ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mulexp Unicode version

Theorem mulexp 9664
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 5571 . . . . . 6  |-  ( j  =  0  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
0 ) )
2 oveq2 5571 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
3 oveq2 5571 . . . . . . 7  |-  ( j  =  0  ->  ( B ^ j )  =  ( B ^ 0 ) )
42, 3oveq12d 5581 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) )
51, 4eqeq12d 2097 . . . . 5  |-  ( j  =  0  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ 0 )  =  ( ( A ^
0 )  x.  ( B ^ 0 ) ) ) )
65imbi2d 228 . . . 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 5571 . . . . . 6  |-  ( j  =  k  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
k ) )
8 oveq2 5571 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
9 oveq2 5571 . . . . . . 7  |-  ( j  =  k  ->  ( B ^ j )  =  ( B ^ k
) )
108, 9oveq12d 5581 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ k )  x.  ( B ^ k
) ) )
117, 10eqeq12d 2097 . . . . 5  |-  ( j  =  k  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ k )  =  ( ( A ^
k )  x.  ( B ^ k ) ) ) )
1211imbi2d 228 . . . 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 5571 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
( k  +  1 ) ) )
14 oveq2 5571 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
15 oveq2 5571 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( B ^ j )  =  ( B ^ (
k  +  1 ) ) )
1614, 15oveq12d 5581 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ ( k  +  1 ) )  x.  ( B ^ (
k  +  1 ) ) ) )
1713, 16eqeq12d 2097 . . . . 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 228 . . . 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 5571 . . . . . 6  |-  ( j  =  N  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^ N ) )
20 oveq2 5571 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
21 oveq2 5571 . . . . . . 7  |-  ( j  =  N  ->  ( B ^ j )  =  ( B ^ N
) )
2220, 21oveq12d 5581 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
2319, 22eqeq12d 2097 . . . . 5  |-  ( j  =  N  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N
)  x.  ( B ^ N ) ) ) )
2423imbi2d 228 . . . 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 7214 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
26 exp0 9629 . . . . . 6  |-  ( ( A  x.  B )  e.  CC  ->  (
( A  x.  B
) ^ 0 )  =  1 )
2725, 26syl 14 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ 0 )  =  1 )
28 exp0 9629 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
29 exp0 9629 . . . . . . 7  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3028, 29oveqan12d 5582 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  ( 1  x.  1 ) )
31 1t1e1 8303 . . . . . 6  |-  ( 1  x.  1 )  =  1
3230, 31syl6eq 2131 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  1 )
3327, 32eqtr4d 2118 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ 0 )  =  ( ( A ^ 0 )  x.  ( B ^
0 ) ) )
34 expp1 9632 . . . . . . . . . 10  |-  ( ( ( A  x.  B
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( A  x.  B ) ^ (
k  +  1 ) )  =  ( ( ( A  x.  B
) ^ k )  x.  ( A  x.  B ) ) )
3525, 34sylan 277 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A  x.  B ) ^
( k  +  1 ) )  =  ( ( ( A  x.  B ) ^ k
)  x.  ( A  x.  B ) ) )
3635adantr 270 . . . . . . . 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 5570 . . . . . . . . 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 9643 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
39 expcl 9643 . . . . . . . . . . . . 13  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  CC )
4038, 39anim12i 331 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( B  e.  CC  /\  k  e.  NN0 )
)  ->  ( ( A ^ k )  e.  CC  /\  ( B ^ k )  e.  CC ) )
4140anandirs 558 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A ^ k )  e.  CC  /\  ( B ^ k )  e.  CC ) )
42 simpl 107 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A  e.  CC  /\  B  e.  CC ) )
43 mul4 7359 . . . . . . . . . . 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 403 . . . . . . . . . 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 9632 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
4645adantlr 461 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
47 expp1 9632 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
4847adantll 460 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( B ^
( k  +  1 ) )  =  ( ( B ^ k
)  x.  B ) )
4946, 48oveq12d 5581 . . . . . . . . . 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 2118 . . . . . . . . 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 2137 . . . . . . . 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 2115 . . . . . . 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 356 . . . . . 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 30 . . . . 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 8594 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) ) )
5756expdcom 1372 . 2  |-  ( A  e.  CC  ->  ( B  e.  CC  ->  ( N  e.  NN0  ->  ( ( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) ) ) )
58573imp 1133 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 102    /\ w3a 920    = wceq 1285    e. wcel 1434  (class class class)co 5563   CCcc 7093   0cc0 7095   1c1 7096    + caddc 7098    x. cmul 7100   NN0cn0 8407   ^cexp 9624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753  df-iseq 9574  df-iexp 9625
This theorem is referenced by:  mulexpzap  9665  expdivap  9676  expubnd  9682  sqmul  9687  mulexpd  9769
  Copyright terms: Public domain W3C validator