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Theorem expaddz 12322
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 10958 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 elznn0nn 10958 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )
3 expadd 12320 . . . . . . . 8  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
433expia 1207 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
54adantlr 719 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0 )  ->  ( N  e. 
NN0  ->  ( A ^
( M  +  N
) )  =  ( ( A ^ M
)  x.  ( A ^ N ) ) ) )
6 expaddzlem 12321 . . . . . . 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 1207 . . . . . 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 792 . . . . 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 12321 . . . . . . . . 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 1007 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  M  e.  NN0 )
1110nn0cnd 10934 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  M  e.  CC )
12 simp2l 1031 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  N  e.  RR )
1312recnd 9676 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  N  e.  CC )
1411, 13addcomd 9842 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( M  +  N )  =  ( N  +  M ) )
1514oveq2d 6321 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( A ^ ( N  +  M )
) )
16 simp1l 1029 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  A  e.  CC )
17 expcl 12296 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
1816, 10, 17syl2anc 665 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^ M )  e.  CC )
19 simp1r 1030 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  A  =/=  0 )
2013negnegd 9984 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u -u N  =  N )
21 simp2r 1032 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u N  e.  NN )
2221nnnn0d 10932 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  -u N  e.  NN0 )
23 nn0negz 10982 . . . . . . . . . . . . 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 2508 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  N  e.  ZZ )
26 expclz 12303 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
2716, 19, 25, 26syl3anc 1264 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  RR  /\  -u N  e.  NN )  /\  M  e.  NN0 )  ->  ( A ^ N )  e.  CC )
2818, 27mulcomd 9671 . . . . . . . . 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 2473 . . . . . . . 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 1207 . . . . . . 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 441 . . . . . 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 1031 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  RR )
3332recnd 9676 . . . . . . . . . . . . . 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 1033 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
3534recnd 9676 . . . . . . . . . . . . . 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 10006 . . . . . . . . . . . . 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 6321 . . . . . . . . . . . 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 1029 . . . . . . . . . . . . 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 1032 . . . . . . . . . . . . . 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 10932 . . . . . . . . . . . . 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 1034 . . . . . . . . . . . . . 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 10932 . . . . . . . . . . . . 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 12320 . . . . . . . . . . . . 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 1264 . . . . . . . . . . . 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 2463 . . . . . . . . . . 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 6321 . . . . . . . . . 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 10764 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
4847oveq1i 6315 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  /  ( ( A ^ -u M )  x.  ( A ^ -u N ) ) )  =  ( 1  / 
( ( A ^ -u M )  x.  ( A ^ -u N ) ) )
4946, 48syl6eqr 2481 . . . . . . . . 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 12296 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
5138, 40, 50syl2anc 665 . . . . . . . . . 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 1030 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  =/=  0 )
5340nn0zd 11045 . . . . . . . . . . 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 12310 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M )  =/=  0 )
5538, 52, 53, 54syl3anc 1264 . . . . . . . . . 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 12296 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ -u N
)  e.  CC )
5738, 42, 56syl2anc 665 . . . . . . . . . 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 11045 . . . . . . . . . . 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 12310 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u N  e.  ZZ )  ->  ( A ^ -u N )  =/=  0 )
6038, 52, 58, 59syl3anc 1264 . . . . . . . . . 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 9604 . . . . . . . . . . 11  |-  1  e.  CC
62 divmuldiv 10314 . . . . . . . . . . 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 686 . . . . . . . . . 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 1265 . . . . . . . . 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 2466 . . . . . . . 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 9669 . . . . . . . . 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 10936 . . . . . . . . . 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 2507 . . . . . . . . 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 12287 . . . . . . . . 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 1264 . . . . . . . 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 12287 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  M  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
7238, 33, 40, 71syl3anc 1264 . . . . . . . . 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 12287 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
7438, 35, 42, 73syl3anc 1264 . . . . . . . . 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 6323 . . . . . . . 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 2473 . . . . . . 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 1207 . . . . . 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 792 . . . . 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 381 . . . 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 477 . . 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 220 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  ZZ )  ->  ( N  e.  ZZ  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) ) )
8281impr 623 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  +  N )
)  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614  (class class class)co 6305   CCcc 9544   RRcr 9545   0cc0 9546   1c1 9547    + caddc 9549    x. cmul 9551   -ucneg 9868    / cdiv 10276   NNcn 10616   NN0cn0 10876   ZZcz 10944   ^cexp 12278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-seq 12220  df-exp 12279
This theorem is referenced by:  m1expeven  12325  expsub  12326  expp1z  12327  iseraltlem2  13748  iseraltlem3  13749  pcaddlem  14832  m1expaddsub  17138  expghm  19065  aaliou3lem2  23297  aaliou3lem6  23302  dchrptlem1  24190  dchrptlem2  24191  lgseisenlem4  24278  lgsquadlem1  24280  lgsquad2lem1  24284  padicabv  24466  pellfund14  35716  rmxyadd  35739
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