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Theorem mulexp 11419
Description: Natural number exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)
Assertion
Ref Expression
mulexp  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )

Proof of Theorem mulexp
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6089 . . . . . 6  |-  ( j  =  0  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
0 ) )
2 oveq2 6089 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
3 oveq2 6089 . . . . . . 7  |-  ( j  =  0  ->  ( B ^ j )  =  ( B ^ 0 ) )
42, 3oveq12d 6099 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) )
51, 4eqeq12d 2450 . . . . 5  |-  ( j  =  0  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ 0 )  =  ( ( A ^
0 )  x.  ( B ^ 0 ) ) ) )
65imbi2d 308 . . . 4  |-  ( j  =  0  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) ^ j )  =  ( ( A ^ j )  x.  ( B ^ j
) ) )  <->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) ^ 0 )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) ) ) )
7 oveq2 6089 . . . . . 6  |-  ( j  =  k  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
k ) )
8 oveq2 6089 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
9 oveq2 6089 . . . . . . 7  |-  ( j  =  k  ->  ( B ^ j )  =  ( B ^ k
) )
108, 9oveq12d 6099 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ k )  x.  ( B ^ k
) ) )
117, 10eqeq12d 2450 . . . . 5  |-  ( j  =  k  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ k )  =  ( ( A ^
k )  x.  ( B ^ k ) ) ) )
1211imbi2d 308 . . . 4  |-  ( j  =  k  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) ^ j )  =  ( ( A ^ j )  x.  ( B ^ j
) ) )  <->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) ^ k )  =  ( ( A ^ k )  x.  ( B ^ k
) ) ) ) )
13 oveq2 6089 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
( k  +  1 ) ) )
14 oveq2 6089 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
15 oveq2 6089 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( B ^ j )  =  ( B ^ (
k  +  1 ) ) )
1614, 15oveq12d 6099 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ ( k  +  1 ) )  x.  ( B ^ (
k  +  1 ) ) ) )
1713, 16eqeq12d 2450 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ ( k  +  1 ) )  =  ( ( A ^
( k  +  1 ) )  x.  ( B ^ ( k  +  1 ) ) ) ) )
1817imbi2d 308 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) ^ j )  =  ( ( A ^ j )  x.  ( B ^ j
) ) )  <->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) ^ ( k  +  1 ) )  =  ( ( A ^ ( k  +  1 ) )  x.  ( B ^ (
k  +  1 ) ) ) ) ) )
19 oveq2 6089 . . . . . 6  |-  ( j  =  N  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^ N ) )
20 oveq2 6089 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
21 oveq2 6089 . . . . . . 7  |-  ( j  =  N  ->  ( B ^ j )  =  ( B ^ N
) )
2220, 21oveq12d 6099 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
2319, 22eqeq12d 2450 . . . . 5  |-  ( j  =  N  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N
)  x.  ( B ^ N ) ) ) )
2423imbi2d 308 . . . 4  |-  ( j  =  N  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) ^ j )  =  ( ( A ^ j )  x.  ( B ^ j
) ) )  <->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) ) ) )
25 mulcl 9074 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
26 exp0 11386 . . . . . 6  |-  ( ( A  x.  B )  e.  CC  ->  (
( A  x.  B
) ^ 0 )  =  1 )
2725, 26syl 16 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ 0 )  =  1 )
28 exp0 11386 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
29 exp0 11386 . . . . . . 7  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3028, 29oveqan12d 6100 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  ( 1  x.  1 ) )
31 1t1e1 10126 . . . . . 6  |-  ( 1  x.  1 )  =  1
3230, 31syl6eq 2484 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  1 )
3327, 32eqtr4d 2471 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ 0 )  =  ( ( A ^ 0 )  x.  ( B ^
0 ) ) )
34 expp1 11388 . . . . . . . . . 10  |-  ( ( ( A  x.  B
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( A  x.  B ) ^ (
k  +  1 ) )  =  ( ( ( A  x.  B
) ^ k )  x.  ( A  x.  B ) ) )
3525, 34sylan 458 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A  x.  B ) ^
( k  +  1 ) )  =  ( ( ( A  x.  B ) ^ k
)  x.  ( A  x.  B ) ) )
3635adantr 452 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  (
( A  x.  B
) ^ k )  =  ( ( A ^ k )  x.  ( B ^ k
) ) )  -> 
( ( A  x.  B ) ^ (
k  +  1 ) )  =  ( ( ( A  x.  B
) ^ k )  x.  ( A  x.  B ) ) )
37 oveq1 6088 . . . . . . . . 9  |-  ( ( ( A  x.  B
) ^ k )  =  ( ( A ^ k )  x.  ( B ^ k
) )  ->  (
( ( A  x.  B ) ^ k
)  x.  ( A  x.  B ) )  =  ( ( ( A ^ k )  x.  ( B ^
k ) )  x.  ( A  x.  B
) ) )
38 expcl 11399 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
39 expcl 11399 . . . . . . . . . . . . 13  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  CC )
4038, 39anim12i 550 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( B  e.  CC  /\  k  e.  NN0 )
)  ->  ( ( A ^ k )  e.  CC  /\  ( B ^ k )  e.  CC ) )
4140anandirs 805 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A ^ k )  e.  CC  /\  ( B ^ k )  e.  CC ) )
42 simpl 444 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A  e.  CC  /\  B  e.  CC ) )
43 mul4 9235 . . . . . . . . . . 11  |-  ( ( ( ( A ^
k )  e.  CC  /\  ( B ^ k
)  e.  CC )  /\  ( A  e.  CC  /\  B  e.  CC ) )  -> 
( ( ( A ^ k )  x.  ( B ^ k
) )  x.  ( A  x.  B )
)  =  ( ( ( A ^ k
)  x.  A )  x.  ( ( B ^ k )  x.  B ) ) )
4441, 42, 43syl2anc 643 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( ( A ^ k )  x.  ( B ^
k ) )  x.  ( A  x.  B
) )  =  ( ( ( A ^
k )  x.  A
)  x.  ( ( B ^ k )  x.  B ) ) )
45 expp1 11388 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
4645adantlr 696 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
47 expp1 11388 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
4847adantll 695 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( B ^
( k  +  1 ) )  =  ( ( B ^ k
)  x.  B ) )
4946, 48oveq12d 6099 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A ^ ( k  +  1 ) )  x.  ( B ^ (
k  +  1 ) ) )  =  ( ( ( A ^
k )  x.  A
)  x.  ( ( B ^ k )  x.  B ) ) )
5044, 49eqtr4d 2471 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( ( A ^ k )  x.  ( B ^
k ) )  x.  ( A  x.  B
) )  =  ( ( A ^ (
k  +  1 ) )  x.  ( B ^ ( k  +  1 ) ) ) )
5137, 50sylan9eqr 2490 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  (
( A  x.  B
) ^ k )  =  ( ( A ^ k )  x.  ( B ^ k
) ) )  -> 
( ( ( A  x.  B ) ^
k )  x.  ( A  x.  B )
)  =  ( ( A ^ ( k  +  1 ) )  x.  ( B ^
( k  +  1 ) ) ) )
5236, 51eqtrd 2468 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  (
( A  x.  B
) ^ k )  =  ( ( A ^ k )  x.  ( B ^ k
) ) )  -> 
( ( A  x.  B ) ^ (
k  +  1 ) )  =  ( ( A ^ ( k  +  1 ) )  x.  ( B ^
( k  +  1 ) ) ) )
5352exp31 588 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( k  e.  NN0  ->  ( ( ( A  x.  B ) ^
k )  =  ( ( A ^ k
)  x.  ( B ^ k ) )  ->  ( ( A  x.  B ) ^
( k  +  1 ) )  =  ( ( A ^ (
k  +  1 ) )  x.  ( B ^ ( k  +  1 ) ) ) ) ) )
5453com12 29 . . . . 5  |-  ( k  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B ) ^
k )  =  ( ( A ^ k
)  x.  ( B ^ k ) )  ->  ( ( A  x.  B ) ^
( k  +  1 ) )  =  ( ( A ^ (
k  +  1 ) )  x.  ( B ^ ( k  +  1 ) ) ) ) ) )
5554a2d 24 . . . 4  |-  ( k  e.  NN0  ->  ( ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^
k )  =  ( ( A ^ k
)  x.  ( B ^ k ) ) )  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) ^ ( k  +  1 ) )  =  ( ( A ^ ( k  +  1 ) )  x.  ( B ^ (
k  +  1 ) ) ) ) ) )
566, 12, 18, 24, 33, 55nn0ind 10366 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) ) )
5756exp3acom3r 1379 . 2  |-  ( A  e.  CC  ->  ( B  e.  CC  ->  ( N  e.  NN0  ->  ( ( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) ) ) )
58573imp 1147 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995   NN0cn0 10221   ^cexp 11382
This theorem is referenced by:  mulexpz  11420  expdiv  11430  expubnd  11440  sqmul  11445  mulexpd  11538  efi4p  12738  logtayl2  20553  ipidsq  22209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324  df-exp 11383
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