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Theorem cxpmul2 20533
Description: Product of exponents law for complex exponentiation. Variation on cxpmul 20532 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 6048 . . . . . . 7  |-  ( x  =  0  ->  ( B  x.  x )  =  ( B  x.  0 ) )
21oveq2d 6056 . . . . . 6  |-  ( x  =  0  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  0 ) ) )
3 oveq2 6048 . . . . . 6  |-  ( x  =  0  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ 0 ) )
42, 3eqeq12d 2418 . . . . 5  |-  ( x  =  0  ->  (
( A  ^ c 
( B  x.  x
) )  =  ( ( A  ^ c  B ) ^ x
)  <->  ( A  ^ c  ( B  x.  0 ) )  =  ( ( A  ^ c  B ) ^ 0 ) ) )
54imbi2d 308 . . . 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 6048 . . . . . . 7  |-  ( x  =  k  ->  ( B  x.  x )  =  ( B  x.  k ) )
76oveq2d 6056 . . . . . 6  |-  ( x  =  k  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  k ) ) )
8 oveq2 6048 . . . . . 6  |-  ( x  =  k  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ k ) )
97, 8eqeq12d 2418 . . . . 5  |-  ( x  =  k  ->  (
( A  ^ c 
( B  x.  x
) )  =  ( ( A  ^ c  B ) ^ x
)  <->  ( A  ^ c  ( B  x.  k ) )  =  ( ( A  ^ c  B ) ^ k
) ) )
109imbi2d 308 . . . 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 6048 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  x.  x )  =  ( B  x.  ( k  +  1 ) ) )
1211oveq2d 6056 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  ( k  +  1 ) ) ) )
13 oveq2 6048 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ ( k  +  1 ) ) )
1412, 13eqeq12d 2418 . . . . 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 308 . . . 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 6048 . . . . . . 7  |-  ( x  =  C  ->  ( B  x.  x )  =  ( B  x.  C ) )
1716oveq2d 6056 . . . . . 6  |-  ( x  =  C  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  C ) ) )
18 oveq2 6048 . . . . . 6  |-  ( x  =  C  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ C ) )
1917, 18eqeq12d 2418 . . . . 5  |-  ( x  =  C  ->  (
( A  ^ c 
( B  x.  x
) )  =  ( ( A  ^ c  B ) ^ x
)  <->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A  ^ c  B ) ^ C
) ) )
2019imbi2d 308 . . . 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 20514 . . . . . 6  |-  ( A  e.  CC  ->  ( A  ^ c  0 )  =  1 )
2221adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
0 )  =  1 )
23 mul01 9201 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  x.  0 )  =  0 )
2423adantl 453 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  0 )  =  0 )
2524oveq2d 6056 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  0 ) )  =  ( A  ^ c  0 ) )
26 cxpcl 20518 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  e.  CC )
2726exp0d 11472 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  ^ c  B ) ^ 0 )  =  1 )
2822, 25, 273eqtr4d 2446 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  0 ) )  =  ( ( A  ^ c  B ) ^ 0 ) )
29 oveq1 6047 . . . . . . 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 9040 . . . . . . . . . . . . 13  |-  0  e.  CC
31 cxp0 20514 . . . . . . . . . . . . 13  |-  ( 0  e.  CC  ->  (
0  ^ c  0 )  =  1 )
3230, 31ax-mp 8 . . . . . . . . . . . 12  |-  ( 0  ^ c  0 )  =  1
33 1t1e1 10082 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
3432, 33eqtr4i 2427 . . . . . . . . . . 11  |-  ( 0  ^ c  0 )  =  ( 1  x.  1 )
35 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  A  =  0 )
36 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  B  =  0 )
3736oveq1d 6055 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( 0  x.  ( k  +  1 ) ) )
38 nn0p1nn 10215 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
3938adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  NN )
4039nncnd 9972 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  CC )
4140ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( k  +  1 )  e.  CC )
4241mul02d 9220 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  ( k  +  1 ) )  =  0 )
4337, 42eqtrd 2436 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  0 )
4435, 43oveq12d 6058 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  ( k  +  1 ) ) )  =  ( 0  ^ c 
0 ) )
4536oveq1d 6055 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  ( 0  x.  k ) )
46 nn0cn 10187 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  CC )
4746adantl 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  k  e.  CC )
4847ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  k  e.  CC )
4948mul02d 9220 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  k )  =  0 )
5045, 49eqtrd 2436 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  0 )
5135, 50oveq12d 6058 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  k ) )  =  ( 0  ^ c 
0 ) )
5251, 32syl6eq 2452 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  k ) )  =  1 )
5335, 36oveq12d 6058 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  B )  =  ( 0  ^ c  0 ) )
5453, 32syl6eq 2452 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  B )  =  1 )
5552, 54oveq12d 6058 . . . . . . . . . . 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 2462 . . . . . . . . . 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 731 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  A  e.  CC )
5857ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  A  e.  CC )
59 simplr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  B  e.  CC )
6059, 47mulcld 9064 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( B  x.  k )  e.  CC )
6160ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  k
)  e.  CC )
62 cxpcl 20518 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( B  x.  k
)  e.  CC )  ->  ( A  ^ c  ( B  x.  k ) )  e.  CC )
6358, 61, 62syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c 
( B  x.  k
) )  e.  CC )
6463mul01d 9221 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( ( A  ^ c  ( B  x.  k ) )  x.  0 )  =  0 )
65 simplr 732 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  A  =  0 )
6665oveq1d 6055 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c  B )  =  ( 0  ^ c  B
) )
6759ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  B  e.  CC )
68 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  B  =/=  0 )
69 0cxp 20510 . . . . . . . . . . . . . 14  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 0  ^ c  B )  =  0 )
7067, 68, 69syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( 0  ^ c  B )  =  0 )
7166, 70eqtrd 2436 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c  B )  =  0 )
7271oveq2d 6056 . . . . . . . . . . 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 6055 . . . . . . . . . . . 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 707 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( k  +  1 )  e.  CC )
7567, 74mulcld 9064 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  e.  CC )
7639nnne0d 10000 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  =/=  0
)
7776ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( k  +  1 )  =/=  0 )
7867, 74, 68, 77mulne0d 9630 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  =/=  0 )
79 0cxp 20510 . . . . . . . . . . . . 13  |-  ( ( ( B  x.  (
k  +  1 ) )  e.  CC  /\  ( B  x.  (
k  +  1 ) )  =/=  0 )  ->  ( 0  ^ c  ( B  x.  ( k  +  1 ) ) )  =  0 )
8075, 78, 79syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( 0  ^ c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8173, 80eqtrd 2436 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8264, 72, 813eqtr4rd 2447 . . . . . . . . . 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 2645 . . . . . . . . 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 452 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  B  e.  CC )
8547adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  k  e.  CC )
86 ax-1cn 9004 . . . . . . . . . . . . . 14  |-  1  e.  CC
8786a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  1  e.  CC )
8884, 85, 87adddid 9068 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  ( B  x.  1 ) ) )
8984mulid1d 9061 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  1 )  =  B )
9089oveq2d 6056 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  (
( B  x.  k
)  +  ( B  x.  1 ) )  =  ( ( B  x.  k )  +  B ) )
9188, 90eqtrd 2436 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  B ) )
9291oveq2d 6056 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( A  ^ c  ( B  x.  ( k  +  1 ) ) )  =  ( A  ^ c  ( ( B  x.  k )  +  B ) ) )
9357adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  A  e.  CC )
94 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  A  =/=  0 )
9560adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  k )  e.  CC )
96 cxpadd 20523 . . . . . . . . . . 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 ) ) )
9793, 94, 95, 84, 96syl211anc 1190 . . . . . . . . . 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 ) ) )
9892, 97eqtrd 2436 . . . . . . . . 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 ) ) )
9983, 98pm2.61dane 2645 . . . . . . . 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 ) ) )
100 expp1 11343 . . . . . . . . 9  |-  ( ( ( A  ^ c  B )  e.  CC  /\  k  e.  NN0 )  ->  ( ( A  ^ c  B ) ^ (
k  +  1 ) )  =  ( ( ( A  ^ c  B ) ^ k
)  x.  ( A  ^ c  B ) ) )
10126, 100sylan 458 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A  ^ c  B ) ^ ( k  +  1 ) )  =  ( ( ( A  ^ c  B ) ^ k )  x.  ( A  ^ c  B ) ) )
10299, 101eqeq12d 2418 . . . . . . 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 ) ) ) )
10329, 102syl5ibr 213 . . . . . 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 ) ) ) )
104103expcom 425 . . . . 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 ) ) ) ) )
105104a2d 24 . . . 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 ) ) ) ) )
1065, 10, 15, 20, 28, 105nn0ind 10322 . . 3  |-  ( C  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  C
) )  =  ( ( A  ^ c  B ) ^ C
) ) )
107106com12 29 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( C  e.  NN0  ->  ( A  ^ c 
( B  x.  C
) )  =  ( ( A  ^ c  B ) ^ C
) ) )
1081073impia 1150 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  NN0 )  ->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A  ^ c  B ) ^ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   NNcn 9956   NN0cn0 10177   ^cexp 11337    ^ c ccxp 20406
This theorem is referenced by:  cxproot  20534  cxpmul2z  20535  cxpmul2d  20553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408
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