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Theorem mulexp 12169
Description: Positive integer 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 6290 . . . . . 6  |-  ( j  =  0  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
0 ) )
2 oveq2 6290 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
3 oveq2 6290 . . . . . . 7  |-  ( j  =  0  ->  ( B ^ j )  =  ( B ^ 0 ) )
42, 3oveq12d 6300 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) )
51, 4eqeq12d 2489 . . . . 5  |-  ( j  =  0  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ 0 )  =  ( ( A ^
0 )  x.  ( B ^ 0 ) ) ) )
65imbi2d 316 . . . 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 6290 . . . . . 6  |-  ( j  =  k  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
k ) )
8 oveq2 6290 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
9 oveq2 6290 . . . . . . 7  |-  ( j  =  k  ->  ( B ^ j )  =  ( B ^ k
) )
108, 9oveq12d 6300 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ k )  x.  ( B ^ k
) ) )
117, 10eqeq12d 2489 . . . . 5  |-  ( j  =  k  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ k )  =  ( ( A ^
k )  x.  ( B ^ k ) ) ) )
1211imbi2d 316 . . . 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 6290 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
( k  +  1 ) ) )
14 oveq2 6290 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
15 oveq2 6290 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( B ^ j )  =  ( B ^ (
k  +  1 ) ) )
1614, 15oveq12d 6300 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ ( k  +  1 ) )  x.  ( B ^ (
k  +  1 ) ) ) )
1713, 16eqeq12d 2489 . . . . 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 316 . . . 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 6290 . . . . . 6  |-  ( j  =  N  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^ N ) )
20 oveq2 6290 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
21 oveq2 6290 . . . . . . 7  |-  ( j  =  N  ->  ( B ^ j )  =  ( B ^ N
) )
2220, 21oveq12d 6300 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
2319, 22eqeq12d 2489 . . . . 5  |-  ( j  =  N  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N
)  x.  ( B ^ N ) ) ) )
2423imbi2d 316 . . . 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 9572 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
26 exp0 12134 . . . . . 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 12134 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
29 exp0 12134 . . . . . . 7  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3028, 29oveqan12d 6301 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  ( 1  x.  1 ) )
31 1t1e1 10679 . . . . . 6  |-  ( 1  x.  1 )  =  1
3230, 31syl6eq 2524 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  1 )
3327, 32eqtr4d 2511 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ 0 )  =  ( ( A ^ 0 )  x.  ( B ^
0 ) ) )
34 expp1 12137 . . . . . . . . . 10  |-  ( ( ( A  x.  B
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( A  x.  B ) ^ (
k  +  1 ) )  =  ( ( ( A  x.  B
) ^ k )  x.  ( A  x.  B ) ) )
3525, 34sylan 471 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A  x.  B ) ^
( k  +  1 ) )  =  ( ( ( A  x.  B ) ^ k
)  x.  ( A  x.  B ) ) )
3635adantr 465 . . . . . . . 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 6289 . . . . . . . . 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 12148 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
39 expcl 12148 . . . . . . . . . . . . 13  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  CC )
4038, 39anim12i 566 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( B  e.  CC  /\  k  e.  NN0 )
)  ->  ( ( A ^ k )  e.  CC  /\  ( B ^ k )  e.  CC ) )
4140anandirs 829 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A ^ k )  e.  CC  /\  ( B ^ k )  e.  CC ) )
42 simpl 457 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A  e.  CC  /\  B  e.  CC ) )
43 mul4 9744 . . . . . . . . . . 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 661 . . . . . . . . . 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 12137 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
4645adantlr 714 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
47 expp1 12137 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
4847adantll 713 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( B ^
( k  +  1 ) )  =  ( ( B ^ k
)  x.  B ) )
4946, 48oveq12d 6300 . . . . . . . . . 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 2511 . . . . . . . . 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 2530 . . . . . . . 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 2508 . . . . . . 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 604 . . . . . 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 31 . . . . 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 26 . . . 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 10953 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) ) )
5756expdcom 439 . 2  |-  ( A  e.  CC  ->  ( B  e.  CC  ->  ( N  e.  NN0  ->  ( ( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) ) ) )
58573imp 1190 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  (
( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493   NN0cn0 10791   ^cexp 12130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-seq 12072  df-exp 12131
This theorem is referenced by:  mulexpz  12170  expdiv  12180  expubnd  12190  sqmul  12195  mulexpd  12289  efi4p  13729  logtayl2  22771  ipidsq  25299
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