Step |
Hyp |
Ref |
Expression |
1 |
|
elznn0nn |
|- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) ) |
2 |
|
elznn0nn |
|- ( M e. ZZ <-> ( M e. NN0 \/ ( M e. RR /\ -u M e. NN ) ) ) |
3 |
|
expmul |
|- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) |
4 |
3
|
3expia |
|- ( ( A e. CC /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) ) |
5 |
4
|
adantlr |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 ) -> ( N e. NN0 -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) ) |
6 |
|
simp2l |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. RR ) |
7 |
6
|
recnd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> M e. CC ) |
8 |
|
simp3 |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. NN0 ) |
9 |
8
|
nn0cnd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. CC ) |
10 |
7 9
|
mulneg1d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( -u M x. N ) = -u ( M x. N ) ) |
11 |
10
|
oveq2d |
|- ( ( ( 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 |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A e. CC ) |
13 |
|
simp2r |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN ) |
14 |
13
|
nnnn0d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. NN0 ) |
15 |
|
expmul |
|- ( ( A e. CC /\ -u M e. NN0 /\ N e. NN0 ) -> ( A ^ ( -u M x. N ) ) = ( ( A ^ -u M ) ^ N ) ) |
16 |
12 14 8 15
|
syl3anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( -u M x. N ) ) = ( ( A ^ -u M ) ^ N ) ) |
17 |
11 16
|
eqtr3d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u ( M x. N ) ) = ( ( A ^ -u M ) ^ N ) ) |
18 |
17
|
oveq2d |
|- ( ( ( 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 |
|- ( ( A e. CC /\ -u M e. NN0 ) -> ( A ^ -u M ) e. CC ) |
20 |
12 14 19
|
syl2anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) e. CC ) |
21 |
|
simp1r |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> A =/= 0 ) |
22 |
13
|
nnzd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u M e. ZZ ) |
23 |
|
expne0i |
|- ( ( A e. CC /\ A =/= 0 /\ -u M e. ZZ ) -> ( A ^ -u M ) =/= 0 ) |
24 |
12 21 22 23
|
syl3anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ -u M ) =/= 0 ) |
25 |
8
|
nn0zd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> N e. ZZ ) |
26 |
|
exprec |
|- ( ( ( A ^ -u M ) e. CC /\ ( A ^ -u M ) =/= 0 /\ N e. ZZ ) -> ( ( 1 / ( A ^ -u M ) ) ^ N ) = ( 1 / ( ( A ^ -u M ) ^ N ) ) ) |
27 |
20 24 25 26
|
syl3anc |
|- ( ( ( 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 ) ) ) |
28 |
18 27
|
eqtr4d |
|- ( ( ( 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 ) ) |
29 |
7 9
|
mulcld |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( M x. N ) e. CC ) |
30 |
14 8
|
nn0mulcld |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( -u M x. N ) e. NN0 ) |
31 |
10 30
|
eqeltrrd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> -u ( M x. N ) e. NN0 ) |
32 |
|
expneg2 |
|- ( ( A e. CC /\ ( M x. N ) e. CC /\ -u ( M x. N ) e. NN0 ) -> ( A ^ ( M x. N ) ) = ( 1 / ( A ^ -u ( M x. N ) ) ) ) |
33 |
12 29 31 32
|
syl3anc |
|- ( ( ( 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 |
|- ( ( A e. CC /\ M e. CC /\ -u M e. NN0 ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) ) |
35 |
12 7 14 34
|
syl3anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ M ) = ( 1 / ( A ^ -u M ) ) ) |
36 |
35
|
oveq1d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( ( A ^ M ) ^ N ) = ( ( 1 / ( A ^ -u M ) ) ^ N ) ) |
37 |
28 33 36
|
3eqtr4d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) |
38 |
37
|
3expia |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) ) -> ( N e. NN0 -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) ) |
39 |
5 38
|
jaodan |
|- ( ( ( 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 |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> M e. NN0 ) |
41 |
40
|
nn0cnd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> M e. CC ) |
42 |
|
simp3l |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR ) |
43 |
42
|
recnd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC ) |
44 |
41 43
|
mulneg2d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( M x. -u N ) = -u ( M x. N ) ) |
45 |
44
|
oveq2d |
|- ( ( ( 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 |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC ) |
47 |
|
simp3r |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN ) |
48 |
47
|
nnnn0d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 ) |
49 |
|
expmul |
|- ( ( A e. CC /\ M e. NN0 /\ -u N e. NN0 ) -> ( A ^ ( M x. -u N ) ) = ( ( A ^ M ) ^ -u N ) ) |
50 |
46 40 48 49
|
syl3anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M x. -u N ) ) = ( ( A ^ M ) ^ -u N ) ) |
51 |
45 50
|
eqtr3d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u ( M x. N ) ) = ( ( A ^ M ) ^ -u N ) ) |
52 |
51
|
oveq2d |
|- ( ( ( 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 |
41 43
|
mulcld |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( M x. N ) e. CC ) |
54 |
40 48
|
nn0mulcld |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( M x. -u N ) e. NN0 ) |
55 |
44 54
|
eqeltrrd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> -u ( M x. N ) e. NN0 ) |
56 |
46 53 55 32
|
syl3anc |
|- ( ( ( 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 |
|- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC ) |
58 |
46 40 57
|
syl2anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ M ) e. CC ) |
59 |
|
expneg2 |
|- ( ( ( A ^ M ) e. CC /\ N e. CC /\ -u N e. NN0 ) -> ( ( A ^ M ) ^ N ) = ( 1 / ( ( A ^ M ) ^ -u N ) ) ) |
60 |
58 43 48 59
|
syl3anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A ^ M ) ^ N ) = ( 1 / ( ( A ^ M ) ^ -u N ) ) ) |
61 |
52 56 60
|
3eqtr4d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) |
62 |
61
|
3expia |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. NN0 ) -> ( ( N e. RR /\ -u N e. NN ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) ) |
63 |
|
simp1l |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC ) |
64 |
|
simp2l |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> M e. RR ) |
65 |
64
|
recnd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> M e. CC ) |
66 |
|
simp2r |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u M e. NN ) |
67 |
66
|
nnnn0d |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u M e. NN0 ) |
68 |
63 65 67 34
|
syl3anc |
|- ( ( ( 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 ) ) ) |
69 |
68
|
oveq1d |
|- ( ( ( 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 ) ) |
70 |
63 67 19
|
syl2anc |
|- ( ( ( 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 |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> A =/= 0 ) |
72 |
66
|
nnzd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u M e. ZZ ) |
73 |
63 71 72 23
|
syl3anc |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u M ) =/= 0 ) |
74 |
70 73
|
reccld |
|- ( ( ( 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 |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR ) |
76 |
75
|
recnd |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC ) |
77 |
|
simp3r |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN ) |
78 |
77
|
nnnn0d |
|- ( ( ( 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 |
|- ( ( ( 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 ) ) ) |
80 |
74 76 78 79
|
syl3anc |
|- ( ( ( 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 ) ) ) |
81 |
77
|
nnzd |
|- ( ( ( 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 |
|- ( ( ( 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 ) ) ) |
83 |
70 73 81 82
|
syl3anc |
|- ( ( ( 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 ) ) ) |
84 |
83
|
oveq2d |
|- ( ( ( 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 |
|- ( ( ( A ^ -u M ) e. CC /\ -u N e. NN0 ) -> ( ( A ^ -u M ) ^ -u N ) e. CC ) |
86 |
70 78 85
|
syl2anc |
|- ( ( ( 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 |
|- ( ( ( A ^ -u M ) e. CC /\ ( A ^ -u M ) =/= 0 /\ -u N e. ZZ ) -> ( ( A ^ -u M ) ^ -u N ) =/= 0 ) |
88 |
70 73 81 87
|
syl3anc |
|- ( ( ( 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 ) |
89 |
86 88
|
recrecd |
|- ( ( ( 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 |
|- ( ( A e. CC /\ -u M e. NN0 /\ -u N e. NN0 ) -> ( A ^ ( -u M x. -u N ) ) = ( ( A ^ -u M ) ^ -u N ) ) |
91 |
63 67 78 90
|
syl3anc |
|- ( ( ( 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 ) ) |
92 |
65 76
|
mul2negd |
|- ( ( ( 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 ) ) |
93 |
92
|
oveq2d |
|- ( ( ( 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 ) ) ) |
94 |
91 93
|
eqtr3d |
|- ( ( ( 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 ) ) ) |
95 |
84 89 94
|
3eqtrd |
|- ( ( ( 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 ) ) ) |
96 |
69 80 95
|
3eqtrrd |
|- ( ( ( 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 ) ) |
97 |
96
|
3expia |
|- ( ( ( 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 ) ) ) |
98 |
62 97
|
jaodan |
|- ( ( ( 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 ) ) ) |
99 |
39 98
|
jaod |
|- ( ( ( 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 ) ) ) |
100 |
2 99
|
sylan2b |
|- ( ( ( 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 ) ) ) |
101 |
1 100
|
syl5bi |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ M e. ZZ ) -> ( N e. ZZ -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) ) |
102 |
101
|
impr |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) |