22633  fallrisefac  27531  fsumcube  28206
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