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Theorem cxpmul2 23157
Description: Product of exponents law for complex exponentiation. Variation on cxpmul 23156 with more general conditions on  A and  B when  C is an integer. (Contributed by Mario Carneiro, 9-Aug-2014.)
Assertion
Ref Expression
cxpmul2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  NN0 )  ->  ( A  ^c  ( B  x.  C ) )  =  ( ( A  ^c  B ) ^ C ) )

Proof of Theorem cxpmul2
Dummy variables  x  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6204 . . . . . . 7  |-  ( x  =  0  ->  ( B  x.  x )  =  ( B  x.  0 ) )
21oveq2d 6212 . . . . . 6  |-  ( x  =  0  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  0 ) ) )
3 oveq2 6204 . . . . . 6  |-  ( x  =  0  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ 0 ) )
42, 3eqeq12d 2404 . . . . 5  |-  ( x  =  0  ->  (
( A  ^c 
( B  x.  x
) )  =  ( ( A  ^c  B ) ^ x
)  <->  ( A  ^c  ( B  x.  0 ) )  =  ( ( A  ^c  B ) ^ 0 ) ) )
54imbi2d 314 . . . 4  |-  ( x  =  0  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  x ) )  =  ( ( A  ^c  B ) ^ x ) )  <-> 
( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  0 ) )  =  ( ( A  ^c  B ) ^ 0 ) ) ) )
6 oveq2 6204 . . . . . . 7  |-  ( x  =  k  ->  ( B  x.  x )  =  ( B  x.  k ) )
76oveq2d 6212 . . . . . 6  |-  ( x  =  k  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  k ) ) )
8 oveq2 6204 . . . . . 6  |-  ( x  =  k  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ k ) )
97, 8eqeq12d 2404 . . . . 5  |-  ( x  =  k  ->  (
( A  ^c 
( B  x.  x
) )  =  ( ( A  ^c  B ) ^ x
)  <->  ( A  ^c  ( B  x.  k ) )  =  ( ( A  ^c  B ) ^ k
) ) )
109imbi2d 314 . . . 4  |-  ( x  =  k  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  x ) )  =  ( ( A  ^c  B ) ^ x ) )  <-> 
( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  k ) )  =  ( ( A  ^c  B ) ^ k ) ) ) )
11 oveq2 6204 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  x.  x )  =  ( B  x.  ( k  +  1 ) ) )
1211oveq2d 6212 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  ( k  +  1 ) ) ) )
13 oveq2 6204 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ ( k  +  1 ) ) )
1412, 13eqeq12d 2404 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( A  ^c 
( B  x.  x
) )  =  ( ( A  ^c  B ) ^ x
)  <->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( ( A  ^c  B ) ^ (
k  +  1 ) ) ) )
1514imbi2d 314 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  x ) )  =  ( ( A  ^c  B ) ^ x ) )  <-> 
( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( ( A  ^c  B ) ^ ( k  +  1 ) ) ) ) )
16 oveq2 6204 . . . . . . 7  |-  ( x  =  C  ->  ( B  x.  x )  =  ( B  x.  C ) )
1716oveq2d 6212 . . . . . 6  |-  ( x  =  C  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  C ) ) )
18 oveq2 6204 . . . . . 6  |-  ( x  =  C  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ C ) )
1917, 18eqeq12d 2404 . . . . 5  |-  ( x  =  C  ->  (
( A  ^c 
( B  x.  x
) )  =  ( ( A  ^c  B ) ^ x
)  <->  ( A  ^c  ( B  x.  C ) )  =  ( ( A  ^c  B ) ^ C
) ) )
2019imbi2d 314 . . . 4  |-  ( x  =  C  ->  (
( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  x ) )  =  ( ( A  ^c  B ) ^ x ) )  <-> 
( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  C ) )  =  ( ( A  ^c  B ) ^ C ) ) ) )
21 cxp0 23138 . . . . . 6  |-  ( A  e.  CC  ->  ( A  ^c  0 )  =  1 )
2221adantr 463 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
0 )  =  1 )
23 mul01 9670 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  x.  0 )  =  0 )
2423adantl 464 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  0 )  =  0 )
2524oveq2d 6212 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
( B  x.  0 ) )  =  ( A  ^c  0 ) )
26 cxpcl 23142 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  B )  e.  CC )
2726exp0d 12206 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  ^c  B ) ^ 0 )  =  1 )
2822, 25, 273eqtr4d 2433 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
( B  x.  0 ) )  =  ( ( A  ^c  B ) ^ 0 ) )
29 oveq1 6203 . . . . . . 7  |-  ( ( A  ^c  ( B  x.  k ) )  =  ( ( A  ^c  B ) ^ k )  ->  ( ( A  ^c  ( B  x.  k ) )  x.  ( A  ^c  B ) )  =  ( ( ( A  ^c  B ) ^ k )  x.  ( A  ^c  B ) ) )
30 0cn 9499 . . . . . . . . . . . . 13  |-  0  e.  CC
31 cxp0 23138 . . . . . . . . . . . . 13  |-  ( 0  e.  CC  ->  (
0  ^c  0 )  =  1 )
3230, 31ax-mp 5 . . . . . . . . . . . 12  |-  ( 0  ^c  0 )  =  1
33 1t1e1 10600 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
3432, 33eqtr4i 2414 . . . . . . . . . . 11  |-  ( 0  ^c  0 )  =  ( 1  x.  1 )
35 simplr 753 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  A  =  0 )
36 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  B  =  0 )
3736oveq1d 6211 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( 0  x.  ( k  +  1 ) ) )
38 nn0p1nn 10752 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
3938adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  NN )
4039nncnd 10468 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  CC )
4140ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( k  +  1 )  e.  CC )
4241mul02d 9689 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  ( k  +  1 ) )  =  0 )
4337, 42eqtrd 2423 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  0 )
4435, 43oveq12d 6214 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( 0  ^c 
0 ) )
4536oveq1d 6211 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  ( 0  x.  k ) )
46 nn0cn 10722 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  CC )
4746adantl 464 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  k  e.  CC )
4847ad2antrr 723 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  k  e.  CC )
4948mul02d 9689 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  k )  =  0 )
5045, 49eqtrd 2423 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  0 )
5135, 50oveq12d 6214 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  k ) )  =  ( 0  ^c 
0 ) )
5251, 32syl6eq 2439 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  k ) )  =  1 )
5335, 36oveq12d 6214 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  B )  =  ( 0  ^c  0 ) )
5453, 32syl6eq 2439 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  B )  =  1 )
5552, 54oveq12d 6214 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( ( A  ^c  ( B  x.  k ) )  x.  ( A  ^c  B ) )  =  ( 1  x.  1 ) )
5634, 44, 553eqtr4a 2449 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( ( A  ^c  ( B  x.  k ) )  x.  ( A  ^c  B ) ) )
57 simpll 751 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  A  e.  CC )
5857ad2antrr 723 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  A  e.  CC )
59 simplr 753 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  B  e.  CC )
6059, 47mulcld 9527 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( B  x.  k )  e.  CC )
6160ad2antrr 723 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  k
)  e.  CC )
62 cxpcl 23142 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( B  x.  k
)  e.  CC )  ->  ( A  ^c  ( B  x.  k ) )  e.  CC )
6358, 61, 62syl2anc 659 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c 
( B  x.  k
) )  e.  CC )
6463mul01d 9690 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( ( A  ^c  ( B  x.  k ) )  x.  0 )  =  0 )
65 simplr 753 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  A  =  0 )
6665oveq1d 6211 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c  B )  =  ( 0  ^c  B ) )
6759ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  B  e.  CC )
68 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  B  =/=  0 )
69 0cxp 23134 . . . . . . . . . . . . . 14  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 0  ^c  B )  =  0 )
7067, 68, 69syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( 0  ^c  B )  =  0 )
7166, 70eqtrd 2423 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c  B )  =  0 )
7271oveq2d 6212 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( ( A  ^c  ( B  x.  k ) )  x.  ( A  ^c  B ) )  =  ( ( A  ^c  ( B  x.  k ) )  x.  0 ) )
7365oveq1d 6211 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c 
( B  x.  (
k  +  1 ) ) )  =  ( 0  ^c  ( B  x.  ( k  +  1 ) ) ) )
7440ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( k  +  1 )  e.  CC )
7567, 74mulcld 9527 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  e.  CC )
7639nnne0d 10497 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  =/=  0
)
7776ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( k  +  1 )  =/=  0 )
7867, 74, 68, 77mulne0d 10118 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  =/=  0 )
79 0cxp 23134 . . . . . . . . . . . . 13  |-  ( ( ( B  x.  (
k  +  1 ) )  e.  CC  /\  ( B  x.  (
k  +  1 ) )  =/=  0 )  ->  ( 0  ^c  ( B  x.  ( k  +  1 ) ) )  =  0 )
8075, 78, 79syl2anc 659 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( 0  ^c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8173, 80eqtrd 2423 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8264, 72, 813eqtr4rd 2434 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c 
( B  x.  (
k  +  1 ) ) )  =  ( ( A  ^c 
( B  x.  k
) )  x.  ( A  ^c  B ) ) )
8356, 82pm2.61dane 2700 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  -> 
( A  ^c 
( B  x.  (
k  +  1 ) ) )  =  ( ( A  ^c 
( B  x.  k
) )  x.  ( A  ^c  B ) ) )
8459adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  B  e.  CC )
8547adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  k  e.  CC )
86 1cnd 9523 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  1  e.  CC )
8784, 85, 86adddid 9531 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  ( B  x.  1 ) ) )
8884mulid1d 9524 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  1 )  =  B )
8988oveq2d 6212 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  (
( B  x.  k
)  +  ( B  x.  1 ) )  =  ( ( B  x.  k )  +  B ) )
9087, 89eqtrd 2423 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  B ) )
9190oveq2d 6212 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( A  ^c  ( ( B  x.  k )  +  B ) ) )
9257adantr 463 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  A  e.  CC )
93 simpr 459 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  A  =/=  0 )
9460adantr 463 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  k )  e.  CC )
95 cxpadd 23147 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  x.  k )  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( ( B  x.  k )  +  B ) )  =  ( ( A  ^c  ( B  x.  k ) )  x.  ( A  ^c  B ) ) )
9692, 93, 94, 84, 95syl211anc 1232 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( A  ^c  ( ( B  x.  k )  +  B ) )  =  ( ( A  ^c  ( B  x.  k ) )  x.  ( A  ^c  B ) ) )
9791, 96eqtrd 2423 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( ( A  ^c  ( B  x.  k ) )  x.  ( A  ^c  B ) ) )
9883, 97pm2.61dane 2700 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( ( A  ^c  ( B  x.  k ) )  x.  ( A  ^c  B ) ) )
99 expp1 12076 . . . . . . . . 9  |-  ( ( ( A  ^c  B )  e.  CC  /\  k  e.  NN0 )  ->  ( ( A  ^c  B ) ^ (
k  +  1 ) )  =  ( ( ( A  ^c  B ) ^ k
)  x.  ( A  ^c  B ) ) )
10026, 99sylan 469 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A  ^c  B ) ^ ( k  +  1 ) )  =  ( ( ( A  ^c  B ) ^ k )  x.  ( A  ^c  B ) ) )
10198, 100eqeq12d 2404 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( ( A  ^c  B ) ^ ( k  +  1 ) )  <->  ( ( A  ^c  ( B  x.  k ) )  x.  ( A  ^c  B ) )  =  ( ( ( A  ^c  B ) ^ k )  x.  ( A  ^c  B ) ) ) )
10229, 101syl5ibr 221 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A  ^c  ( B  x.  k ) )  =  ( ( A  ^c  B ) ^ k )  -> 
( A  ^c 
( B  x.  (
k  +  1 ) ) )  =  ( ( A  ^c  B ) ^ (
k  +  1 ) ) ) )
103102expcom 433 . . . . 5  |-  ( k  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  ^c  ( B  x.  k ) )  =  ( ( A  ^c  B ) ^ k
)  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( ( A  ^c  B ) ^ (
k  +  1 ) ) ) ) )
104103a2d 26 . . . 4  |-  ( k  e.  NN0  ->  ( ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  k ) )  =  ( ( A  ^c  B ) ^ k
) )  ->  (
( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( ( A  ^c  B ) ^ (
k  +  1 ) ) ) ) )
1055, 10, 15, 20, 28, 104nn0ind 10874 . . 3  |-  ( C  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
( B  x.  C
) )  =  ( ( A  ^c  B ) ^ C
) ) )
106105com12 31 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( C  e.  NN0  ->  ( A  ^c 
( B  x.  C
) )  =  ( ( A  ^c  B ) ^ C
) ) )
1071063impia 1191 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  NN0 )  ->  ( A  ^c  ( B  x.  C ) )  =  ( ( A  ^c  B ) ^ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577  (class class class)co 6196   CCcc 9401   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408   NNcn 10452   NN0cn0 10712   ^cexp 12069    ^c ccxp 23028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ioc 11455  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-fac 12256  df-bc 12283  df-hash 12308  df-shft 12902  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-limsup 13296  df-clim 13313  df-rlim 13314  df-sum 13511  df-ef 13805  df-sin 13807  df-cos 13808  df-pi 13810  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-fbas 18529  df-fg 18530  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-lp 19723  df-perf 19724  df-cn 19814  df-cnp 19815  df-haus 19902  df-tx 20148  df-hmeo 20341  df-fil 20432  df-fm 20524  df-flim 20525  df-flf 20526  df-xms 20908  df-ms 20909  df-tms 20910  df-cncf 21467  df-limc 22355  df-dv 22356  df-log 23029  df-cxp 23030
This theorem is referenced by:  cxproot  23158  cxpmul2z  23159  cxpmul2d  23177
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