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Theorem cxpmul2 20036
Description: Product of exponents law for complex exponentiation. Variation on cxpmul 20035 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 5866 . . . . . . 7  |-  ( x  =  0  ->  ( B  x.  x )  =  ( B  x.  0 ) )
21oveq2d 5874 . . . . . 6  |-  ( x  =  0  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  0 ) ) )
3 oveq2 5866 . . . . . 6  |-  ( x  =  0  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ 0 ) )
42, 3eqeq12d 2297 . . . . 5  |-  ( x  =  0  ->  (
( A  ^ c 
( B  x.  x
) )  =  ( ( A  ^ c  B ) ^ x
)  <->  ( A  ^ c  ( B  x.  0 ) )  =  ( ( A  ^ c  B ) ^ 0 ) ) )
54imbi2d 307 . . . 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 5866 . . . . . . 7  |-  ( x  =  k  ->  ( B  x.  x )  =  ( B  x.  k ) )
76oveq2d 5874 . . . . . 6  |-  ( x  =  k  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  k ) ) )
8 oveq2 5866 . . . . . 6  |-  ( x  =  k  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ k ) )
97, 8eqeq12d 2297 . . . . 5  |-  ( x  =  k  ->  (
( A  ^ c 
( B  x.  x
) )  =  ( ( A  ^ c  B ) ^ x
)  <->  ( A  ^ c  ( B  x.  k ) )  =  ( ( A  ^ c  B ) ^ k
) ) )
109imbi2d 307 . . . 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 5866 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  x.  x )  =  ( B  x.  ( k  +  1 ) ) )
1211oveq2d 5874 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  ( k  +  1 ) ) ) )
13 oveq2 5866 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ ( k  +  1 ) ) )
1412, 13eqeq12d 2297 . . . . 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 307 . . . 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 5866 . . . . . . 7  |-  ( x  =  C  ->  ( B  x.  x )  =  ( B  x.  C ) )
1716oveq2d 5874 . . . . . 6  |-  ( x  =  C  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  C ) ) )
18 oveq2 5866 . . . . . 6  |-  ( x  =  C  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ C ) )
1917, 18eqeq12d 2297 . . . . 5  |-  ( x  =  C  ->  (
( A  ^ c 
( B  x.  x
) )  =  ( ( A  ^ c  B ) ^ x
)  <->  ( A  ^ c  ( B  x.  C ) )  =  ( ( A  ^ c  B ) ^ C
) ) )
2019imbi2d 307 . . . 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 20017 . . . . . 6  |-  ( A  e.  CC  ->  ( A  ^ c  0 )  =  1 )
2221adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
0 )  =  1 )
23 mul01 8991 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  x.  0 )  =  0 )
2423adantl 452 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  0 )  =  0 )
2524oveq2d 5874 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  0 ) )  =  ( A  ^ c  0 ) )
26 cxpcl 20021 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  e.  CC )
2726exp0d 11239 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  ^ c  B ) ^ 0 )  =  1 )
2822, 25, 273eqtr4d 2325 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  0 ) )  =  ( ( A  ^ c  B ) ^ 0 ) )
29 oveq1 5865 . . . . . . 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 8831 . . . . . . . . . . . . 13  |-  0  e.  CC
31 cxp0 20017 . . . . . . . . . . . . 13  |-  ( 0  e.  CC  ->  (
0  ^ c  0 )  =  1 )
3230, 31ax-mp 8 . . . . . . . . . . . 12  |-  ( 0  ^ c  0 )  =  1
33 1t1e1 9870 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
3432, 33eqtr4i 2306 . . . . . . . . . . 11  |-  ( 0  ^ c  0 )  =  ( 1  x.  1 )
35 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  A  =  0 )
36 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  B  =  0 )
3736oveq1d 5873 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( 0  x.  ( k  +  1 ) ) )
38 nn0p1nn 10003 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
3938adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  NN )
4039nncnd 9762 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  CC )
4140ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( k  +  1 )  e.  CC )
4241mul02d 9010 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  ( k  +  1 ) )  =  0 )
4337, 42eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  0 )
4435, 43oveq12d 5876 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  ( k  +  1 ) ) )  =  ( 0  ^ c 
0 ) )
4536oveq1d 5873 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  ( 0  x.  k ) )
46 nn0cn 9975 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  CC )
4746adantl 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  k  e.  CC )
4847ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  k  e.  CC )
4948mul02d 9010 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  k )  =  0 )
5045, 49eqtrd 2315 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  0 )
5135, 50oveq12d 5876 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  k ) )  =  ( 0  ^ c 
0 ) )
5251, 32syl6eq 2331 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  k ) )  =  1 )
5335, 36oveq12d 5876 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  B )  =  ( 0  ^ c  0 ) )
5453, 32syl6eq 2331 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  B )  =  1 )
5552, 54oveq12d 5876 . . . . . . . . . . 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 2341 . . . . . . . . . 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 730 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  A  e.  CC )
5857ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  A  e.  CC )
59 simplr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  B  e.  CC )
6059, 47mulcld 8855 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( B  x.  k )  e.  CC )
6160ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  k
)  e.  CC )
62 cxpcl 20021 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( B  x.  k
)  e.  CC )  ->  ( A  ^ c  ( B  x.  k ) )  e.  CC )
6358, 61, 62syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c 
( B  x.  k
) )  e.  CC )
6463mul01d 9011 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( ( A  ^ c  ( B  x.  k ) )  x.  0 )  =  0 )
65 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  A  =  0 )
6665oveq1d 5873 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c  B )  =  ( 0  ^ c  B
) )
6759ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  B  e.  CC )
68 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  B  =/=  0 )
69 0cxp 20013 . . . . . . . . . . . . . 14  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 0  ^ c  B )  =  0 )
7067, 68, 69syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( 0  ^ c  B )  =  0 )
7166, 70eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c  B )  =  0 )
7271oveq2d 5874 . . . . . . . . . . 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 5873 . . . . . . . . . . . 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 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( k  +  1 )  e.  CC )
7567, 74mulcld 8855 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  e.  CC )
7639nnne0d 9790 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  =/=  0
)
7776ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( k  +  1 )  =/=  0 )
7867, 74, 68, 77mulne0d 9420 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  =/=  0 )
79 0cxp 20013 . . . . . . . . . . . . 13  |-  ( ( ( B  x.  (
k  +  1 ) )  e.  CC  /\  ( B  x.  (
k  +  1 ) )  =/=  0 )  ->  ( 0  ^ c  ( B  x.  ( k  +  1 ) ) )  =  0 )
8075, 78, 79syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( 0  ^ c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8173, 80eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8264, 72, 813eqtr4rd 2326 . . . . . . . . . 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 2524 . . . . . . . . 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 451 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  B  e.  CC )
8547adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  k  e.  CC )
86 ax-1cn 8795 . . . . . . . . . . . . . 14  |-  1  e.  CC
8786a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  1  e.  CC )
8884, 85, 87adddid 8859 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  ( B  x.  1 ) ) )
8984mulid1d 8852 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  1 )  =  B )
9089oveq2d 5874 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  (
( B  x.  k
)  +  ( B  x.  1 ) )  =  ( ( B  x.  k )  +  B ) )
9188, 90eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  B ) )
9291oveq2d 5874 . . . . . . . . . 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 451 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  A  e.  CC )
94 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  A  =/=  0 )
9560adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  k )  e.  CC )
96 cxpadd 20026 . . . . . . . . . . 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 1188 . . . . . . . . . 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 2315 . . . . . . . . 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 2524 . . . . . . . 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 11110 . . . . . . . . 9  |-  ( ( ( A  ^ c  B )  e.  CC  /\  k  e.  NN0 )  ->  ( ( A  ^ c  B ) ^ (
k  +  1 ) )  =  ( ( ( A  ^ c  B ) ^ k
)  x.  ( A  ^ c  B ) ) )
10126, 100sylan 457 . . . . . . . 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 2297 . . . . . . 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 212 . . . . . 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 424 . . . . 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 23 . . . 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 10108 . . 3  |-  ( C  e.  NN0  ->  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  C
) )  =  ( ( A  ^ c  B ) ^ C
) ) )
107106com12 27 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( C  e.  NN0  ->  ( A  ^ c 
( B  x.  C
) )  =  ( ( A  ^ c  B ) ^ C
) ) )
1081073impia 1148 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   NN0cn0 9965   ^cexp 11104    ^ c ccxp 19913
This theorem is referenced by:  cxproot  20037  cxpmul2z  20038  cxpmul2d  20056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
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