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Theorem cxpmul2 20258
Description: Product of exponents law for complex exponentiation. Variation on cxpmul 20257 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 5989 . . . . . . 7  |-  ( x  =  0  ->  ( B  x.  x )  =  ( B  x.  0 ) )
21oveq2d 5997 . . . . . 6  |-  ( x  =  0  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  0 ) ) )
3 oveq2 5989 . . . . . 6  |-  ( x  =  0  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ 0 ) )
42, 3eqeq12d 2380 . . . . 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 5989 . . . . . . 7  |-  ( x  =  k  ->  ( B  x.  x )  =  ( B  x.  k ) )
76oveq2d 5997 . . . . . 6  |-  ( x  =  k  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  k ) ) )
8 oveq2 5989 . . . . . 6  |-  ( x  =  k  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ k ) )
97, 8eqeq12d 2380 . . . . 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 5989 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  x.  x )  =  ( B  x.  ( k  +  1 ) ) )
1211oveq2d 5997 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  ( k  +  1 ) ) ) )
13 oveq2 5989 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ ( k  +  1 ) ) )
1412, 13eqeq12d 2380 . . . . 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 5989 . . . . . . 7  |-  ( x  =  C  ->  ( B  x.  x )  =  ( B  x.  C ) )
1716oveq2d 5997 . . . . . 6  |-  ( x  =  C  ->  ( A  ^ c  ( B  x.  x ) )  =  ( A  ^ c  ( B  x.  C ) ) )
18 oveq2 5989 . . . . . 6  |-  ( x  =  C  ->  (
( A  ^ c  B ) ^ x
)  =  ( ( A  ^ c  B
) ^ C ) )
1917, 18eqeq12d 2380 . . . . 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 20239 . . . . . 6  |-  ( A  e.  CC  ->  ( A  ^ c  0 )  =  1 )
2221adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
0 )  =  1 )
23 mul01 9138 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  x.  0 )  =  0 )
2423adantl 452 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  0 )  =  0 )
2524oveq2d 5997 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  0 ) )  =  ( A  ^ c  0 ) )
26 cxpcl 20243 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c  B )  e.  CC )
2726exp0d 11404 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  ^ c  B ) ^ 0 )  =  1 )
2822, 25, 273eqtr4d 2408 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^ c 
( B  x.  0 ) )  =  ( ( A  ^ c  B ) ^ 0 ) )
29 oveq1 5988 . . . . . . 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 8978 . . . . . . . . . . . . 13  |-  0  e.  CC
31 cxp0 20239 . . . . . . . . . . . . 13  |-  ( 0  e.  CC  ->  (
0  ^ c  0 )  =  1 )
3230, 31ax-mp 8 . . . . . . . . . . . 12  |-  ( 0  ^ c  0 )  =  1
33 1t1e1 10019 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
3432, 33eqtr4i 2389 . . . . . . . . . . 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 5996 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( 0  x.  ( k  +  1 ) ) )
38 nn0p1nn 10152 . . . . . . . . . . . . . . . . 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 9909 . . . . . . . . . . . . . . 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 9157 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  ( k  +  1 ) )  =  0 )
4337, 42eqtrd 2398 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  0 )
4435, 43oveq12d 5999 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  ( k  +  1 ) ) )  =  ( 0  ^ c 
0 ) )
4536oveq1d 5996 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  ( 0  x.  k ) )
46 nn0cn 10124 . . . . . . . . . . . . . . . . . 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 9157 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  k )  =  0 )
5045, 49eqtrd 2398 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  0 )
5135, 50oveq12d 5999 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  k ) )  =  ( 0  ^ c 
0 ) )
5251, 32syl6eq 2414 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  ( B  x.  k ) )  =  1 )
5335, 36oveq12d 5999 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  B )  =  ( 0  ^ c  0 ) )
5453, 32syl6eq 2414 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^ c  B )  =  1 )
5552, 54oveq12d 5999 . . . . . . . . . . 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 2424 . . . . . . . . . 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 9002 . . . . . . . . . . . . . 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 20243 . . . . . . . . . . . . 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 9158 . . . . . . . . . . 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 5996 . . . . . . . . . . . . 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 20235 . . . . . . . . . . . . . 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 2398 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c  B )  =  0 )
7271oveq2d 5997 . . . . . . . . . . 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 5996 . . . . . . . . . . . 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 9002 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  e.  CC )
7639nnne0d 9937 . . . . . . . . . . . . . . 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 9567 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  =/=  0 )
79 0cxp 20235 . . . . . . . . . . . . 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 2398 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^ c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8264, 72, 813eqtr4rd 2409 . . . . . . . . . 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 2607 . . . . . . . . 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 8942 . . . . . . . . . . . . . 14  |-  1  e.  CC
8786a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  1  e.  CC )
8884, 85, 87adddid 9006 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  ( B  x.  1 ) ) )
8984mulid1d 8999 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  1 )  =  B )
9089oveq2d 5997 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  (
( B  x.  k
)  +  ( B  x.  1 ) )  =  ( ( B  x.  k )  +  B ) )
9188, 90eqtrd 2398 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  B ) )
9291oveq2d 5997 . . . . . . . . . 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 20248 . . . . . . . . . . 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 1189 . . . . . . . . . 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 2398 . . . . . . . . 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 2607 . . . . . . . 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 11275 . . . . . . . . 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 2380 . . . . . . 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 10259 . . 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 1149 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 935    = wceq 1647    e. wcel 1715    =/= wne 2529  (class class class)co 5981   CCcc 8882   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889   NNcn 9893   NN0cn0 10114   ^cexp 11269    ^ c ccxp 20131
This theorem is referenced by:  cxproot  20259  cxpmul2z  20260  cxpmul2d  20278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-ioc 10814  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-fac 11454  df-bc 11481  df-hash 11506  df-shft 11769  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-limsup 12152  df-clim 12169  df-rlim 12170  df-sum 12367  df-ef 12557  df-sin 12559  df-cos 12560  df-pi 12562  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-hom 13440  df-cco 13441  df-rest 13537  df-topn 13538  df-topgen 13554  df-pt 13555  df-prds 13558  df-xrs 13613  df-0g 13614  df-gsum 13615  df-qtop 13620  df-imas 13621  df-xps 13623  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-submnd 14626  df-mulg 14702  df-cntz 15003  df-cmn 15301  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-fbas 16590  df-fg 16591  df-cnfld 16594  df-top 16853  df-bases 16855  df-topon 16856  df-topsp 16857  df-cld 16973  df-ntr 16974  df-cls 16975  df-nei 17052  df-lp 17085  df-perf 17086  df-cn 17174  df-cnp 17175  df-haus 17260  df-tx 17474  df-hmeo 17663  df-fil 17754  df-fm 17846  df-flim 17847  df-flf 17848  df-xms 18098  df-ms 18099  df-tms 18100  df-cncf 18596  df-limc 19431  df-dv 19432  df-log 20132  df-cxp 20133
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