MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  expmul Unicode version

Theorem expmul 11384
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
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6052 . . . . . . 7  |-  ( j  =  0  ->  ( M  x.  j )  =  ( M  x.  0 ) )
21oveq2d 6060 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  0 ) ) )
3 oveq2 6052 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
0 ) )
42, 3eqeq12d 2422 . . . . 5  |-  ( j  =  0  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) ) )
54imbi2d 308 . . . 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 6052 . . . . . . 7  |-  ( j  =  k  ->  ( M  x.  j )  =  ( M  x.  k ) )
76oveq2d 6060 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  k )
) )
8 oveq2 6052 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
k ) )
97, 8eqeq12d 2422 . . . . 5  |-  ( j  =  k  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^ k
) ) )
109imbi2d 308 . . . 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 6052 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  x.  j )  =  ( M  x.  ( k  +  1 ) ) )
1211oveq2d 6060 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  ( k  +  1 ) ) ) )
13 oveq2 6052 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2422 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  ( k  +  1 ) ) )  =  ( ( A ^ M ) ^ (
k  +  1 ) ) ) )
1514imbi2d 308 . . . 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 6052 . . . . . . 7  |-  ( j  =  N  ->  ( M  x.  j )  =  ( M  x.  N ) )
1716oveq2d 6060 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  N )
) )
18 oveq2 6052 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^ N ) )
1917, 18eqeq12d 2422 . . . . 5  |-  ( j  =  N  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
2019imbi2d 308 . . . 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 10191 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  CC )
2221mul01d 9225 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  x.  0 )  =  0 )
2322oveq2d 6060 . . . . . 6  |-  ( M  e.  NN0  ->  ( A ^ ( M  x.  0 ) )  =  ( A ^ 0 ) )
24 exp0 11345 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2523, 24sylan9eqr 2462 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  1 )
26 expcl 11358 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
27 exp0 11345 . . . . . 6  |-  ( ( A ^ M )  e.  CC  ->  (
( A ^ M
) ^ 0 )  =  1 )
2826, 27syl 16 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M ) ^ 0 )  =  1 )
2925, 28eqtr4d 2443 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) )
30 oveq1 6051 . . . . . . 7  |-  ( ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^
k )  ->  (
( A ^ ( M  x.  k )
)  x.  ( A ^ M ) )  =  ( ( ( A ^ M ) ^ k )  x.  ( A ^ M
) ) )
31 nn0cn 10191 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
32 ax-1cn 9008 . . . . . . . . . . . . . 14  |-  1  e.  CC
33 adddi 9039 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( M  x.  ( k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
3432, 33mp3an3 1268 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
35 mulid1 9048 . . . . . . . . . . . . . . 15  |-  ( M  e.  CC  ->  ( M  x.  1 )  =  M )
3635adantr 452 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  1 )  =  M )
3736oveq2d 6060 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  x.  k )  +  ( M  x.  1 ) )  =  ( ( M  x.  k )  +  M ) )
3834, 37eqtrd 2440 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  M ) )
3921, 31, 38syl2an 464 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  M ) )
4039adantll 695 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  x.  ( k  +  1 ) )  =  ( ( M  x.  k
)  +  M ) )
4140oveq2d 6060 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( A ^ ( ( M  x.  k )  +  M ) ) )
42 simpll 731 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  A  e.  CC )
43 nn0mulcl 10216 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  x.  k
)  e.  NN0 )
4443adantll 695 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  x.  k )  e.  NN0 )
45 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  M  e.  NN0 )
46 expadd 11381 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  x.  k )  +  M
) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
4841, 47eqtrd 2440 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
49 expp1 11347 . . . . . . . . 9  |-  ( ( ( A ^ M
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^ M ) ^ (
k  +  1 ) )  =  ( ( ( A ^ M
) ^ k )  x.  ( A ^ M ) ) )
5026, 49sylan 458 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M ) ^
( k  +  1 ) )  =  ( ( ( A ^ M ) ^ k
)  x.  ( A ^ M ) ) )
5148, 50eqeq12d 2422 . . . . . . 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 213 . . . . . 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 425 . . . . 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 24 . . . 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 10326 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
5655exp3acom3r 1376 . 2  |-  ( A  e.  CC  ->  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) ) )
57563imp 1147 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721  (class class class)co 6044   CCcc 8948   0cc0 8950   1c1 8951    + caddc 8953    x. cmul 8955   NN0cn0 10181   ^cexp 11341
This theorem is referenced by:  expmulz  11385  expnass  11445  expmuld  11485  mcubic  20644  quart1  20653  log2cnv  20741  log2ublem2  20744  log2ub  20746  basellem3  20822  bclbnd  21021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-n0 10182  df-z 10243  df-uz 10449  df-seq 11283  df-exp 11342
  Copyright terms: Public domain W3C validator