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Theorem expaddz 11077
Description: Sum of exponents law for integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expaddz  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) )

Proof of Theorem expaddz
StepHypRef Expression
1 elznn0nn 9969 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 elznn0nn 9969 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )
3 expadd 11075 . . . . . . . 8  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
433expia 1158 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
54adantlr 698 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0 )  ->  ( N  e. 
NN0  ->  ( A ^
( M  +  N
) )  =  ( ( A ^ M
)  x.  ( A ^ N ) ) ) )
6 expaddzlem 11076 . . . . . . 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 1158 . . . . . 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 763 . . . . 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 expaddzlem 11076 . . . . . . . . 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 962 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  M  e.  NN0 )
1110nn0cnd 9952 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  M  e.  CC )
12 simp2l 986 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  N  e.  RR )
1312recnd 8794 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  N  e.  CC )
1411, 13addcomd 8947 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( M  +  N )  =  ( N  +  M ) )
1514oveq2d 5773 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( A ^ ( N  +  M )
) )
16 simp1l 984 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  A  e.  CC )
17 expcl 11052 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
1816, 10, 17syl2anc 645 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^ M )  e.  CC )
19 simp1r 985 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  A  =/=  0 )
2013negnegd 9081 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u -u N  =  N )
21 simp2r 987 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u N  e.  NN )
2221nnnn0d 9950 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u N  e.  NN0 )
23 nn0negz 9989 . . . . . . . . . . . . 13  |-  ( -u N  e.  NN0  ->  -u -u N  e.  ZZ )
2422, 23syl 17 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u -u N  e.  ZZ )
2520, 24eqeltrrd 2331 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  N  e.  ZZ )
26 expclz 11059 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
2716, 19, 25, 26syl3anc 1187 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^ N )  e.  CC )
2818, 27mulcomd 8789 . . . . . . . . 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 2298 . . . . . . . 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 1158 . . . . . . 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 429 . . . . . 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 986 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  RR )
3332recnd 8794 . . . . . . . . . . . . . 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 988 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
3534recnd 8794 . . . . . . . . . . . . . 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 9103 . . . . . . . . . . . . 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 5773 . . . . . . . . . . . 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 984 . . . . . . . . . . . . 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 987 . . . . . . . . . . . . . 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 9950 . . . . . . . . . . . . 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 989 . . . . . . . . . . . . . 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 9950 . . . . . . . . . . . . 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 11075 . . . . . . . . . . . . 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 1187 . . . . . . . . . . . 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 2288 . . . . . . . . . . 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 5773 . . . . . . . . . 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 9802 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
4847oveq1i 5767 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  /  ( ( A ^ -u M )  x.  ( A ^ -u N ) ) )  =  ( 1  / 
( ( A ^ -u M )  x.  ( A ^ -u N ) ) )
4946, 48syl6eqr 2306 . . . . . . . . 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 11052 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
5138, 40, 50syl2anc 645 . . . . . . . . . 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 985 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  =/=  0 )
5340nn0zd 10047 . . . . . . . . . . 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 expne0i 11065 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M )  =/=  0 )
5538, 52, 53, 54syl3anc 1187 . . . . . . . . . 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 11052 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ -u N
)  e.  CC )
5738, 42, 56syl2anc 645 . . . . . . . . . 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 10047 . . . . . . . . . . 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 expne0i 11065 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u N  e.  ZZ )  ->  ( A ^ -u N )  =/=  0 )
6038, 52, 58, 59syl3anc 1187 . . . . . . . . . 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 8728 . . . . . . . . . . 11  |-  1  e.  CC
62 divmuldiv 9393 . . . . . . . . . . 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 666 . . . . . . . . . 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 1188 . . . . . . . . 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 2291 . . . . . . . 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 8787 . . . . . . . . 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 9954 . . . . . . . . . 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 2330 . . . . . . . . 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 expneg2 11043 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( M  +  N
)  e.  CC  /\  -u ( M  +  N
)  e.  NN0 )  ->  ( A ^ ( M  +  N )
)  =  ( 1  /  ( A ^ -u ( M  +  N
) ) ) )
7038, 66, 68, 69syl3anc 1187 . . . . . . . 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 expneg2 11043 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  M  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
7238, 33, 40, 71syl3anc 1187 . . . . . . . . 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 expneg2 11043 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
7438, 35, 42, 73syl3anc 1187 . . . . . . . . 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 5775 . . . . . . . 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 2298 . . . . . . 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 1158 . . . . . 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 763 . . . . 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 371 . . . 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 463 . . 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 210 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  ZZ )  ->  ( N  e.  ZZ  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
8281impr 605 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 6    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675   -ucneg 8971    / cdiv 9356   NNcn 9679   NN0cn0 9897   ZZcz 9956   ^cexp 11035
This theorem is referenced by:  expsub  11080  expp1z  11081  iseraltlem2  12085  iseraltlem3  12086  pcaddlem  12863  expghm  16377  aaliou3lem2  19650  aaliou3lem6  19655  dchrptlem1  20430  dchrptlem2  20431  lgseisenlem4  20518  lgsquadlem1  20520  lgsquad2lem1  20524  padicabv  20706  pellfund14  26315  rmxyadd  26338  m1expaddsub  26753
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-n0 9898  df-z 9957  df-uz 10163  df-seq 10978  df-exp 11036
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