MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulexp Unicode version

Theorem mulexp 11136
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
) ) )
Dummy variables  j 
k are mutually distinct and distinct from all other variables.

Proof of Theorem mulexp
StepHypRef Expression
1 oveq2 5828 . . . . . 6  |-  ( j  =  0  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
0 ) )
2 oveq2 5828 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
3 oveq2 5828 . . . . . . 7  |-  ( j  =  0  ->  ( B ^ j )  =  ( B ^ 0 ) )
42, 3oveq12d 5838 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) )
51, 4eqeq12d 2299 . . . . 5  |-  ( j  =  0  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ 0 )  =  ( ( A ^
0 )  x.  ( B ^ 0 ) ) ) )
65imbi2d 309 . . . 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 5828 . . . . . 6  |-  ( j  =  k  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
k ) )
8 oveq2 5828 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
9 oveq2 5828 . . . . . . 7  |-  ( j  =  k  ->  ( B ^ j )  =  ( B ^ k
) )
108, 9oveq12d 5838 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ k )  x.  ( B ^ k
) ) )
117, 10eqeq12d 2299 . . . . 5  |-  ( j  =  k  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ k )  =  ( ( A ^
k )  x.  ( B ^ k ) ) ) )
1211imbi2d 309 . . . 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 5828 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
( k  +  1 ) ) )
14 oveq2 5828 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
15 oveq2 5828 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( B ^ j )  =  ( B ^ (
k  +  1 ) ) )
1614, 15oveq12d 5838 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ ( k  +  1 ) )  x.  ( B ^ (
k  +  1 ) ) ) )
1713, 16eqeq12d 2299 . . . . 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 309 . . . 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 5828 . . . . . 6  |-  ( j  =  N  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^ N ) )
20 oveq2 5828 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
21 oveq2 5828 . . . . . . 7  |-  ( j  =  N  ->  ( B ^ j )  =  ( B ^ N
) )
2220, 21oveq12d 5838 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
2319, 22eqeq12d 2299 . . . . 5  |-  ( j  =  N  ->  (
( ( A  x.  B ) ^ j
)  =  ( ( A ^ j )  x.  ( B ^
j ) )  <->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N
)  x.  ( B ^ N ) ) ) )
2423imbi2d 309 . . . 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 8817 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
26 exp0 11103 . . . . . 6  |-  ( ( A  x.  B )  e.  CC  ->  (
( A  x.  B
) ^ 0 )  =  1 )
2725, 26syl 17 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ 0 )  =  1 )
28 exp0 11103 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
29 exp0 11103 . . . . . . 7  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3028, 29oveqan12d 5839 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  ( 1  x.  1 ) )
31 1t1e1 9866 . . . . . 6  |-  ( 1  x.  1 )  =  1
3230, 31syl6eq 2333 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  1 )
3327, 32eqtr4d 2320 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ 0 )  =  ( ( A ^ 0 )  x.  ( B ^
0 ) ) )
34 expp1 11105 . . . . . . . . . 10  |-  ( ( ( A  x.  B
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( A  x.  B ) ^ (
k  +  1 ) )  =  ( ( ( A  x.  B
) ^ k )  x.  ( A  x.  B ) ) )
3525, 34sylan 459 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A  x.  B ) ^
( k  +  1 ) )  =  ( ( ( A  x.  B ) ^ k
)  x.  ( A  x.  B ) ) )
3635adantr 453 . . . . . . . 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 5827 . . . . . . . . 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 11116 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
39 expcl 11116 . . . . . . . . . . . . 13  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  CC )
4038, 39anim12i 551 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( B  e.  CC  /\  k  e.  NN0 )
)  ->  ( ( A ^ k )  e.  CC  /\  ( B ^ k )  e.  CC ) )
4140anandirs 806 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A ^ k )  e.  CC  /\  ( B ^ k )  e.  CC ) )
42 simpl 445 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A  e.  CC  /\  B  e.  CC ) )
43 mul4 8977 . . . . . . . . . . 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 644 . . . . . . . . . 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 11105 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
4645adantlr 697 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
47 expp1 11105 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
4847adantll 696 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( B ^
( k  +  1 ) )  =  ( ( B ^ k
)  x.  B ) )
4946, 48oveq12d 5838 . . . . . . . . . 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 2320 . . . . . . . . 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 2339 . . . . . . . 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 2317 . . . . . . 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 589 . . . . . 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 25 . . . 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 10104 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) ) )
5756exp3acom3r 1362 . 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 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685  (class class class)co 5820   CCcc 8731   0cc0 8733   1c1 8734    + caddc 8736    x. cmul 8738   NN0cn0 9961   ^cexp 11099
This theorem is referenced by:  mulexpz  11137  expdiv  11147  expubnd  11157  sqmul  11162  mulexpd  11255  efi4p  12412  logtayl2  20004  ipidsq  21279  stoweidlem1  27150  stoweidlem24  27173
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-n0 9962  df-z 10021  df-uz 10227  df-seq 11042  df-exp 11100
  Copyright terms: Public domain W3C validator