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Theorem expmul 12168
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 6285 . . . . . . 7  |-  ( j  =  0  ->  ( M  x.  j )  =  ( M  x.  0 ) )
21oveq2d 6293 . . . . . 6  |-  ( j  =  0  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  0 ) ) )
3 oveq2 6285 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
0 ) )
42, 3eqeq12d 2484 . . . . 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 6285 . . . . . . 7  |-  ( j  =  k  ->  ( M  x.  j )  =  ( M  x.  k ) )
76oveq2d 6293 . . . . . 6  |-  ( j  =  k  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  k )
) )
8 oveq2 6285 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
k ) )
97, 8eqeq12d 2484 . . . . 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 6285 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( M  x.  j )  =  ( M  x.  ( k  +  1 ) ) )
1211oveq2d 6293 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  ( k  +  1 ) ) ) )
13 oveq2 6285 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2484 . . . . 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 6285 . . . . . . 7  |-  ( j  =  N  ->  ( M  x.  j )  =  ( M  x.  N ) )
1716oveq2d 6293 . . . . . 6  |-  ( j  =  N  ->  ( A ^ ( M  x.  j ) )  =  ( A ^ ( M  x.  N )
) )
18 oveq2 6285 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ M
) ^ j )  =  ( ( A ^ M ) ^ N ) )
1917, 18eqeq12d 2484 . . . . 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 10796 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  CC )
2221mul01d 9769 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  x.  0 )  =  0 )
2322oveq2d 6293 . . . . . 6  |-  ( M  e.  NN0  ->  ( A ^ ( M  x.  0 ) )  =  ( A ^ 0 ) )
24 exp0 12128 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2523, 24sylan9eqr 2525 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  1 )
26 expcl 12142 . . . . . 6  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
27 exp0 12128 . . . . . 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 2506 . . . 4  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ ( M  x.  0 ) )  =  ( ( A ^ M ) ^ 0 ) )
30 oveq1 6284 . . . . . . 7  |-  ( ( A ^ ( M  x.  k ) )  =  ( ( A ^ M ) ^
k )  ->  (
( A ^ ( M  x.  k )
)  x.  ( A ^ M ) )  =  ( ( ( A ^ M ) ^ k )  x.  ( A ^ M
) ) )
31 nn0cn 10796 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
32 ax-1cn 9541 . . . . . . . . . . . . . 14  |-  1  e.  CC
33 adddi 9572 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( M  x.  ( k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
3432, 33mp3an3 1308 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  (
k  +  1 ) )  =  ( ( M  x.  k )  +  ( M  x.  1 ) ) )
35 mulid1 9584 . . . . . . . . . . . . . . 15  |-  ( M  e.  CC  ->  ( M  x.  1 )  =  M )
3635adantr 465 . . . . . . . . . . . . . 14  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( M  x.  1 )  =  M )
3736oveq2d 6293 . . . . . . . . . . . . 13  |-  ( ( M  e.  CC  /\  k  e.  CC )  ->  ( ( M  x.  k )  +  ( M  x.  1 ) )  =  ( ( M  x.  k )  +  M ) )
3834, 37eqtrd 2503 . . . . . . . . . . . 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 6293 . . . . . . . . 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 10823 . . . . . . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  M  e.  NN0 )
46 expadd 12165 . . . . . . . . . 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 1223 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( ( M  x.  k )  +  M
) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
4841, 47eqtrd 2503 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  M  e.  NN0 )  /\  k  e.  NN0 )  ->  ( A ^
( M  x.  (
k  +  1 ) ) )  =  ( ( A ^ ( M  x.  k )
)  x.  ( A ^ M ) ) )
49 expp1 12131 . . . . . . . . 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 2484 . . . . . . 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 10948 . . 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 1185 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 968    = wceq 1374    e. wcel 1762  (class class class)co 6277   CCcc 9481   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488   NN0cn0 10786   ^cexp 12124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-seq 12066  df-exp 12125
This theorem is referenced by:  expmulz  12169  expnass  12230  expmuld  12270  mcubic  22901  quart1  22910  log2cnv  22998  log2ublem2  23001  log2ub  23003  basellem3  23079  bclbnd  23278
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