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Theorem expmul 10538
Description: Product of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
Assertion
Ref Expression
expmul  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )

Proof of Theorem expmul
Dummy variables  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5876 . . . . . . 7  |-  ( j  =  0  ->  ( M  x.  j )  =  ( M  x.  0 ) )
21oveq2d 5884 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  0 ) ) )
3 oveq2 5876 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
0 ) )
42, 3eqeq12d 2192 . . . . 5  |-  ( j  =  0  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) ) )
54imbi2d 230 . . . 4  |-  ( j  =  0  ->  (
( ( A  e.  CC  /\  M  e. 
NN0 )  ->  ( A ^ ( M  x.  j ) )  =  ( ( A ^ M ) ^ j
) )  <->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) ) ) )
6 oveq2 5876 . . . . . . 7  |-  ( j  =  k  ->  ( M  x.  j )  =  ( M  x.  k ) )
76oveq2d 5884 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  k )
) )
8 oveq2 5876 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
k ) )
97, 8eqeq12d 2192 . . . . 5  |-  ( j  =  k  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^ k
) ) )
109imbi2d 230 . . . 4  |-  ( j  =  k  ->  (
( ( A  e.  CC  /\  M  e. 
NN0 )  ->  ( A ^ ( M  x.  j ) )  =  ( ( A ^ M ) ^ j
) )  <->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^ k
) ) ) )
11 oveq2 5876 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  x.  j )  =  ( M  x.  ( k  +  1 ) ) )
1211oveq2d 5884 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  ( k  +  1 ) ) ) )
13 oveq2 5876 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2192 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  ( k  +  1 ) ) )  =  ( ( A ^ M ) ^ (
k  +  1 ) ) ) )
1514imbi2d 230 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( A  e.  CC  /\  M  e. 
NN0 )  ->  ( A ^ ( M  x.  j ) )  =  ( ( A ^ M ) ^ j
) )  <->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  x.  ( k  +  1 ) ) )  =  ( ( A ^ M ) ^ (
k  +  1 ) ) ) ) )
16 oveq2 5876 . . . . . . 7  |-  ( j  =  N  ->  ( M  x.  j )  =  ( M  x.  N ) )
1716oveq2d 5884 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  N )
) )
18 oveq2 5876 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^ N ) )
1917, 18eqeq12d 2192 . . . . 5  |-  ( j  =  N  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
2019imbi2d 230 . . . 4  |-  ( j  =  N  ->  (
( ( A  e.  CC  /\  M  e. 
NN0 )  ->  ( A ^ ( M  x.  j ) )  =  ( ( A ^ M ) ^ j
) )  <->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) ) )
21 nn0cn 9162 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  CC )
2221mul01d 8327 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  x.  0 )  =  0 )
2322oveq2d 5884 . . . . . 6  |-  ( M  e.  NN0  ->  ( A ^ ( M  x.  0 ) )  =  ( A ^ 0 ) )
24 exp0 10497 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2523, 24sylan9eqr 2232 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  1 )
26 expcl 10511 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
27 exp0 10497 . . . . . 6  |-  ( ( A ^ M )  e.  CC  ->  (
( A ^ M
) ^ 0 )  =  1 )
2826, 27syl 14 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M ) ^ 0 )  =  1 )
2925, 28eqtr4d 2213 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) )
30 oveq1 5875 . . . . . . 7  |-  ( ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^
k )  ->  (
( A ^ ( M  x.  k )
)  x.  ( A ^ M ) )  =  ( ( ( A ^ M ) ^ k )  x.  ( A ^ M
) ) )
31 nn0cn 9162 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
32 ax-1cn 7882 . . . . . . . . . . . . . 14  |-  1  e.  CC
33 adddi 7921 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( M  x.  ( k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
3432, 33mp3an3 1326 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
35 mulid1 7932 . . . . . . . . . . . . . . 15  |-  ( M  e.  CC  ->  ( M  x.  1 )  =  M )
3635adantr 276 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  1 )  =  M )
3736oveq2d 5884 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  x.  k )  +  ( M  x.  1 ) )  =  ( ( M  x.  k )  +  M ) )
3834, 37eqtrd 2210 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  M ) )
3921, 31, 38syl2an 289 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  M ) )
4039adantll 476 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  x.  ( k  +  1 ) )  =  ( ( M  x.  k
)  +  M ) )
4140oveq2d 5884 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( A ^ ( ( M  x.  k )  +  M ) ) )
42 simpll 527 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  A  e.  CC )
43 nn0mulcl 9188 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  x.  k
)  e.  NN0 )
4443adantll 476 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  x.  k )  e.  NN0 )
45 simplr 528 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  M  e.  NN0 )
46 expadd 10535 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( M  x.  k
)  e.  NN0  /\  M  e.  NN0 )  -> 
( A ^ (
( M  x.  k
)  +  M ) )  =  ( ( A ^ ( M  x.  k ) )  x.  ( A ^ M ) ) )
4742, 44, 45, 46syl3anc 1238 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  x.  k )  +  M
) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
4841, 47eqtrd 2210 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
49 expp1 10500 . . . . . . . . 9  |-  ( ( ( A ^ M
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^ M ) ^ (
k  +  1 ) )  =  ( ( ( A ^ M
) ^ k )  x.  ( A ^ M ) ) )
5026, 49sylan 283 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M ) ^
( k  +  1 ) )  =  ( ( ( A ^ M ) ^ k
)  x.  ( A ^ M ) ) )
5148, 50eqeq12d 2192 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ ( M  x.  ( k  +  1 ) ) )  =  ( ( A ^ M ) ^ (
k  +  1 ) )  <->  ( ( A ^ ( M  x.  k ) )  x.  ( A ^ M
) )  =  ( ( ( A ^ M ) ^ k
)  x.  ( A ^ M ) ) ) )
5230, 51syl5ibr 156 . . . . . 6  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^ k
)  ->  ( A ^ ( M  x.  ( k  +  1 ) ) )  =  ( ( A ^ M ) ^ (
k  +  1 ) ) ) )
5352expcom 116 . . . . 5  |-  ( k  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^
( M  x.  k
) )  =  ( ( A ^ M
) ^ k )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( ( A ^ M
) ^ ( k  +  1 ) ) ) ) )
5453a2d 26 . . . 4  |-  ( k  e.  NN0  ->  ( ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  x.  k )
)  =  ( ( A ^ M ) ^ k ) )  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  ( A ^ ( M  x.  ( k  +  1 ) ) )  =  ( ( A ^ M ) ^ (
k  +  1 ) ) ) ) )
555, 10, 15, 20, 29, 54nn0ind 9343 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
5655expdcom 1442 . 2  |-  ( A  e.  CC  ->  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) ) )
57563imp 1193 1  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148  (class class class)co 5868   CCcc 7787   0cc0 7789   1c1 7790    + caddc 7792    x. cmul 7794   NN0cn0 9152   ^cexp 10492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-n0 9153  df-z 9230  df-uz 9505  df-seqfrec 10419  df-exp 10493
This theorem is referenced by:  expmulzap  10539  expnass  10598  expmuld  10629
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