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Theorem expmul 12213
Description: Product of exponents law for positive 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 6304 . . . . . . 7  |-  ( j  =  0  ->  ( M  x.  j )  =  ( M  x.  0 ) )
21oveq2d 6312 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  0 ) ) )
3 oveq2 6304 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
0 ) )
42, 3eqeq12d 2479 . . . . 5  |-  ( j  =  0  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) ) )
54imbi2d 316 . . . 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 6304 . . . . . . 7  |-  ( j  =  k  ->  ( M  x.  j )  =  ( M  x.  k ) )
76oveq2d 6312 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  k )
) )
8 oveq2 6304 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
k ) )
97, 8eqeq12d 2479 . . . . 5  |-  ( j  =  k  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^ k
) ) )
109imbi2d 316 . . . 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 6304 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  x.  j )  =  ( M  x.  ( k  +  1 ) ) )
1211oveq2d 6312 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  ( k  +  1 ) ) ) )
13 oveq2 6304 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2479 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  ( k  +  1 ) ) )  =  ( ( A ^ M ) ^ (
k  +  1 ) ) ) )
1514imbi2d 316 . . . 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 6304 . . . . . . 7  |-  ( j  =  N  ->  ( M  x.  j )  =  ( M  x.  N ) )
1716oveq2d 6312 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  N )
) )
18 oveq2 6304 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^ N ) )
1917, 18eqeq12d 2479 . . . . 5  |-  ( j  =  N  ->  (
( A ^ ( M  x.  j )
)  =  ( ( A ^ M ) ^ j )  <->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
2019imbi2d 316 . . . 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 10826 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  CC )
2221mul01d 9796 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  x.  0 )  =  0 )
2322oveq2d 6312 . . . . . 6  |-  ( M  e.  NN0  ->  ( A ^ ( M  x.  0 ) )  =  ( A ^ 0 ) )
24 exp0 12172 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2523, 24sylan9eqr 2520 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  1 )
26 expcl 12186 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
27 exp0 12172 . . . . . 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 2501 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) )
30 oveq1 6303 . . . . . . 7  |-  ( ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^
k )  ->  (
( A ^ ( M  x.  k )
)  x.  ( A ^ M ) )  =  ( ( ( A ^ M ) ^ k )  x.  ( A ^ M
) ) )
31 nn0cn 10826 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
32 ax-1cn 9567 . . . . . . . . . . . . . 14  |-  1  e.  CC
33 adddi 9598 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( M  x.  ( k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
3432, 33mp3an3 1313 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
35 mulid1 9610 . . . . . . . . . . . . . . 15  |-  ( M  e.  CC  ->  ( M  x.  1 )  =  M )
3635adantr 465 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  1 )  =  M )
3736oveq2d 6312 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  x.  k )  +  ( M  x.  1 ) )  =  ( ( M  x.  k )  +  M ) )
3834, 37eqtrd 2498 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  M ) )
3921, 31, 38syl2an 477 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  M ) )
4039adantll 713 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  x.  ( k  +  1 ) )  =  ( ( M  x.  k
)  +  M ) )
4140oveq2d 6312 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( A ^ ( ( M  x.  k )  +  M ) ) )
42 simpll 753 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  A  e.  CC )
43 nn0mulcl 10853 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  x.  k
)  e.  NN0 )
4443adantll 713 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( M  x.  k )  e.  NN0 )
45 simplr 755 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  M  e.  NN0 )
46 expadd 12210 . . . . . . . . . 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 1228 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  x.  k )  +  M
) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
4841, 47eqtrd 2498 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
49 expp1 12175 . . . . . . . . 9  |-  ( ( ( A ^ M
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^ M ) ^ (
k  +  1 ) )  =  ( ( ( A ^ M
) ^ k )  x.  ( A ^ M ) ) )
5026, 49sylan 471 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( A ^ M ) ^
( k  +  1 ) )  =  ( ( ( A ^ M ) ^ k
)  x.  ( A ^ M ) ) )
5148, 50eqeq12d 2479 . . . . . . 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 221 . . . . . 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 435 . . . . 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 10980 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
5655expdcom 439 . 2  |-  ( A  e.  CC  ->  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) ) )
57563imp 1190 1  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   NN0cn0 10816   ^cexp 12168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-seq 12110  df-exp 12169
This theorem is referenced by:  expmulz  12214  expnass  12275  expmuld  12315  mcubic  23303  quart1  23312  log2cnv  23400  log2ublem2  23403  log2ub  23405  basellem3  23481  bclbnd  23680
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