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Theorem expmulz 11150
Description: Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.)
Assertion
Ref Expression
expmulz  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) )

Proof of Theorem expmulz
StepHypRef Expression
1 elznn0nn 10039 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 elznn0nn 10039 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )
3 expmul 11149 . . . . . . . 8  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
433expia 1153 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
54adantlr 695 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0 )  ->  ( N  e. 
NN0  ->  ( A ^
( M  x.  N
) )  =  ( ( A ^ M
) ^ N ) ) )
6 simp2l 981 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
76recnd 8863 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
8 simp3 957 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
98nn0cnd 10022 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
107, 9mulneg1d 9234 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u M  x.  N )  =  -u ( M  x.  N ) )
1110oveq2d 5876 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( -u M  x.  N ) )  =  ( A ^ -u ( M  x.  N )
) )
12 simp1l 979 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
13 simp2r 982 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
1413nnnn0d 10020 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
15 expmul 11149 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  -u M  e.  NN0  /\  N  e.  NN0 )  -> 
( A ^ ( -u M  x.  N ) )  =  ( ( A ^ -u M
) ^ N ) )
1612, 14, 8, 15syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( -u M  x.  N ) )  =  ( ( A ^ -u M ) ^ N
) )
1711, 16eqtr3d 2319 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u ( M  x.  N ) )  =  ( ( A ^ -u M ) ^ N ) )
1817oveq2d 5876 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  (
1  /  ( A ^ -u ( M  x.  N ) ) )  =  ( 1  /  ( ( A ^ -u M ) ^ N ) ) )
19 expcl 11123 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
2012, 14, 19syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
21 simp1r 980 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  =/=  0 )
2213nnzd 10118 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
23 expne0i 11136 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M )  =/=  0 )
2412, 21, 22, 23syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  =/=  0 )
258nn0zd 10117 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
26 exprec 11145 . . . . . . . . . 10  |-  ( ( ( A ^ -u M
)  e.  CC  /\  ( A ^ -u M
)  =/=  0  /\  N  e.  ZZ )  ->  ( ( 1  /  ( A ^ -u M ) ) ^ N )  =  ( 1  /  ( ( A ^ -u M
) ^ N ) ) )
2720, 24, 25, 26syl3anc 1182 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  (
( 1  /  ( A ^ -u M ) ) ^ N )  =  ( 1  / 
( ( A ^ -u M ) ^ N
) ) )
2818, 27eqtr4d 2320 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  (
1  /  ( A ^ -u ( M  x.  N ) ) )  =  ( ( 1  /  ( A ^ -u M ) ) ^ N ) )
297, 9mulcld 8857 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  x.  N )  e.  CC )
3014, 8nn0mulcld 10025 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u M  x.  N )  e.  NN0 )
3110, 30eqeltrrd 2360 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  x.  N )  e.  NN0 )
32 expneg2 11114 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( M  x.  N
)  e.  CC  /\  -u ( M  x.  N
)  e.  NN0 )  ->  ( A ^ ( M  x.  N )
)  =  ( 1  /  ( A ^ -u ( M  x.  N
) ) ) )
3312, 29, 31, 32syl3anc 1182 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( 1  /  ( A ^ -u ( M  x.  N ) ) ) )
34 expneg2 11114 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  M  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3512, 7, 14, 34syl3anc 1182 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3635oveq1d 5875 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  (
( A ^ M
) ^ N )  =  ( ( 1  /  ( A ^ -u M ) ) ^ N ) )
3728, 33, 363eqtr4d 2327 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
38373expia 1153 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN ) )  -> 
( N  e.  NN0  ->  ( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
395, 38jaodan 760 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e. 
NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )  ->  ( N  e. 
NN0  ->  ( A ^
( M  x.  N
) )  =  ( ( A ^ M
) ^ N ) ) )
40 simp2 956 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  NN0 )
4140nn0cnd 10022 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  CC )
42 simp3l 983 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
4342recnd 8863 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4441, 43mulneg2d 9235 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( M  x.  -u N )  =  -u ( M  x.  N ) )
4544oveq2d 5876 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ ( M  x.  -u N ) )  =  ( A ^ -u ( M  x.  N )
) )
46 simp1l 979 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
47 simp3r 984 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
4847nnnn0d 10020 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
49 expmul 11149 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( A ^ ( M  x.  -u N ) )  =  ( ( A ^ M ) ^ -u N
) )
5046, 40, 48, 49syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ ( M  x.  -u N ) )  =  ( ( A ^ M ) ^ -u N
) )
5145, 50eqtr3d 2319 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ -u ( M  x.  N ) )  =  ( ( A ^ M ) ^ -u N ) )
5251oveq2d 5876 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
1  /  ( A ^ -u ( M  x.  N ) ) )  =  ( 1  /  ( ( A ^ M ) ^ -u N ) ) )
5341, 43mulcld 8857 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( M  x.  N )  e.  CC )
5440, 48nn0mulcld 10025 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( M  x.  -u N )  e.  NN0 )
5544, 54eqeltrrd 2360 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u ( M  x.  N )  e.  NN0 )
5646, 53, 55, 32syl3anc 1182 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ ( M  x.  N ) )  =  ( 1  /  ( A ^ -u ( M  x.  N ) ) ) )
57 expcl 11123 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
5846, 40, 57syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ M )  e.  CC )
59 expneg2 11114 . . . . . . . . 9  |-  ( ( ( A ^ M
)  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  (
( A ^ M
) ^ N )  =  ( 1  / 
( ( A ^ M ) ^ -u N
) ) )
6058, 43, 48, 59syl3anc 1182 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
( A ^ M
) ^ N )  =  ( 1  / 
( ( A ^ M ) ^ -u N
) ) )
6152, 56, 603eqtr4d 2327 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
62613expia 1153 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0 )  ->  ( ( N  e.  RR  /\  -u N  e.  NN )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
63 simp1l 979 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
64 simp2l 981 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  RR )
6564recnd 8863 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  CC )
66 simp2r 982 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u M  e.  NN )
6766nnnn0d 10020 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u M  e.  NN0 )
6863, 65, 67, 34syl3anc 1182 . . . . . . . . 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 ) ) )
6968oveq1d 5875 . . . . . . . 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 ) )
7063, 67, 19syl2anc 642 . . . . . . . . . 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 )
71 simp1r 980 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  =/=  0 )
7266nnzd 10118 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u M  e.  ZZ )
7363, 71, 72, 23syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u M
)  =/=  0 )
7470, 73reccld 9531 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( 1  /  ( A ^ -u M ) )  e.  CC )
75 simp3l 983 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
7675recnd 8863 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
77 simp3r 984 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
7877nnnn0d 10020 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
79 expneg2 11114 . . . . . . . . 9  |-  ( ( ( 1  /  ( A ^ -u M ) )  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  (
( 1  /  ( A ^ -u M ) ) ^ N )  =  ( 1  / 
( ( 1  / 
( A ^ -u M
) ) ^ -u N
) ) )
8074, 76, 78, 79syl3anc 1182 . . . . . . . 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  /  ( ( 1  /  ( A ^ -u M ) ) ^ -u N ) ) )
8177nnzd 10118 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
82 exprec 11145 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  ( A ^ -u M
)  =/=  0  /\  -u N  e.  ZZ )  ->  ( ( 1  /  ( A ^ -u M ) ) ^ -u N )  =  ( 1  /  ( ( A ^ -u M
) ^ -u N
) ) )
8370, 73, 81, 82syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( 1  / 
( A ^ -u M
) ) ^ -u N
)  =  ( 1  /  ( ( A ^ -u M ) ^ -u N ) ) )
8483oveq2d 5876 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( 1  /  (
( 1  /  ( A ^ -u M ) ) ^ -u N
) )  =  ( 1  /  ( 1  /  ( ( A ^ -u M ) ^ -u N ) ) ) )
85 expcl 11123 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  -u N  e.  NN0 )  ->  ( ( A ^ -u M ) ^ -u N
)  e.  CC )
8670, 78, 85syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( A ^ -u M ) ^ -u N
)  e.  CC )
87 expne0i 11136 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  ( A ^ -u M
)  =/=  0  /\  -u N  e.  ZZ )  ->  ( ( A ^ -u M ) ^ -u N )  =/=  0 )
8870, 73, 81, 87syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( A ^ -u M ) ^ -u N
)  =/=  0 )
8986, 88recrecd 9535 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( 1  /  (
1  /  ( ( A ^ -u M
) ^ -u N
) ) )  =  ( ( A ^ -u M ) ^ -u N
) )
90 expmul 11149 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( A ^ ( -u M  x.  -u N
) )  =  ( ( A ^ -u M
) ^ -u N
) )
9163, 67, 78, 90syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( -u M  x.  -u N
) )  =  ( ( A ^ -u M
) ^ -u N
) )
9265, 76mul2negd 9236 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( -u M  x.  -u N
)  =  ( M  x.  N ) )
9392oveq2d 5876 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( -u M  x.  -u N
) )  =  ( A ^ ( M  x.  N ) ) )
9491, 93eqtr3d 2319 . . . . . . . . 9  |-  ( ( ( 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 ^ ( M  x.  N ) ) )
9584, 89, 943eqtrd 2321 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( 1  /  (
( 1  /  ( A ^ -u M ) ) ^ -u N
) )  =  ( A ^ ( M  x.  N ) ) )
9669, 80, 953eqtrrd 2322 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) )
97963expia 1153 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN ) )  -> 
( ( N  e.  RR  /\  -u N  e.  NN )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
9862, 97jaodan 760 . . . . 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  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
9939, 98jaod 369 . . . 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  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
1002, 99sylan2b 461 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  ZZ )  ->  ( ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
1011, 100syl5bi 208 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  ZZ )  ->  ( N  e.  ZZ  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
102101impr 602 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    x. cmul 8744   -ucneg 9040    / cdiv 9425   NNcn 9748   NN0cn0 9967   ZZcz 10026   ^cexp 11106
This theorem is referenced by:  iexpcyc  11209  iseraltlem2  12157  iseraltlem3  12158  dvexp3  19327  cxpeq  20099  atantayl2  20236  basellem3  20322  lgseisenlem1  20590  lgseisenlem4  20593  lgsquadlem1  20595  lgsquad2lem1  20599  m1lgs  20603  jm2.21  27098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-n0 9968  df-z 10027  df-uz 10233  df-seq 11049  df-exp 11107
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