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Theorem expadd 11022
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
StepHypRef Expression
1 oveq2 5718 . . . . . . 7  |-  ( j  =  0  ->  ( M  +  j )  =  ( M  + 
0 ) )
21oveq2d 5726 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  0 ) ) )
3 oveq2 5718 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
43oveq2d 5726 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) )
52, 4eqeq12d 2267 . . . . 5  |-  ( j  =  0  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  + 
0 ) )  =  ( ( A ^ M )  x.  ( A ^ 0 ) ) ) )
65imbi2d 309 . . . 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 5718 . . . . . . 7  |-  ( j  =  k  ->  ( M  +  j )  =  ( M  +  k ) )
87oveq2d 5726 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  k )
) )
9 oveq2 5718 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
109oveq2d 5726 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ k
) ) )
118, 10eqeq12d 2267 . . . . 5  |-  ( j  =  k  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k ) ) ) )
1211imbi2d 309 . . . 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 5718 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  +  j )  =  ( M  +  ( k  +  1 ) ) )
1413oveq2d 5726 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
15 oveq2 5718 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1615oveq2d 5726 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) ) )
1714, 16eqeq12d 2267 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  ( k  +  1 ) ) )  =  ( ( A ^ M )  x.  ( A ^ ( k  +  1 ) ) ) ) )
1817imbi2d 309 . . . 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 5718 . . . . . . 7  |-  ( j  =  N  ->  ( M  +  j )  =  ( M  +  N ) )
2019oveq2d 5726 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  +  j ) )  =  ( A ^ ( M  +  N )
) )
21 oveq2 5718 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
2221oveq2d 5726 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
)  x.  ( A ^ j ) )  =  ( ( A ^ M )  x.  ( A ^ N
) ) )
2320, 22eqeq12d 2267 . . . . 5  |-  ( j  =  N  ->  (
( A ^ ( M  +  j )
)  =  ( ( A ^ M )  x.  ( A ^
j ) )  <->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
2423imbi2d 309 . . . 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 9854 . . . . . . . . 9  |-  ( M  e.  NN0  ->  M  e.  CC )
2625addid1d 8892 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( M  +  0 )  =  M )
2726adantl 454 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
2827oveq2d 5726 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( A ^ M ) )
29 expcl 10999 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
3029mulid1d 8732 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  1 )  =  ( A ^ M ) )
3128, 30eqtr4d 2288 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  1 ) )
32 exp0 10986 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3332adantr 453 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ 0 )  =  1 )
3433oveq2d 5726 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  x.  ( A ^ 0 ) )  =  ( ( A ^ M )  x.  1 ) )
3531, 34eqtr4d 2288 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  0 ) )  =  ( ( A ^ M )  x.  ( A ^
0 ) ) )
36 oveq1 5717 . . . . . . 7  |-  ( ( A ^ ( M  +  k ) )  =  ( ( A ^ M )  x.  ( A ^ k
) )  ->  (
( A ^ ( M  +  k )
)  x.  A )  =  ( ( ( A ^ M )  x.  ( A ^
k ) )  x.  A ) )
37 nn0cn 9854 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 8675 . . . . . . . . . . . . 13  |-  1  e.  CC
39 addass 8704 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4038, 39mp3an3 1271 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4125, 37, 40syl2an 465 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4241adantll 697 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
4342oveq2d 5726 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( A ^ ( M  +  ( k  +  1 ) ) ) )
44 simpll 733 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  A  e.  CC )
45 nn0addcl 9878 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  +  k )  e.  NN0 )
4645adantll 697 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  +  k )  e.  NN0 )
47 expp1 10988 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( M  +  k
)  e.  NN0 )  ->  ( A ^ (
( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k ) )  x.  A ) )
4844, 46, 47syl2anc 645 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  +  k )  +  1 ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
4943, 48eqtr3d 2287 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  +  ( k  +  1 ) ) )  =  ( ( A ^ ( M  +  k )
)  x.  A ) )
50 expp1 10988 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5150adantlr 698 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
5251oveq2d 5726 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M )  x.  ( A ^ (
k  +  1 ) ) )  =  ( ( A ^ M
)  x.  ( ( A ^ k )  x.  A ) ) )
5329adantr 453 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^ M )  e.  CC )
54 expcl 10999 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
5554adantlr 698 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
k )  e.  CC )
5653, 55, 44mulassd 8738 . . . . . . . . 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 2288 . . . . . . . 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 2267 . . . . . . 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 214 . . . . . 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 426 . . . . 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 25 . . . 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 9987 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
6362exp3acom3r 1366 . 2  |-  ( A  e.  CC  ->  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) ) )
64633imp 1150 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621  (class class class)co 5710   CCcc 8615   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622   NN0cn0 9844   ^cexp 10982
This theorem is referenced by:  expaddzlem  11023  expaddz  11024  expmul  11025  i4  11083  expaddd  11125  faclbnd4lem1  11184  ef01bndlem  12338  modxai  12957  numexp2x  12968  expmhm  16281  quart1lem  19983  log2ublem2  20075  bposlem8  20362  fsumcube  23969
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-n0 9845  df-z 9904  df-uz 10110  df-seq 10925  df-exp 10983
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