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Theorem cxpmul2 20541
Description: Product of exponents law for complex exponentiation. Variation on cxpmul 20540 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 6056 . . . . . . 7  |-  ( x  =  0  ->  ( B  x.  x )  =  ( B  x.  0 ) )
21oveq2d 6064 . . . . . 6  |-  ( x  =  0  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  0 ) ) )
3 oveq2 6056 . . . . . 6  |-  ( x  =  0  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ 0 ) )
42, 3eqeq12d 2426 . . . . 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 6056 . . . . . . 7  |-  ( x  =  k  ->  ( B  x.  x )  =  ( B  x.  k ) )
76oveq2d 6064 . . . . . 6  |-  ( x  =  k  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  k ) ) )
8 oveq2 6056 . . . . . 6  |-  ( x  =  k  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ k ) )
97, 8eqeq12d 2426 . . . . 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 6056 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  x.  x )  =  ( B  x.  ( k  +  1 ) ) )
1211oveq2d 6064 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  ( k  +  1 ) ) ) )
13 oveq2 6056 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ ( k  +  1 ) ) )
1412, 13eqeq12d 2426 . . . . 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 6056 . . . . . . 7  |-  ( x  =  C  ->  ( B  x.  x )  =  ( B  x.  C ) )
1716oveq2d 6064 . . . . . 6  |-  ( x  =  C  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  C ) ) )
18 oveq2 6056 . . . . . 6  |-  ( x  =  C  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ C ) )
1917, 18eqeq12d 2426 . . . . 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 20522 . . . . . 6  |-  ( A  e.  CC  ->  ( A  ^ c  0 )  =  1 )
2221adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
0 )  =  1 )
23 mul01 9209 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  x.  0 )  =  0 )
2423adantl 453 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  0 )  =  0 )
2524oveq2d 6064 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  0 ) )  =  ( A  ^ c  0 ) )
26 cxpcl 20526 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  e.  CC )
2726exp0d 11480 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  ^ c  B ) ^ 0 )  =  1 )
2822, 25, 273eqtr4d 2454 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  0 ) )  =  ( ( A  ^ c  B ) ^ 0 ) )
29 oveq1 6055 . . . . . . 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 9048 . . . . . . . . . . . . 13  |-  0  e.  CC
31 cxp0 20522 . . . . . . . . . . . . 13  |-  ( 0  e.  CC  ->  (
0  ^ c  0 )  =  1 )
3230, 31ax-mp 8 . . . . . . . . . . . 12  |-  ( 0  ^ c  0 )  =  1
33 1t1e1 10090 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
3432, 33eqtr4i 2435 . . . . . . . . . . 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 6063 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( 0  x.  ( k  +  1 ) ) )
38 nn0p1nn 10223 . . . . . . . . . . . . . . . . 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 9980 . . . . . . . . . . . . . . 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 9228 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  ( k  +  1 ) )  =  0 )
4337, 42eqtrd 2444 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  0 )
4435, 43oveq12d 6066 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  ( k  +  1 ) ) )  =  ( 0  ^ c 
0 ) )
4536oveq1d 6063 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  ( 0  x.  k ) )
46 nn0cn 10195 . . . . . . . . . . . . . . . . . 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 9228 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  k )  =  0 )
5045, 49eqtrd 2444 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  0 )
5135, 50oveq12d 6066 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  k ) )  =  ( 0  ^ c 
0 ) )
5251, 32syl6eq 2460 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  k ) )  =  1 )
5335, 36oveq12d 6066 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  B )  =  ( 0  ^ c  0 ) )
5453, 32syl6eq 2460 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  B )  =  1 )
5552, 54oveq12d 6066 . . . . . . . . . . 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 2470 . . . . . . . . . 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 9072 . . . . . . . . . . . . . 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 20526 . . . . . . . . . . . . 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 9229 . . . . . . . . . . 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 6063 . . . . . . . . . . . . 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 20518 . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c  B )  =  0 )
7271oveq2d 6064 . . . . . . . . . . 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 6063 . . . . . . . . . . . 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 9072 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  e.  CC )
7639nnne0d 10008 . . . . . . . . . . . . . . 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 9638 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  =/=  0 )
79 0cxp 20518 . . . . . . . . . . . . 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 2444 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8264, 72, 813eqtr4rd 2455 . . . . . . . . . 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 2653 . . . . . . . . 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 9012 . . . . . . . . . . . . . 14  |-  1  e.  CC
8786a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  1  e.  CC )
8884, 85, 87adddid 9076 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  ( B  x.  1 ) ) )
8984mulid1d 9069 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  1 )  =  B )
9089oveq2d 6064 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  (
( B  x.  k
)  +  ( B  x.  1 ) )  =  ( ( B  x.  k )  +  B ) )
9188, 90eqtrd 2444 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  B ) )
9291oveq2d 6064 . . . . . . . . . 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 20531 . . . . . . . . . . 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 2444 . . . . . . . . 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 2653 . . . . . . . 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 11351 . . . . . . . . 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 2426 . . . . . . 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 10330 . . 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 2575  (class class class)co 6048   CCcc 8952   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959   NNcn 9964   NN0cn0 10185   ^cexp 11345    ^ c ccxp 20414
This theorem is referenced by:  cxproot  20542  cxpmul2z  20543  cxpmul2d  20561
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ioc 10885  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633  df-sin 12635  df-cos 12636  df-pi 12638  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-limc 19714  df-dv 19715  df-log 20415  df-cxp 20416
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