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Theorem cxpmul2 23046
Description: Product of exponents law for complex exponentiation. Variation on cxpmul 23045 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 6289 . . . . . . 7  |-  ( x  =  0  ->  ( B  x.  x )  =  ( B  x.  0 ) )
21oveq2d 6297 . . . . . 6  |-  ( x  =  0  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  0 ) ) )
3 oveq2 6289 . . . . . 6  |-  ( x  =  0  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ 0 ) )
42, 3eqeq12d 2465 . . . . 5  |-  ( x  =  0  ->  (
( A  ^c 
( B  x.  x
) )  =  ( ( A  ^c  B ) ^ x
)  <->  ( A  ^c  ( B  x.  0 ) )  =  ( ( A  ^c  B ) ^ 0 ) ) )
54imbi2d 316 . . . 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 6289 . . . . . . 7  |-  ( x  =  k  ->  ( B  x.  x )  =  ( B  x.  k ) )
76oveq2d 6297 . . . . . 6  |-  ( x  =  k  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  k ) ) )
8 oveq2 6289 . . . . . 6  |-  ( x  =  k  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ k ) )
97, 8eqeq12d 2465 . . . . 5  |-  ( x  =  k  ->  (
( A  ^c 
( B  x.  x
) )  =  ( ( A  ^c  B ) ^ x
)  <->  ( A  ^c  ( B  x.  k ) )  =  ( ( A  ^c  B ) ^ k
) ) )
109imbi2d 316 . . . 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 6289 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  x.  x )  =  ( B  x.  ( k  +  1 ) ) )
1211oveq2d 6297 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  ( k  +  1 ) ) ) )
13 oveq2 6289 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ ( k  +  1 ) ) )
1412, 13eqeq12d 2465 . . . . 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 316 . . . 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 6289 . . . . . . 7  |-  ( x  =  C  ->  ( B  x.  x )  =  ( B  x.  C ) )
1716oveq2d 6297 . . . . . 6  |-  ( x  =  C  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  C ) ) )
18 oveq2 6289 . . . . . 6  |-  ( x  =  C  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ C ) )
1917, 18eqeq12d 2465 . . . . 5  |-  ( x  =  C  ->  (
( A  ^c 
( B  x.  x
) )  =  ( ( A  ^c  B ) ^ x
)  <->  ( A  ^c  ( B  x.  C ) )  =  ( ( A  ^c  B ) ^ C
) ) )
2019imbi2d 316 . . . 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 23027 . . . . . 6  |-  ( A  e.  CC  ->  ( A  ^c  0 )  =  1 )
2221adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
0 )  =  1 )
23 mul01 9762 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  x.  0 )  =  0 )
2423adantl 466 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  0 )  =  0 )
2524oveq2d 6297 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
( B  x.  0 ) )  =  ( A  ^c  0 ) )
26 cxpcl 23031 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  B )  e.  CC )
2726exp0d 12285 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  ^c  B ) ^ 0 )  =  1 )
2822, 25, 273eqtr4d 2494 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
( B  x.  0 ) )  =  ( ( A  ^c  B ) ^ 0 ) )
29 oveq1 6288 . . . . . . 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 9591 . . . . . . . . . . . . 13  |-  0  e.  CC
31 cxp0 23027 . . . . . . . . . . . . 13  |-  ( 0  e.  CC  ->  (
0  ^c  0 )  =  1 )
3230, 31ax-mp 5 . . . . . . . . . . . 12  |-  ( 0  ^c  0 )  =  1
33 1t1e1 10690 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
3432, 33eqtr4i 2475 . . . . . . . . . . 11  |-  ( 0  ^c  0 )  =  ( 1  x.  1 )
35 simplr 755 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  A  =  0 )
36 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  B  =  0 )
3736oveq1d 6296 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( 0  x.  ( k  +  1 ) ) )
38 nn0p1nn 10842 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
3938adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  NN )
4039nncnd 10559 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  CC )
4140ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( k  +  1 )  e.  CC )
4241mul02d 9781 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  ( k  +  1 ) )  =  0 )
4337, 42eqtrd 2484 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  0 )
4435, 43oveq12d 6299 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( 0  ^c 
0 ) )
4536oveq1d 6296 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  ( 0  x.  k ) )
46 nn0cn 10812 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  CC )
4746adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  k  e.  CC )
4847ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  k  e.  CC )
4948mul02d 9781 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  k )  =  0 )
5045, 49eqtrd 2484 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  0 )
5135, 50oveq12d 6299 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  k ) )  =  ( 0  ^c 
0 ) )
5251, 32syl6eq 2500 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  k ) )  =  1 )
5335, 36oveq12d 6299 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  B )  =  ( 0  ^c  0 ) )
5453, 32syl6eq 2500 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  B )  =  1 )
5552, 54oveq12d 6299 . . . . . . . . . . 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 2510 . . . . . . . . . 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 753 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  A  e.  CC )
5857ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  A  e.  CC )
59 simplr 755 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  B  e.  CC )
6059, 47mulcld 9619 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( B  x.  k )  e.  CC )
6160ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  k
)  e.  CC )
62 cxpcl 23031 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( B  x.  k
)  e.  CC )  ->  ( A  ^c  ( B  x.  k ) )  e.  CC )
6358, 61, 62syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c 
( B  x.  k
) )  e.  CC )
6463mul01d 9782 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( ( A  ^c  ( B  x.  k ) )  x.  0 )  =  0 )
65 simplr 755 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  A  =  0 )
6665oveq1d 6296 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c  B )  =  ( 0  ^c  B ) )
6759ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  B  e.  CC )
68 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  B  =/=  0 )
69 0cxp 23023 . . . . . . . . . . . . . 14  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 0  ^c  B )  =  0 )
7067, 68, 69syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( 0  ^c  B )  =  0 )
7166, 70eqtrd 2484 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c  B )  =  0 )
7271oveq2d 6297 . . . . . . . . . . 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 6296 . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( k  +  1 )  e.  CC )
7567, 74mulcld 9619 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  e.  CC )
7639nnne0d 10587 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  ( k  +  1 )  =/=  0
)
7776ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( k  +  1 )  =/=  0 )
7867, 74, 68, 77mulne0d 10208 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  =/=  0 )
79 0cxp 23023 . . . . . . . . . . . . 13  |-  ( ( ( B  x.  (
k  +  1 ) )  e.  CC  /\  ( B  x.  (
k  +  1 ) )  =/=  0 )  ->  ( 0  ^c  ( B  x.  ( k  +  1 ) ) )  =  0 )
8075, 78, 79syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( 0  ^c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8173, 80eqtrd 2484 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8264, 72, 813eqtr4rd 2495 . . . . . . . . . 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 2761 . . . . . . . . 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 465 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  B  e.  CC )
8547adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  k  e.  CC )
86 1cnd 9615 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  1  e.  CC )
8784, 85, 86adddid 9623 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  ( B  x.  1 ) ) )
8884mulid1d 9616 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  1 )  =  B )
8988oveq2d 6297 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  (
( B  x.  k
)  +  ( B  x.  1 ) )  =  ( ( B  x.  k )  +  B ) )
9087, 89eqtrd 2484 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  B ) )
9190oveq2d 6297 . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  A  e.  CC )
93 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  A  =/=  0 )
9460adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  k )  e.  CC )
95 cxpadd 23036 . . . . . . . . . . 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 1235 . . . . . . . . . 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 2484 . . . . . . . . 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 2761 . . . . . . . 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 12154 . . . . . . . . 9  |-  ( ( ( A  ^c  B )  e.  CC  /\  k  e.  NN0 )  ->  ( ( A  ^c  B ) ^ (
k  +  1 ) )  =  ( ( ( A  ^c  B ) ^ k
)  x.  ( A  ^c  B ) ) )
10026, 99sylan 471 . . . . . . . 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 2465 . . . . . . 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 435 . . . . 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 10966 . . 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 1194 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 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   NNcn 10543   NN0cn0 10802   ^cexp 12147    ^c ccxp 22919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-ioo 11543  df-ioc 11544  df-ico 11545  df-icc 11546  df-fz 11683  df-fzo 11806  df-fl 11910  df-mod 11978  df-seq 12089  df-exp 12148  df-fac 12335  df-bc 12362  df-hash 12387  df-shft 12881  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-limsup 13275  df-clim 13292  df-rlim 13293  df-sum 13490  df-ef 13784  df-sin 13786  df-cos 13787  df-pi 13789  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-starv 14693  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-unif 14701  df-hom 14702  df-cco 14703  df-rest 14801  df-topn 14802  df-0g 14820  df-gsum 14821  df-topgen 14822  df-pt 14823  df-prds 14826  df-xrs 14880  df-qtop 14885  df-imas 14886  df-xps 14888  df-mre 14964  df-mrc 14965  df-acs 14967  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-submnd 15945  df-mulg 16038  df-cntz 16333  df-cmn 16778  df-psmet 18389  df-xmet 18390  df-met 18391  df-bl 18392  df-mopn 18393  df-fbas 18394  df-fg 18395  df-cnfld 18399  df-top 19376  df-bases 19378  df-topon 19379  df-topsp 19380  df-cld 19497  df-ntr 19498  df-cls 19499  df-nei 19576  df-lp 19614  df-perf 19615  df-cn 19705  df-cnp 19706  df-haus 19793  df-tx 20040  df-hmeo 20233  df-fil 20324  df-fm 20416  df-flim 20417  df-flf 20418  df-xms 20800  df-ms 20801  df-tms 20802  df-cncf 21359  df-limc 22247  df-dv 22248  df-log 22920  df-cxp 22921
This theorem is referenced by:  cxproot  23047  cxpmul2z  23048  cxpmul2d  23066
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