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Theorem expaddzap 10344
Description: Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
Assertion
Ref Expression
expaddzap  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )

Proof of Theorem expaddzap
StepHypRef Expression
1 elznn0nn 9075 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 elznn0nn 9075 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )
3 expadd 10342 . . . . . . . 8  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
433expia 1183 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
54adantlr 468 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
6 expaddzaplem 10343 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  +  N
) )  =  ( ( A ^ M
)  x.  ( A ^ N ) ) )
763expia 1183 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN ) )  ->  ( N  e.  NN0  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
85, 7jaodan 786 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )  -> 
( N  e.  NN0  ->  ( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
9 expaddzaplem 10343 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^
( N  +  M
) )  =  ( ( A ^ N
)  x.  ( A ^ M ) ) )
10 simp3 983 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  M  e.  NN0 )
1110nn0cnd 9039 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  M  e.  CC )
12 simp2l 1007 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  N  e.  RR )
1312recnd 7801 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  N  e.  CC )
1411, 13addcomd 7920 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( M  +  N )  =  ( N  +  M ) )
1514oveq2d 5790 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^
( M  +  N
) )  =  ( A ^ ( N  +  M ) ) )
16 simp1l 1005 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  A  e.  CC )
17 expcl 10318 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
1816, 10, 17syl2anc 408 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^ M )  e.  CC )
19 simp1r 1006 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  A #  0 )
2013negnegd 8071 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u -u N  =  N )
21 simp2r 1008 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u N  e.  NN )
2221nnnn0d 9037 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u N  e.  NN0 )
23 nn0negz 9095 . . . . . . . . . . . . 13  |-  ( -u N  e.  NN0  ->  -u -u N  e.  ZZ )
2422, 23syl 14 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u -u N  e.  ZZ )
2520, 24eqeltrrd 2217 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  N  e.  ZZ )
26 expclzap 10325 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
2716, 19, 25, 26syl3anc 1216 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^ N )  e.  CC )
2818, 27mulcomd 7794 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( ( A ^ M )  x.  ( A ^ N
) )  =  ( ( A ^ N
)  x.  ( A ^ M ) ) )
299, 15, 283eqtr4d 2182 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^
( M  +  N
) )  =  ( ( A ^ M
)  x.  ( A ^ N ) ) )
30293expia 1183 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( M  e.  NN0  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
3130impancom 258 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0 )  -> 
( ( N  e.  RR  /\  -u N  e.  NN )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
32 simp2l 1007 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  RR )
3332recnd 7801 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  CC )
34 simp3l 1009 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
3534recnd 7801 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
3633, 35negdid 8093 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u ( M  +  N
)  =  ( -u M  +  -u N ) )
3736oveq2d 5790 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u ( M  +  N )
)  =  ( A ^ ( -u M  +  -u N ) ) )
38 simp1l 1005 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
39 simp2r 1008 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u M  e.  NN )
4039nnnn0d 9037 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u M  e.  NN0 )
41 simp3r 1010 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
4241nnnn0d 9037 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
43 expadd 10342 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  -u M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( A ^ ( -u M  +  -u N
) )  =  ( ( A ^ -u M
)  x.  ( A ^ -u N ) ) )
4438, 40, 42, 43syl3anc 1216 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( -u M  +  -u N
) )  =  ( ( A ^ -u M
)  x.  ( A ^ -u N ) ) )
4537, 44eqtrd 2172 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u ( M  +  N )
)  =  ( ( A ^ -u M
)  x.  ( A ^ -u N ) ) )
4645oveq2d 5790 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( 1  /  ( A ^ -u ( M  +  N ) ) )  =  ( 1  /  ( ( A ^ -u M )  x.  ( A ^ -u N ) ) ) )
47 1t1e1 8879 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
4847oveq1i 5784 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  /  ( ( A ^ -u M )  x.  ( A ^ -u N ) ) )  =  ( 1  / 
( ( A ^ -u M )  x.  ( A ^ -u N ) ) )
4946, 48syl6eqr 2190 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( 1  /  ( A ^ -u ( M  +  N ) ) )  =  ( ( 1  x.  1 )  /  ( ( A ^ -u M )  x.  ( A ^ -u N ) ) ) )
50 expcl 10318 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
5138, 40, 50syl2anc 408 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u M
)  e.  CC )
52 simp1r 1006 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A #  0 )
5340nn0zd 9178 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u M  e.  ZZ )
54 expap0i 10332 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A #  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M ) #  0 )
5538, 52, 53, 54syl3anc 1216 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u M
) #  0 )
56 expcl 10318 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ -u N
)  e.  CC )
5738, 42, 56syl2anc 408 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u N
)  e.  CC )
5842nn0zd 9178 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
59 expap0i 10332 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A #  0  /\  -u N  e.  ZZ )  ->  ( A ^ -u N ) #  0 )
6038, 52, 58, 59syl3anc 1216 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u N
) #  0 )
61 ax-1cn 7720 . . . . . . . . . . 11  |-  1  e.  CC
62 divmuldivap 8479 . . . . . . . . . . 11  |-  ( ( ( 1  e.  CC  /\  1  e.  CC )  /\  ( ( ( A ^ -u M
)  e.  CC  /\  ( A ^ -u M
) #  0 )  /\  ( ( A ^ -u N )  e.  CC  /\  ( A ^ -u N
) #  0 ) ) )  ->  ( (
1  /  ( A ^ -u M ) )  x.  ( 1  /  ( A ^ -u N ) ) )  =  ( ( 1  x.  1 )  / 
( ( A ^ -u M )  x.  ( A ^ -u N ) ) ) )
6361, 61, 62mpanl12 432 . . . . . . . . . 10  |-  ( ( ( ( A ^ -u M )  e.  CC  /\  ( A ^ -u M
) #  0 )  /\  ( ( A ^ -u N )  e.  CC  /\  ( A ^ -u N
) #  0 ) )  ->  ( ( 1  /  ( A ^ -u M ) )  x.  ( 1  /  ( A ^ -u N ) ) )  =  ( ( 1  x.  1 )  /  ( ( A ^ -u M
)  x.  ( A ^ -u N ) ) ) )
6451, 55, 57, 60, 63syl22anc 1217 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( 1  / 
( A ^ -u M
) )  x.  (
1  /  ( A ^ -u N ) ) )  =  ( ( 1  x.  1 )  /  ( ( A ^ -u M
)  x.  ( A ^ -u N ) ) ) )
6549, 64eqtr4d 2175 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( 1  /  ( A ^ -u ( M  +  N ) ) )  =  ( ( 1  /  ( A ^ -u M ) )  x.  ( 1  /  ( A ^ -u N ) ) ) )
6633, 35addcld 7792 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( M  +  N
)  e.  CC )
6740, 42nn0addcld 9041 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( -u M  +  -u N )  e.  NN0 )
6836, 67eqeltrd 2216 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u ( M  +  N
)  e.  NN0 )
69 expineg2 10309 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( ( M  +  N )  e.  CC  /\  -u ( M  +  N
)  e.  NN0 )
)  ->  ( A ^ ( M  +  N ) )  =  ( 1  /  ( A ^ -u ( M  +  N ) ) ) )
7038, 52, 66, 68, 69syl22anc 1217 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( M  +  N )
)  =  ( 1  /  ( A ^ -u ( M  +  N
) ) ) )
71 expineg2 10309 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  CC  /\  -u M  e.  NN0 ) )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
7238, 52, 33, 40, 71syl22anc 1217 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ M
)  =  ( 1  /  ( A ^ -u M ) ) )
73 expineg2 10309 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
7438, 52, 35, 42, 73syl22anc 1217 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ N
)  =  ( 1  /  ( A ^ -u N ) ) )
7572, 74oveq12d 5792 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( A ^ M )  x.  ( A ^ N ) )  =  ( ( 1  /  ( A ^ -u M ) )  x.  ( 1  /  ( A ^ -u N ) ) ) )
7665, 70, 753eqtr4d 2182 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
77763expia 1183 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN ) )  ->  (
( N  e.  RR  /\  -u N  e.  NN )  ->  ( A ^
( M  +  N
) )  =  ( ( A ^ M
)  x.  ( A ^ N ) ) ) )
7831, 77jaodan 786 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )  -> 
( ( N  e.  RR  /\  -u N  e.  NN )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
798, 78jaod 706 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )  -> 
( ( N  e. 
NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
802, 79sylan2b 285 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  ZZ )  ->  ( ( N  e. 
NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
811, 80syl5bi 151 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  ZZ )  ->  ( N  e.  ZZ  ->  ( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
8281impr 376 1  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627   1c1 7628    + caddc 7630    x. cmul 7632   -ucneg 7941   # cap 8350    / cdiv 8439   NNcn 8727   NN0cn0 8984   ZZcz 9061   ^cexp 10299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-seqfrec 10226  df-exp 10300
This theorem is referenced by:  m1expeven  10347  expsubap  10348  expp1zap  10349
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