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Theorem cxpmul2 22936
Description: Product of exponents law for complex exponentiation. Variation on cxpmul 22935 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 6303 . . . . . . 7  |-  ( x  =  0  ->  ( B  x.  x )  =  ( B  x.  0 ) )
21oveq2d 6311 . . . . . 6  |-  ( x  =  0  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  0 ) ) )
3 oveq2 6303 . . . . . 6  |-  ( x  =  0  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ 0 ) )
42, 3eqeq12d 2489 . . . . 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 6303 . . . . . . 7  |-  ( x  =  k  ->  ( B  x.  x )  =  ( B  x.  k ) )
76oveq2d 6311 . . . . . 6  |-  ( x  =  k  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  k ) ) )
8 oveq2 6303 . . . . . 6  |-  ( x  =  k  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ k ) )
97, 8eqeq12d 2489 . . . . 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 6303 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  x.  x )  =  ( B  x.  ( k  +  1 ) ) )
1211oveq2d 6311 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  ( k  +  1 ) ) ) )
13 oveq2 6303 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ ( k  +  1 ) ) )
1412, 13eqeq12d 2489 . . . . 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 6303 . . . . . . 7  |-  ( x  =  C  ->  ( B  x.  x )  =  ( B  x.  C ) )
1716oveq2d 6311 . . . . . 6  |-  ( x  =  C  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  C ) ) )
18 oveq2 6303 . . . . . 6  |-  ( x  =  C  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ C ) )
1917, 18eqeq12d 2489 . . . . 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 22917 . . . . . 6  |-  ( A  e.  CC  ->  ( A  ^c  0 )  =  1 )
2221adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
0 )  =  1 )
23 mul01 9770 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  x.  0 )  =  0 )
2423adantl 466 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  0 )  =  0 )
2524oveq2d 6311 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
( B  x.  0 ) )  =  ( A  ^c  0 ) )
26 cxpcl 22921 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  B )  e.  CC )
2726exp0d 12284 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  ^c  B ) ^ 0 )  =  1 )
2822, 25, 273eqtr4d 2518 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
( B  x.  0 ) )  =  ( ( A  ^c  B ) ^ 0 ) )
29 oveq1 6302 . . . . . . 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 9600 . . . . . . . . . . . . 13  |-  0  e.  CC
31 cxp0 22917 . . . . . . . . . . . . 13  |-  ( 0  e.  CC  ->  (
0  ^c  0 )  =  1 )
3230, 31ax-mp 5 . . . . . . . . . . . 12  |-  ( 0  ^c  0 )  =  1
33 1t1e1 10695 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
3432, 33eqtr4i 2499 . . . . . . . . . . 11  |-  ( 0  ^c  0 )  =  ( 1  x.  1 )
35 simplr 754 . . . . . . . . . . . 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 6310 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( 0  x.  ( k  +  1 ) ) )
38 nn0p1nn 10847 . . . . . . . . . . . . . . . . 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 10564 . . . . . . . . . . . . . . 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 9789 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  ( k  +  1 ) )  =  0 )
4337, 42eqtrd 2508 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  0 )
4435, 43oveq12d 6313 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( 0  ^c 
0 ) )
4536oveq1d 6310 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  ( 0  x.  k ) )
46 nn0cn 10817 . . . . . . . . . . . . . . . . . 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 9789 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  k )  =  0 )
5045, 49eqtrd 2508 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  0 )
5135, 50oveq12d 6313 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  k ) )  =  ( 0  ^c 
0 ) )
5251, 32syl6eq 2524 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  k ) )  =  1 )
5335, 36oveq12d 6313 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  B )  =  ( 0  ^c  0 ) )
5453, 32syl6eq 2524 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  B )  =  1 )
5552, 54oveq12d 6313 . . . . . . . . . . 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 2534 . . . . . . . . . 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 754 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  ->  B  e.  CC )
6059, 47mulcld 9628 . . . . . . . . . . . . . 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 22921 . . . . . . . . . . . . 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 9790 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( ( A  ^c  ( B  x.  k ) )  x.  0 )  =  0 )
65 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  ->  A  =  0 )
6665oveq1d 6310 . . . . . . . . . . . . 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 22913 . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c  B )  =  0 )
7271oveq2d 6311 . . . . . . . . . . 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 6310 . . . . . . . . . . . 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 9628 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  e.  CC )
7639nnne0d 10592 . . . . . . . . . . . . . . 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 10213 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  =/=  0 )
79 0cxp 22913 . . . . . . . . . . . . 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 2508 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8264, 72, 813eqtr4rd 2519 . . . . . . . . . 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 2785 . . . . . . . . 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 9624 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  1  e.  CC )
8784, 85, 86adddid 9632 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  ( B  x.  1 ) ) )
8884mulid1d 9625 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  1 )  =  B )
8988oveq2d 6311 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  (
( B  x.  k
)  +  ( B  x.  1 ) )  =  ( ( B  x.  k )  +  B ) )
9087, 89eqtrd 2508 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  B ) )
9190oveq2d 6311 . . . . . . . . . 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 22926 . . . . . . . . . . 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 1234 . . . . . . . . . 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 2508 . . . . . . . . 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 2785 . . . . . . . 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 12153 . . . . . . . . 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 2489 . . . . . . 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 10969 . . 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 1193 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509   NNcn 10548   NN0cn0 10807   ^cexp 12146    ^c ccxp 22809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684  df-cos 13685  df-pi 13687  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139  df-log 22810  df-cxp 22811
This theorem is referenced by:  cxproot  22937  cxpmul2z  22938  cxpmul2d  22956
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