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Theorem expmulzap 9966
Description: Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
Assertion
Ref Expression
expmulzap  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )

Proof of Theorem expmulzap
StepHypRef Expression
1 elznn0nn 8734 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 elznn0nn 8734 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )
3 expmul 9965 . . . . . . . 8  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
433expia 1145 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
54adantlr 461 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
6 simp2l 969 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
76recnd 7495 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
8 simp3 945 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
98nn0cnd 8698 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
107, 9mulneg1d 7868 . . . . . . . . . . . 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 5650 . . . . . . . . . . 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 967 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
13 simp2r 970 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
1413nnnn0d 8696 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
15 expmul 9965 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  -u M  e.  NN0  /\  N  e.  NN0 )  -> 
( A ^ ( -u M  x.  N ) )  =  ( ( A ^ -u M
) ^ N ) )
1612, 14, 8, 15syl3anc 1174 . . . . . . . . . . 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 2122 . . . . . . . . . 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 5650 . . . . . . . . 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 9938 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
2012, 14, 19syl2anc 403 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
21 simp1r 968 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A #  0 )
2213nnzd 8837 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
23 expap0i 9952 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A #  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M ) #  0 )
2412, 21, 22, 23syl3anc 1174 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M ) #  0 )
258nn0zd 8836 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
26 exprecap 9961 . . . . . . . . . 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 1174 . . . . . . . . 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 2123 . . . . . . . 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 7487 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  x.  N )  e.  CC )
3014, 8nn0mulcld 8701 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u M  x.  N )  e.  NN0 )
3110, 30eqeltrrd 2165 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  x.  N )  e.  NN0 )
32 expineg2 9929 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( ( M  x.  N )  e.  CC  /\  -u ( M  x.  N
)  e.  NN0 )
)  ->  ( A ^ ( M  x.  N ) )  =  ( 1  /  ( A ^ -u ( M  x.  N ) ) ) )
3312, 21, 29, 31, 32syl22anc 1175 . . . . . . . 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 expineg2 9929 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  CC  /\  -u M  e.  NN0 ) )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3512, 21, 7, 14, 34syl22anc 1175 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3635oveq1d 5649 . . . . . . . 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 2130 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  x.  N
) )  =  ( ( A ^ M
) ^ N ) )
38373expia 1145 . . . . . 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 746 . . . . 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 944 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  NN0 )
4140nn0cnd 8698 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  CC )
42 simp3l 971 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
4342recnd 7495 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4441, 43mulneg2d 7869 . . . . . . . . . . 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 5650 . . . . . . . . . 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 967 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
47 simp3r 972 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
4847nnnn0d 8696 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
49 expmul 9965 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( A ^ ( M  x.  -u N ) )  =  ( ( A ^ M ) ^ -u N
) )
5046, 40, 48, 49syl3anc 1174 . . . . . . . . . 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 2122 . . . . . . . . 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 5650 . . . . . . . 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 ) ) )
53 simp1r 968 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A #  0 )
5441, 43mulcld 7487 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( M  x.  N
)  e.  CC )
5540, 48nn0mulcld 8701 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( M  x.  -u N
)  e.  NN0 )
5644, 55eqeltrrd 2165 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u ( M  x.  N
)  e.  NN0 )
5746, 53, 54, 56, 32syl22anc 1175 . . . . . . . 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
) ) ) )
58 expcl 9938 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
5946, 40, 58syl2anc 403 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ M
)  e.  CC )
6040nn0zd 8836 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  ZZ )
61 expap0i 9952 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  M  e.  ZZ )  ->  ( A ^ M ) #  0 )
6246, 53, 60, 61syl3anc 1174 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ M
) #  0 )
63 expineg2 9929 . . . . . . . . 9  |-  ( ( ( ( A ^ M )  e.  CC  /\  ( A ^ M
) #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  ->  (
( A ^ M
) ^ N )  =  ( 1  / 
( ( A ^ M ) ^ -u N
) ) )
6459, 62, 43, 48, 63syl22anc 1175 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( A ^ M ) ^ N
)  =  ( 1  /  ( ( A ^ M ) ^ -u N ) ) )
6552, 57, 643eqtr4d 2130 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) )
66653expia 1145 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0 )  -> 
( ( N  e.  RR  /\  -u N  e.  NN )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
67 simp1l 967 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
68 simp1r 968 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A #  0 )
69 simp2l 969 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  RR )
7069recnd 7495 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  CC )
71 simp2r 970 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u M  e.  NN )
7271nnnn0d 8696 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u M  e.  NN0 )
7367, 68, 70, 72, 34syl22anc 1175 . . . . . . . . 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 ) ) )
7473oveq1d 5649 . . . . . . . 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 ) )
7567, 72, 19syl2anc 403 . . . . . . . . . 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 )
7671nnzd 8837 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u M  e.  ZZ )
7767, 68, 76, 23syl3anc 1174 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ -u M
) #  0 )
7875, 77recclapd 8222 . . . . . . . . 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 )
7975, 77recap0d 8223 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( 1  /  ( A ^ -u M ) ) #  0 )
80 simp3l 971 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
8180recnd 7495 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
82 simp3r 972 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
8382nnnn0d 8696 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
84 expineg2 9929 . . . . . . . . 9  |-  ( ( ( ( 1  / 
( A ^ -u M
) )  e.  CC  /\  ( 1  /  ( A ^ -u M ) ) #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  ->  (
( 1  /  ( A ^ -u M ) ) ^ N )  =  ( 1  / 
( ( 1  / 
( A ^ -u M
) ) ^ -u N
) ) )
8578, 79, 81, 83, 84syl22anc 1175 . . . . . . . 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 ) ) )
8682nnzd 8837 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
87 exprecap 9961 . . . . . . . . . . 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 ) ) )
8875, 77, 86, 87syl3anc 1174 . . . . . . . . . 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 ) ) )
8988oveq2d 5650 . . . . . . . . 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 ) ) ) )
90 expcl 9938 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  -u N  e.  NN0 )  ->  ( ( A ^ -u M ) ^ -u N
)  e.  CC )
9175, 83, 90syl2anc 403 . . . . . . . . . 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 )
92 expap0i 9952 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  ( A ^ -u M
) #  0  /\  -u N  e.  ZZ )  ->  (
( A ^ -u M
) ^ -u N
) #  0 )
9375, 77, 86, 92syl3anc 1174 . . . . . . . . . 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 )
9491, 93recrecapd 8226 . . . . . . . . 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
) )
95 expmul 9965 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( A ^ ( -u M  x.  -u N
) )  =  ( ( A ^ -u M
) ^ -u N
) )
9667, 72, 83, 95syl3anc 1174 . . . . . . . . . 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
) )
9770, 81mul2negd 7870 . . . . . . . . . . 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 ) )
9897oveq2d 5650 . . . . . . . . . 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 ) ) )
9996, 98eqtr3d 2122 . . . . . . . . 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 ) ) )
10089, 94, 993eqtrd 2124 . . . . . . . 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 ) ) )
10174, 85, 1003eqtrrd 2125 . . . . . . 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 ) )
1021013expia 1145 . . . . . 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 ) ) )
10366, 102jaodan 746 . . . . 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
) ) )
10439, 103jaod 672 . . . 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 ) ) )
1052, 104sylan2b 281 . . 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 ) ) )
1061, 105syl5bi 150 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  ZZ )  ->  ( N  e.  ZZ  ->  ( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
107106impr 371 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    /\ wa 102    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327   RRcr 7328   0cc0 7329   1c1 7330    x. cmul 7334   -ucneg 7633   # cap 8034    / cdiv 8113   NNcn 8394   NN0cn0 8643   ZZcz 8720   ^cexp 9919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-iseq 9818  df-seq3 9819  df-exp 9920
This theorem is referenced by:  iexpcyc  10024
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