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Theorem expmulz 11902
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 10652 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 elznn0nn 10652 . . . 4  |-  ( M  e.  ZZ  <->  ( M  e.  NN0  \/  ( M  e.  RR  /\  -u M  e.  NN ) ) )
3 expmul 11901 . . . . . . . 8  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
433expia 1189 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) ) )
54adantlr 714 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0 )  ->  ( N  e. 
NN0  ->  ( A ^
( M  x.  N
) )  =  ( ( A ^ M
) ^ N ) ) )
6 simp2l 1014 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  RR )
76recnd 9404 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  M  e.  CC )
8 simp3 990 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
98nn0cnd 10630 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  CC )
107, 9mulneg1d 9789 . . . . . . . . . . . 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 6102 . . . . . . . . . . 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 1012 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  e.  CC )
13 simp2r 1015 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN )
1413nnnn0d 10628 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  NN0 )
15 expmul 11901 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  -u M  e.  NN0  /\  N  e.  NN0 )  -> 
( A ^ ( -u M  x.  N ) )  =  ( ( A ^ -u M
) ^ N ) )
1612, 14, 8, 15syl3anc 1218 . . . . . . . . . . 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 2472 . . . . . . . . . 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 6102 . . . . . . . . 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 11875 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ -u M
)  e.  CC )
2012, 14, 19syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  e.  CC )
21 simp1r 1013 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  A  =/=  0 )
2213nnzd 10738 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u M  e.  ZZ )
23 expne0i 11888 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -u M  e.  ZZ )  ->  ( A ^ -u M )  =/=  0 )
2412, 21, 22, 23syl3anc 1218 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ -u M )  =/=  0 )
258nn0zd 10737 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
26 exprec 11897 . . . . . . . . . 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 1218 . . . . . . . . 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 2473 . . . . . . . 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 9398 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( M  x.  N )  e.  CC )
3014, 8nn0mulcld 10633 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( -u M  x.  N )  e.  NN0 )
3110, 30eqeltrrd 2513 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  -u ( M  x.  N )  e.  NN0 )
32 expneg2 11866 . . . . . . . . 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 1218 . . . . . . . 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 11866 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  M  e.  CC  /\  -u M  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3512, 7, 14, 34syl3anc 1218 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ M )  =  ( 1  /  ( A ^ -u M ) ) )
3635oveq1d 6101 . . . . . . . 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 2480 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
38373expia 1189 . . . . . 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 783 . . . . 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 989 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  NN0 )
4140nn0cnd 10630 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  CC )
42 simp3l 1016 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
4342recnd 9404 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4441, 43mulneg2d 9790 . . . . . . . . . . 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 6102 . . . . . . . . . 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 1012 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  e.  CC )
47 simp3r 1017 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
4847nnnn0d 10628 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
49 expmul 11901 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  -u N  e.  NN0 )  ->  ( A ^ ( M  x.  -u N ) )  =  ( ( A ^ M ) ^ -u N
) )
5046, 40, 48, 49syl3anc 1218 . . . . . . . . . 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 2472 . . . . . . . . 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 6102 . . . . . . . 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 9398 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( M  x.  N )  e.  CC )
5440, 48nn0mulcld 10633 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( M  x.  -u N )  e.  NN0 )
5544, 54eqeltrrd 2513 . . . . . . . . 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 1218 . . . . . . . 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 11875 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  CC )
5846, 40, 57syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ M )  e.  CC )
59 expneg2 11866 . . . . . . . . 9  |-  ( ( ( A ^ M
)  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  (
( A ^ M
) ^ N )  =  ( 1  / 
( ( A ^ M ) ^ -u N
) ) )
6058, 43, 48, 59syl3anc 1218 . . . . . . . 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 2480 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  NN0  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) )
62613expia 1189 . . . . . 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 1012 . . . . . . . . . 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 1014 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  M  e.  RR )
6564recnd 9404 . . . . . . . . . 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 1015 . . . . . . . . . . 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 10628 . . . . . . . . . 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 1218 . . . . . . . . 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 6101 . . . . . . . 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 661 . . . . . . . . . 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 1013 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  A  =/=  0 )
7266nnzd 10738 . . . . . . . . . . 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 1218 . . . . . . . . . 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 10092 . . . . . . . . 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 1016 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
7675recnd 9404 . . . . . . . . 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 1017 . . . . . . . . . 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 10628 . . . . . . . . 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 11866 . . . . . . . . 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 1218 . . . . . . . 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 10738 . . . . . . . . . . 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 11897 . . . . . . . . . . 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 1218 . . . . . . . . . 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 6102 . . . . . . . . 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 11875 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  -u N  e.  NN0 )  ->  ( ( A ^ -u M ) ^ -u N
)  e.  CC )
8670, 78, 85syl2anc 661 . . . . . . . . . 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 11888 . . . . . . . . . . 11  |-  ( ( ( A ^ -u M
)  e.  CC  /\  ( A ^ -u M
)  =/=  0  /\  -u N  e.  ZZ )  ->  ( ( A ^ -u M ) ^ -u N )  =/=  0 )
8870, 73, 81, 87syl3anc 1218 . . . . . . . . . 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 10096 . . . . . . . . 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 11901 . . . . . . . . . . 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 1218 . . . . . . . . . 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 9791 . . . . . . . . . . 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 6102 . . . . . . . . . 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 2472 . . . . . . . . 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 2474 . . . . . . . 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 2475 . . . . . . 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 1189 . . . . . 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 783 . . . . 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 380 . . . 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 475 . . 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 217 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  ZZ )  ->  ( N  e.  ZZ  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N
) ) )
102101impr 619 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  x.  N )
)  =  ( ( A ^ M ) ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    x. cmul 9279   -ucneg 9588    / cdiv 9985   NNcn 10314   NN0cn0 10571   ZZcz 10638   ^cexp 11857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-seq 11799  df-exp 11858
This theorem is referenced by:  iexpcyc  11962  iseraltlem2  13152  iseraltlem3  13153  dvexp3  21425  cxpeq  22170  atantayl2  22308  basellem3  22395  lgseisenlem1  22663  lgseisenlem4  22666  lgsquadlem1  22668  lgsquad2lem1  22672  m1lgs  22676  jm2.21  29296
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