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Theorem expmulzap 10279
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 9019 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 elznn0nn 9019 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )
3 expmul 10278 . . . . . . . 8  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
433expia 1166 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
54adantlr 466 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
6 simp2l 990 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
76recnd 7758 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
8 simp3 966 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
98nn0cnd 8983 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
107, 9mulneg1d 8137 . . . . . . . . . . . 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 5756 . . . . . . . . . . 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 988 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
13 simp2r 991 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
1413nnnn0d 8981 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
15 expmul 10278 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  -u M  e.  NN0  /\  N  e.  NN0 )  -> 
( A ^ ( -u M  x.  N ) )  =  ( ( A ^ -u M
) ^ N ) )
1612, 14, 8, 15syl3anc 1199 . . . . . . . . . . 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 2150 . . . . . . . . . 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 5756 . . . . . . . . 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 10251 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
2012, 14, 19syl2anc 406 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
21 simp1r 989 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A #  0 )
2213nnzd 9123 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
23 expap0i 10265 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A #  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M ) #  0 )
2412, 21, 22, 23syl3anc 1199 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M ) #  0 )
258nn0zd 9122 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
26 exprecap 10274 . . . . . . . . . 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 1199 . . . . . . . . 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 2151 . . . . . . . 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 7750 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  x.  N )  e.  CC )
3014, 8nn0mulcld 8986 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u M  x.  N )  e.  NN0 )
3110, 30eqeltrrd 2193 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  x.  N )  e.  NN0 )
32 expineg2 10242 . . . . . . . . 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 1200 . . . . . . . 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 10242 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  CC  /\  -u M  e.  NN0 ) )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3512, 21, 7, 14, 34syl22anc 1200 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3635oveq1d 5755 . . . . . . . 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 2158 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^
( M  x.  N
) )  =  ( ( A ^ M
) ^ N ) )
38373expia 1166 . . . . . 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 769 . . . . 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 965 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  NN0 )
4140nn0cnd 8983 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  CC )
42 simp3l 992 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
4342recnd 7758 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4441, 43mulneg2d 8138 . . . . . . . . . . 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 5756 . . . . . . . . . 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 988 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
47 simp3r 993 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
4847nnnn0d 8981 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
49 expmul 10278 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( A ^ ( M  x.  -u N ) )  =  ( ( A ^ M ) ^ -u N
) )
5046, 40, 48, 49syl3anc 1199 . . . . . . . . . 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 2150 . . . . . . . . 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 5756 . . . . . . . 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 989 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A #  0 )
5441, 43mulcld 7750 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( M  x.  N
)  e.  CC )
5540, 48nn0mulcld 8986 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( M  x.  -u N
)  e.  NN0 )
5644, 55eqeltrrd 2193 . . . . . . . . 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 1200 . . . . . . . 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 10251 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
5946, 40, 58syl2anc 406 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ M
)  e.  CC )
6040nn0zd 9122 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  ZZ )
61 expap0i 10265 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0  /\  M  e.  ZZ )  ->  ( A ^ M ) #  0 )
6246, 53, 60, 61syl3anc 1199 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ M
) #  0 )
63 expineg2 10242 . . . . . . . . 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 1200 . . . . . . . 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 2158 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) )
66653expia 1166 . . . . . 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 988 . . . . . . . . . 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 989 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A #  0 )
69 simp2l 990 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  RR )
7069recnd 7758 . . . . . . . . . 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 991 . . . . . . . . . . 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 8981 . . . . . . . . . 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 1200 . . . . . . . . 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 5755 . . . . . . . 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 406 . . . . . . . . . 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 9123 . . . . . . . . . . 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 1199 . . . . . . . . . 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 8501 . . . . . . . . 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 8502 . . . . . . . . 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 992 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
8180recnd 7758 . . . . . . . . 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 993 . . . . . . . . . 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 8981 . . . . . . . . 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 10242 . . . . . . . . 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 1200 . . . . . . . 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 9123 . . . . . . . . . . 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 10274 . . . . . . . . . . 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 1199 . . . . . . . . . 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 5756 . . . . . . . . 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 10251 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  -u N  e.  NN0 )  ->  ( ( A ^ -u M ) ^ -u N
)  e.  CC )
9175, 83, 90syl2anc 406 . . . . . . . . . 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 10265 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  ( A ^ -u M
) #  0  /\  -u N  e.  ZZ )  ->  (
( A ^ -u M
) ^ -u N
) #  0 )
9375, 77, 86, 92syl3anc 1199 . . . . . . . . . 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 8505 . . . . . . . . 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 10278 . . . . . . . . . . 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 1199 . . . . . . . . . 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 8139 . . . . . . . . . . 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 5756 . . . . . . . . . 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 2150 . . . . . . . . 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 2152 . . . . . . . 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 2153 . . . . . . 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 1166 . . . . . 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 769 . . . . 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 689 . . . 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 283 . . 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 151 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  ZZ )  ->  ( N  e.  ZZ  ->  ( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
107106impr 374 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 103    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584   1c1 7585    x. cmul 7589   -ucneg 7898   # cap 8306    / cdiv 8392   NNcn 8677   NN0cn0 8928   ZZcz 9005   ^cexp 10232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-n0 8929  df-z 9006  df-uz 9276  df-seqfrec 10159  df-exp 10233
This theorem is referenced by:  iexpcyc  10337
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