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Theorem expmulzap 9619
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 8446 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 elznn0nn 8446 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )
3 expmul 9618 . . . . . . . 8  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
433expia 1141 . . . . . . 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 965 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
76recnd 7209 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
8 simp3 941 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
98nn0cnd 8410 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
107, 9mulneg1d 7582 . . . . . . . . . . . 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 5559 . . . . . . . . . . 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 963 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
13 simp2r 966 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
1413nnnn0d 8408 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
15 expmul 9618 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  -u M  e.  NN0  /\  N  e.  NN0 )  -> 
( A ^ ( -u M  x.  N ) )  =  ( ( A ^ -u M
) ^ N ) )
1612, 14, 8, 15syl3anc 1170 . . . . . . . . . . 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 2116 . . . . . . . . . 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 5559 . . . . . . . . 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 9591 . . . . . . . . . . 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 964 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A #  0 )
2213nnzd 8549 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
23 expap0i 9605 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A #  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M ) #  0 )
2412, 21, 22, 23syl3anc 1170 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M ) #  0 )
258nn0zd 8548 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
26 exprecap 9614 . . . . . . . . . 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 1170 . . . . . . . . 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 2117 . . . . . . . 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 7201 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  x.  N )  e.  CC )
3014, 8nn0mulcld 8413 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u M  x.  N )  e.  NN0 )
3110, 30eqeltrrd 2157 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  x.  N )  e.  NN0 )
32 expineg2 9582 . . . . . . . . 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 1171 . . . . . . . 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 9582 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  CC  /\  -u M  e.  NN0 ) )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3512, 21, 7, 14, 34syl22anc 1171 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3635oveq1d 5558 . . . . . . . 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 2124 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  x.  N
) )  =  ( ( A ^ M
) ^ N ) )
38373expia 1141 . . . . . 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 744 . . . . 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 940 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  NN0 )
4140nn0cnd 8410 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  CC )
42 simp3l 967 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
4342recnd 7209 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4441, 43mulneg2d 7583 . . . . . . . . . . 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 5559 . . . . . . . . . 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 963 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
47 simp3r 968 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
4847nnnn0d 8408 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
49 expmul 9618 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( A ^ ( M  x.  -u N ) )  =  ( ( A ^ M ) ^ -u N
) )
5046, 40, 48, 49syl3anc 1170 . . . . . . . . . 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 2116 . . . . . . . . 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 5559 . . . . . . . 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 964 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A #  0 )
5441, 43mulcld 7201 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( M  x.  N
)  e.  CC )
5540, 48nn0mulcld 8413 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( M  x.  -u N
)  e.  NN0 )
5644, 55eqeltrrd 2157 . . . . . . . . 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 1171 . . . . . . . 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 9591 . . . . . . . . . 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 8548 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  ZZ )
61 expap0i 9605 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  M  e.  ZZ )  ->  ( A ^ M ) #  0 )
6246, 53, 60, 61syl3anc 1170 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ M
) #  0 )
63 expineg2 9582 . . . . . . . . 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 1171 . . . . . . . 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 2124 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) )
66653expia 1141 . . . . . 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 963 . . . . . . . . . 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 964 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A #  0 )
69 simp2l 965 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  RR )
7069recnd 7209 . . . . . . . . . 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 966 . . . . . . . . . . 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 8408 . . . . . . . . . 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 1171 . . . . . . . . 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 5558 . . . . . . . 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 8549 . . . . . . . . . . 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 1170 . . . . . . . . . 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 7936 . . . . . . . . 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 7937 . . . . . . . . 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 967 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
8180recnd 7209 . . . . . . . . 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 968 . . . . . . . . . 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 8408 . . . . . . . . 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 9582 . . . . . . . . 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 1171 . . . . . . . 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 8549 . . . . . . . . . . 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 9614 . . . . . . . . . . 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 1170 . . . . . . . . . 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 5559 . . . . . . . . 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 9591 . . . . . . . . . . 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 9605 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  ( A ^ -u M
) #  0  /\  -u N  e.  ZZ )  ->  (
( A ^ -u M
) ^ -u N
) #  0 )
9375, 77, 86, 92syl3anc 1170 . . . . . . . . . 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 7940 . . . . . . . . 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 9618 . . . . . . . . . . 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 1170 . . . . . . . . . 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 7584 . . . . . . . . . . 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 5559 . . . . . . . . . 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 2116 . . . . . . . . 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 2118 . . . . . . . 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 2119 . . . . . . 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 1141 . . . . . 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 744 . . . . 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 670 . . . 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 662    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041   RRcr 7042   0cc0 7043   1c1 7044    x. cmul 7048   -ucneg 7347   # cap 7748    / cdiv 7827   NNcn 8106   NN0cn0 8355   ZZcz 8432   ^cexp 9572
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522  df-iexp 9573
This theorem is referenced by:  iexpcyc  9676
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