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Theorem cxpmul2 22139
Description: Product of exponents law for complex exponentiation. Variation on cxpmul 22138 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 6104 . . . . . . 7  |-  ( x  =  0  ->  ( B  x.  x )  =  ( B  x.  0 ) )
21oveq2d 6112 . . . . . 6  |-  ( x  =  0  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  0 ) ) )
3 oveq2 6104 . . . . . 6  |-  ( x  =  0  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ 0 ) )
42, 3eqeq12d 2457 . . . . 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 6104 . . . . . . 7  |-  ( x  =  k  ->  ( B  x.  x )  =  ( B  x.  k ) )
76oveq2d 6112 . . . . . 6  |-  ( x  =  k  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  k ) ) )
8 oveq2 6104 . . . . . 6  |-  ( x  =  k  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ k ) )
97, 8eqeq12d 2457 . . . . 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 6104 . . . . . . 7  |-  ( x  =  ( k  +  1 )  ->  ( B  x.  x )  =  ( B  x.  ( k  +  1 ) ) )
1211oveq2d 6112 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  ( k  +  1 ) ) ) )
13 oveq2 6104 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ ( k  +  1 ) ) )
1412, 13eqeq12d 2457 . . . . 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 6104 . . . . . . 7  |-  ( x  =  C  ->  ( B  x.  x )  =  ( B  x.  C ) )
1716oveq2d 6112 . . . . . 6  |-  ( x  =  C  ->  ( A  ^c  ( B  x.  x ) )  =  ( A  ^c  ( B  x.  C ) ) )
18 oveq2 6104 . . . . . 6  |-  ( x  =  C  ->  (
( A  ^c  B ) ^ x
)  =  ( ( A  ^c  B ) ^ C ) )
1917, 18eqeq12d 2457 . . . . 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 22120 . . . . . 6  |-  ( A  e.  CC  ->  ( A  ^c  0 )  =  1 )
2221adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
0 )  =  1 )
23 mul01 9553 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  x.  0 )  =  0 )
2423adantl 466 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  0 )  =  0 )
2524oveq2d 6112 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
( B  x.  0 ) )  =  ( A  ^c  0 ) )
26 cxpcl 22124 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c  B )  e.  CC )
2726exp0d 12007 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  ^c  B ) ^ 0 )  =  1 )
2822, 25, 273eqtr4d 2485 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  ^c 
( B  x.  0 ) )  =  ( ( A  ^c  B ) ^ 0 ) )
29 oveq1 6103 . . . . . . 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 9383 . . . . . . . . . . . . 13  |-  0  e.  CC
31 cxp0 22120 . . . . . . . . . . . . 13  |-  ( 0  e.  CC  ->  (
0  ^c  0 )  =  1 )
3230, 31ax-mp 5 . . . . . . . . . . . 12  |-  ( 0  ^c  0 )  =  1
33 1t1e1 10474 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
3432, 33eqtr4i 2466 . . . . . . . . . . 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 6111 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( 0  x.  ( k  +  1 ) ) )
38 nn0p1nn 10624 . . . . . . . . . . . . . . . . 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 10343 . . . . . . . . . . . . . . 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 9572 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  ( k  +  1 ) )  =  0 )
4337, 42eqtrd 2475 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  ( k  +  1 ) )  =  0 )
4435, 43oveq12d 6114 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  ( k  +  1 ) ) )  =  ( 0  ^c 
0 ) )
4536oveq1d 6111 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  ( 0  x.  k ) )
46 nn0cn 10594 . . . . . . . . . . . . . . . . . 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 9572 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( 0  x.  k )  =  0 )
5045, 49eqtrd 2475 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( B  x.  k )  =  0 )
5135, 50oveq12d 6114 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  k ) )  =  ( 0  ^c 
0 ) )
5251, 32syl6eq 2491 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  ( B  x.  k ) )  =  1 )
5335, 36oveq12d 6114 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  B )  =  ( 0  ^c  0 ) )
5453, 32syl6eq 2491 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =  0 )  ->  ( A  ^c  B )  =  1 )
5552, 54oveq12d 6114 . . . . . . . . . . 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 2501 . . . . . . . . . 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 9411 . . . . . . . . . . . . . 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 22124 . . . . . . . . . . . . 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 9573 . . . . . . . . . . 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 6111 . . . . . . . . . . . . 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 22116 . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c  B )  =  0 )
7271oveq2d 6112 . . . . . . . . . . 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 6111 . . . . . . . . . . . 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 9411 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  e.  CC )
7639nnne0d 10371 . . . . . . . . . . . . . . 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 9993 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( B  x.  (
k  +  1 ) )  =/=  0 )
79 0cxp 22116 . . . . . . . . . . . . 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 2475 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =  0 )  /\  B  =/=  0 )  -> 
( A  ^c 
( B  x.  (
k  +  1 ) ) )  =  0 )
8264, 72, 813eqtr4rd 2486 . . . . . . . . . 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 2694 . . . . . . . . 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 9407 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  1  e.  CC )
8784, 85, 86adddid 9415 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  ( B  x.  1 ) ) )
8884mulid1d 9408 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  1 )  =  B )
8988oveq2d 6112 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  (
( B  x.  k
)  +  ( B  x.  1 ) )  =  ( ( B  x.  k )  +  B ) )
9087, 89eqtrd 2475 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  k  e.  NN0 )  /\  A  =/=  0 )  ->  ( B  x.  ( k  +  1 ) )  =  ( ( B  x.  k )  +  B ) )
9190oveq2d 6112 . . . . . . . . . 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 22129 . . . . . . . . . . 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 1224 . . . . . . . . . 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 2475 . . . . . . . . 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 2694 . . . . . . . 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 11877 . . . . . . . . 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 2457 . . . . . . 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 10743 . . 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 1184 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292   NNcn 10327   NN0cn0 10584   ^cexp 11870    ^c ccxp 22012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-shft 12561  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-limsup 12954  df-clim 12971  df-rlim 12972  df-sum 13169  df-ef 13358  df-sin 13360  df-cos 13361  df-pi 13363  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-tx 19140  df-hmeo 19333  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-tms 19902  df-cncf 20459  df-limc 21346  df-dv 21347  df-log 22013  df-cxp 22014
This theorem is referenced by:  cxproot  22140  cxpmul2z  22141  cxpmul2d  22159
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