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Theorem mulexp 11107
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
StepHypRef Expression
1 oveq2 5800 . . . . . 6  |-  ( j  =  0  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
0 ) )
2 oveq2 5800 . . . . . . 7  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
3 oveq2 5800 . . . . . . 7  |-  ( j  =  0  ->  ( B ^ j )  =  ( B ^ 0 ) )
42, 3oveq12d 5810 . . . . . 6  |-  ( j  =  0  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ 0 )  x.  ( B ^ 0 ) ) )
51, 4eqeq12d 2272 . . . . 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 5800 . . . . . 6  |-  ( j  =  k  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
k ) )
8 oveq2 5800 . . . . . . 7  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
9 oveq2 5800 . . . . . . 7  |-  ( j  =  k  ->  ( B ^ j )  =  ( B ^ k
) )
108, 9oveq12d 5810 . . . . . 6  |-  ( j  =  k  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ k )  x.  ( B ^ k
) ) )
117, 10eqeq12d 2272 . . . . 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 5800 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^
( k  +  1 ) ) )
14 oveq2 5800 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
15 oveq2 5800 . . . . . . 7  |-  ( j  =  ( k  +  1 )  ->  ( B ^ j )  =  ( B ^ (
k  +  1 ) ) )
1614, 15oveq12d 5810 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ ( k  +  1 ) )  x.  ( B ^ (
k  +  1 ) ) ) )
1713, 16eqeq12d 2272 . . . . 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 5800 . . . . . 6  |-  ( j  =  N  ->  (
( A  x.  B
) ^ j )  =  ( ( A  x.  B ) ^ N ) )
20 oveq2 5800 . . . . . . 7  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
21 oveq2 5800 . . . . . . 7  |-  ( j  =  N  ->  ( B ^ j )  =  ( B ^ N
) )
2220, 21oveq12d 5810 . . . . . 6  |-  ( j  =  N  ->  (
( A ^ j
)  x.  ( B ^ j ) )  =  ( ( A ^ N )  x.  ( B ^ N
) ) )
2319, 22eqeq12d 2272 . . . . 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 8789 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
26 exp0 11074 . . . . . 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 11074 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
29 exp0 11074 . . . . . . 7  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3028, 29oveqan12d 5811 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  ( 1  x.  1 ) )
31 1t1e1 9837 . . . . . 6  |-  ( 1  x.  1 )  =  1
3230, 31syl6eq 2306 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
0 )  x.  ( B ^ 0 ) )  =  1 )
3327, 32eqtr4d 2293 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ 0 )  =  ( ( A ^ 0 )  x.  ( B ^
0 ) ) )
34 expp1 11076 . . . . . . . . . 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 5799 . . . . . . . . 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 11087 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
39 expcl 11087 . . . . . . . . . . . . 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 807 . . . . . . . . . . 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 8949 . . . . . . . . . . 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 645 . . . . . . . . . 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 11076 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
4645adantlr 698 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
47 expp1 11076 . . . . . . . . . . . 12  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
4847adantll 697 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( B ^
( k  +  1 ) )  =  ( ( B ^ k
)  x.  B ) )
4946, 48oveq12d 5810 . . . . . . . . . 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 2293 . . . . . . . . 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 2312 . . . . . . . 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 2290 . . . . . . 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 590 . . . . . 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 10075 . . 3  |-  ( N  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^ N
)  =  ( ( A ^ N )  x.  ( B ^ N ) ) ) )
5756exp3acom3r 1366 . 2  |-  ( A  e.  CC  ->  ( B  e.  CC  ->  ( N  e.  NN0  ->  ( ( A  x.  B
) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N
) ) ) ) )
58573imp 1150 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 939    = wceq 1619    e. wcel 1621  (class class class)co 5792   CCcc 8703   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710   NN0cn0 9932   ^cexp 11070
This theorem is referenced by:  mulexpz  11108  expdiv  11118  expubnd  11128  sqmul  11133  mulexpd  11226  efi4p  12379  logtayl2  19971  ipidsq  21246  stoweidlem1  27085  stoweidlem24  27108
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-n0 9933  df-z 9992  df-uz 10198  df-seq 11013  df-exp 11071
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