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Theorem expmul 11078
Description: Product of exponents law for natural number 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
StepHypRef Expression
1 oveq2 5765 . . . . . . 7  |-  ( j  =  0  ->  ( M  x.  j )  =  ( M  x.  0 ) )
21oveq2d 5773 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  0 ) ) )
3 oveq2 5765 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
0 ) )
42, 3eqeq12d 2270 . . . . 5  |-  ( j  =  0  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) ) )
54imbi2d 309 . . . 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 5765 . . . . . . 7  |-  ( j  =  k  ->  ( M  x.  j )  =  ( M  x.  k ) )
76oveq2d 5773 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  k )
) )
8 oveq2 5765 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
k ) )
97, 8eqeq12d 2270 . . . . 5  |-  ( j  =  k  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^ k
) ) )
109imbi2d 309 . . . 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 5765 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  x.  j )  =  ( M  x.  ( k  +  1 ) ) )
1211oveq2d 5773 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  ( k  +  1 ) ) ) )
13 oveq2 5765 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2270 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  ( k  +  1 ) ) )  =  ( ( A ^ M ) ^ (
k  +  1 ) ) ) )
1514imbi2d 309 . . . 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 5765 . . . . . . 7  |-  ( j  =  N  ->  ( M  x.  j )  =  ( M  x.  N ) )
1716oveq2d 5773 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  N )
) )
18 oveq2 5765 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^ N ) )
1917, 18eqeq12d 2270 . . . . 5  |-  ( j  =  N  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
2019imbi2d 309 . . . 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 9907 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  CC )
2221mul01d 8944 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  x.  0 )  =  0 )
2322oveq2d 5773 . . . . . 6  |-  ( M  e.  NN0  ->  ( A ^ ( M  x.  0 ) )  =  ( A ^ 0 ) )
24 exp0 11039 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2523, 24sylan9eqr 2310 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  1 )
26 expcl 11052 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
27 exp0 11039 . . . . . 6  |-  ( ( A ^ M )  e.  CC  ->  (
( A ^ M
) ^ 0 )  =  1 )
2826, 27syl 17 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M ) ^ 0 )  =  1 )
2925, 28eqtr4d 2291 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) )
30 oveq1 5764 . . . . . . 7  |-  ( ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^
k )  ->  (
( A ^ ( M  x.  k )
)  x.  ( A ^ M ) )  =  ( ( ( A ^ M ) ^ k )  x.  ( A ^ M
) ) )
31 nn0cn 9907 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
32 ax-1cn 8728 . . . . . . . . . . . . . 14  |-  1  e.  CC
33 adddi 8759 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( M  x.  ( k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
3432, 33mp3an3 1271 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
35 mulid1 8767 . . . . . . . . . . . . . . 15  |-  ( M  e.  CC  ->  ( M  x.  1 )  =  M )
3635adantr 453 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  1 )  =  M )
3736oveq2d 5773 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  x.  k )  +  ( M  x.  1 ) )  =  ( ( M  x.  k )  +  M ) )
3834, 37eqtrd 2288 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  M ) )
3921, 31, 38syl2an 465 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  M ) )
4039adantll 697 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  x.  ( k  +  1 ) )  =  ( ( M  x.  k
)  +  M ) )
4140oveq2d 5773 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( A ^ ( ( M  x.  k )  +  M ) ) )
42 simpll 733 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  A  e.  CC )
43 nn0mulcl 9932 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  x.  k
)  e.  NN0 )
4443adantll 697 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  x.  k )  e.  NN0 )
45 simplr 734 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  M  e.  NN0 )
46 expadd 11075 . . . . . . . . . 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 1187 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  x.  k )  +  M
) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
4841, 47eqtrd 2288 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
49 expp1 11041 . . . . . . . . 9  |-  ( ( ( A ^ M
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^ M ) ^ (
k  +  1 ) )  =  ( ( ( A ^ M
) ^ k )  x.  ( A ^ M ) ) )
5026, 49sylan 459 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M ) ^
( k  +  1 ) )  =  ( ( ( A ^ M ) ^ k
)  x.  ( A ^ M ) ) )
5148, 50eqeq12d 2270 . . . . . . 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 214 . . . . . 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 426 . . . . 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 25 . . . 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 10040 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
5655exp3acom3r 1366 . 2  |-  ( A  e.  CC  ->  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) ) )
57563imp 1150 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621  (class class class)co 5757   CCcc 8668   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675   NN0cn0 9897   ^cexp 11035
This theorem is referenced by:  expmulz  11079  expnass  11139  expmuld  11179  mcubic  20070  quart1  20079  log2cnv  20167  log2ublem2  20170  log2ub  20172  basellem3  20247  bclbnd  20446  stoweidlem1  27050
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-n0 9898  df-z 9957  df-uz 10163  df-seq 10978  df-exp 11036
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