Step |
Hyp |
Ref |
Expression |
1 |
|
elnn0 |
|- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) ) |
2 |
|
elnn0 |
|- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) ) |
3 |
|
expgcd |
|- ( ( A e. NN /\ B e. NN /\ N e. NN0 ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) |
4 |
3
|
3expia |
|- ( ( A e. NN /\ B e. NN ) -> ( N e. NN0 -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
5 |
|
elnn0 |
|- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) ) |
6 |
|
0exp |
|- ( N e. NN -> ( 0 ^ N ) = 0 ) |
7 |
6
|
3ad2ant3 |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( 0 ^ N ) = 0 ) |
8 |
7
|
oveq1d |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( ( 0 ^ N ) gcd ( B ^ N ) ) = ( 0 gcd ( B ^ N ) ) ) |
9 |
|
simp2 |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> B e. NN ) |
10 |
|
nnnn0 |
|- ( N e. NN -> N e. NN0 ) |
11 |
10
|
3ad2ant3 |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> N e. NN0 ) |
12 |
9 11
|
nnexpcld |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( B ^ N ) e. NN ) |
13 |
12
|
nnzd |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( B ^ N ) e. ZZ ) |
14 |
|
gcd0id |
|- ( ( B ^ N ) e. ZZ -> ( 0 gcd ( B ^ N ) ) = ( abs ` ( B ^ N ) ) ) |
15 |
13 14
|
syl |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( 0 gcd ( B ^ N ) ) = ( abs ` ( B ^ N ) ) ) |
16 |
12
|
nnred |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( B ^ N ) e. RR ) |
17 |
|
0red |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> 0 e. RR ) |
18 |
12
|
nngt0d |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> 0 < ( B ^ N ) ) |
19 |
17 16 18
|
ltled |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> 0 <_ ( B ^ N ) ) |
20 |
16 19
|
absidd |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( abs ` ( B ^ N ) ) = ( B ^ N ) ) |
21 |
8 15 20
|
3eqtrrd |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( B ^ N ) = ( ( 0 ^ N ) gcd ( B ^ N ) ) ) |
22 |
|
oveq1 |
|- ( A = 0 -> ( A gcd B ) = ( 0 gcd B ) ) |
23 |
22
|
3ad2ant1 |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( A gcd B ) = ( 0 gcd B ) ) |
24 |
|
nnz |
|- ( B e. NN -> B e. ZZ ) |
25 |
24
|
3ad2ant2 |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> B e. ZZ ) |
26 |
|
gcd0id |
|- ( B e. ZZ -> ( 0 gcd B ) = ( abs ` B ) ) |
27 |
25 26
|
syl |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( 0 gcd B ) = ( abs ` B ) ) |
28 |
|
nnre |
|- ( B e. NN -> B e. RR ) |
29 |
|
0red |
|- ( B e. NN -> 0 e. RR ) |
30 |
|
nngt0 |
|- ( B e. NN -> 0 < B ) |
31 |
29 28 30
|
ltled |
|- ( B e. NN -> 0 <_ B ) |
32 |
28 31
|
absidd |
|- ( B e. NN -> ( abs ` B ) = B ) |
33 |
32
|
3ad2ant2 |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( abs ` B ) = B ) |
34 |
23 27 33
|
3eqtrd |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( A gcd B ) = B ) |
35 |
34
|
oveq1d |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) ^ N ) = ( B ^ N ) ) |
36 |
|
oveq1 |
|- ( A = 0 -> ( A ^ N ) = ( 0 ^ N ) ) |
37 |
36
|
3ad2ant1 |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( A ^ N ) = ( 0 ^ N ) ) |
38 |
37
|
oveq1d |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( ( A ^ N ) gcd ( B ^ N ) ) = ( ( 0 ^ N ) gcd ( B ^ N ) ) ) |
39 |
21 35 38
|
3eqtr4d |
|- ( ( A = 0 /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) |
40 |
39
|
3expia |
|- ( ( A = 0 /\ B e. NN ) -> ( N e. NN -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
41 |
|
1z |
|- 1 e. ZZ |
42 |
|
gcd1 |
|- ( 1 e. ZZ -> ( 1 gcd 1 ) = 1 ) |
43 |
41 42
|
ax-mp |
|- ( 1 gcd 1 ) = 1 |
44 |
43
|
eqcomi |
|- 1 = ( 1 gcd 1 ) |
45 |
|
simp1 |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> A = 0 ) |
46 |
45
|
oveq1d |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( A gcd B ) = ( 0 gcd B ) ) |
47 |
|
simp2 |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> B e. NN ) |
48 |
47
|
nnzd |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> B e. ZZ ) |
49 |
48 26
|
syl |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( 0 gcd B ) = ( abs ` B ) ) |
50 |
32
|
3ad2ant2 |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( abs ` B ) = B ) |
51 |
46 49 50
|
3eqtrd |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( A gcd B ) = B ) |
52 |
|
simp3 |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> N = 0 ) |
53 |
51 52
|
oveq12d |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( ( A gcd B ) ^ N ) = ( B ^ 0 ) ) |
54 |
47
|
nncnd |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> B e. CC ) |
55 |
54
|
exp0d |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( B ^ 0 ) = 1 ) |
56 |
53 55
|
eqtrd |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( ( A gcd B ) ^ N ) = 1 ) |
57 |
45 52
|
oveq12d |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( A ^ N ) = ( 0 ^ 0 ) ) |
58 |
|
0exp0e1 |
|- ( 0 ^ 0 ) = 1 |
59 |
58
|
a1i |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( 0 ^ 0 ) = 1 ) |
60 |
57 59
|
eqtrd |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( A ^ N ) = 1 ) |
61 |
52
|
oveq2d |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( B ^ N ) = ( B ^ 0 ) ) |
62 |
61 55
|
eqtrd |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( B ^ N ) = 1 ) |
63 |
60 62
|
oveq12d |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( ( A ^ N ) gcd ( B ^ N ) ) = ( 1 gcd 1 ) ) |
64 |
44 56 63
|
3eqtr4a |
|- ( ( A = 0 /\ B e. NN /\ N = 0 ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) |
65 |
64
|
3expia |
|- ( ( A = 0 /\ B e. NN ) -> ( N = 0 -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
66 |
40 65
|
jaod |
|- ( ( A = 0 /\ B e. NN ) -> ( ( N e. NN \/ N = 0 ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
67 |
5 66
|
syl5bi |
|- ( ( A = 0 /\ B e. NN ) -> ( N e. NN0 -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
68 |
|
nnnn0 |
|- ( A e. NN -> A e. NN0 ) |
69 |
68
|
3ad2ant1 |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> A e. NN0 ) |
70 |
10
|
3ad2ant3 |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> N e. NN0 ) |
71 |
69 70
|
nn0expcld |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( A ^ N ) e. NN0 ) |
72 |
|
nn0gcdid0 |
|- ( ( A ^ N ) e. NN0 -> ( ( A ^ N ) gcd 0 ) = ( A ^ N ) ) |
73 |
71 72
|
syl |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( ( A ^ N ) gcd 0 ) = ( A ^ N ) ) |
74 |
|
simp2 |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> B = 0 ) |
75 |
74
|
oveq1d |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( B ^ N ) = ( 0 ^ N ) ) |
76 |
6
|
3ad2ant3 |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( 0 ^ N ) = 0 ) |
77 |
75 76
|
eqtrd |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( B ^ N ) = 0 ) |
78 |
77
|
oveq2d |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( ( A ^ N ) gcd ( B ^ N ) ) = ( ( A ^ N ) gcd 0 ) ) |
79 |
74
|
oveq2d |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( A gcd B ) = ( A gcd 0 ) ) |
80 |
|
nn0gcdid0 |
|- ( A e. NN0 -> ( A gcd 0 ) = A ) |
81 |
68 80
|
syl |
|- ( A e. NN -> ( A gcd 0 ) = A ) |
82 |
81
|
3ad2ant1 |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( A gcd 0 ) = A ) |
83 |
79 82
|
eqtrd |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( A gcd B ) = A ) |
84 |
83
|
oveq1d |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( ( A gcd B ) ^ N ) = ( A ^ N ) ) |
85 |
73 78 84
|
3eqtr4rd |
|- ( ( A e. NN /\ B = 0 /\ N e. NN ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) |
86 |
85
|
3expia |
|- ( ( A e. NN /\ B = 0 ) -> ( N e. NN -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
87 |
|
nncn |
|- ( A e. NN -> A e. CC ) |
88 |
87
|
exp0d |
|- ( A e. NN -> ( A ^ 0 ) = 1 ) |
89 |
88 43
|
eqtr4di |
|- ( A e. NN -> ( A ^ 0 ) = ( 1 gcd 1 ) ) |
90 |
81
|
oveq1d |
|- ( A e. NN -> ( ( A gcd 0 ) ^ 0 ) = ( A ^ 0 ) ) |
91 |
58
|
a1i |
|- ( A e. NN -> ( 0 ^ 0 ) = 1 ) |
92 |
88 91
|
oveq12d |
|- ( A e. NN -> ( ( A ^ 0 ) gcd ( 0 ^ 0 ) ) = ( 1 gcd 1 ) ) |
93 |
89 90 92
|
3eqtr4d |
|- ( A e. NN -> ( ( A gcd 0 ) ^ 0 ) = ( ( A ^ 0 ) gcd ( 0 ^ 0 ) ) ) |
94 |
93
|
3ad2ant1 |
|- ( ( A e. NN /\ B = 0 /\ N = 0 ) -> ( ( A gcd 0 ) ^ 0 ) = ( ( A ^ 0 ) gcd ( 0 ^ 0 ) ) ) |
95 |
|
simp2 |
|- ( ( A e. NN /\ B = 0 /\ N = 0 ) -> B = 0 ) |
96 |
95
|
oveq2d |
|- ( ( A e. NN /\ B = 0 /\ N = 0 ) -> ( A gcd B ) = ( A gcd 0 ) ) |
97 |
|
simp3 |
|- ( ( A e. NN /\ B = 0 /\ N = 0 ) -> N = 0 ) |
98 |
96 97
|
oveq12d |
|- ( ( A e. NN /\ B = 0 /\ N = 0 ) -> ( ( A gcd B ) ^ N ) = ( ( A gcd 0 ) ^ 0 ) ) |
99 |
97
|
oveq2d |
|- ( ( A e. NN /\ B = 0 /\ N = 0 ) -> ( A ^ N ) = ( A ^ 0 ) ) |
100 |
95 97
|
oveq12d |
|- ( ( A e. NN /\ B = 0 /\ N = 0 ) -> ( B ^ N ) = ( 0 ^ 0 ) ) |
101 |
99 100
|
oveq12d |
|- ( ( A e. NN /\ B = 0 /\ N = 0 ) -> ( ( A ^ N ) gcd ( B ^ N ) ) = ( ( A ^ 0 ) gcd ( 0 ^ 0 ) ) ) |
102 |
94 98 101
|
3eqtr4d |
|- ( ( A e. NN /\ B = 0 /\ N = 0 ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) |
103 |
102
|
3expia |
|- ( ( A e. NN /\ B = 0 ) -> ( N = 0 -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
104 |
86 103
|
jaod |
|- ( ( A e. NN /\ B = 0 ) -> ( ( N e. NN \/ N = 0 ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
105 |
5 104
|
syl5bi |
|- ( ( A e. NN /\ B = 0 ) -> ( N e. NN0 -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
106 |
|
gcd0val |
|- ( 0 gcd 0 ) = 0 |
107 |
6 106
|
eqtr4di |
|- ( N e. NN -> ( 0 ^ N ) = ( 0 gcd 0 ) ) |
108 |
106
|
a1i |
|- ( N e. NN -> ( 0 gcd 0 ) = 0 ) |
109 |
108
|
oveq1d |
|- ( N e. NN -> ( ( 0 gcd 0 ) ^ N ) = ( 0 ^ N ) ) |
110 |
6 6
|
oveq12d |
|- ( N e. NN -> ( ( 0 ^ N ) gcd ( 0 ^ N ) ) = ( 0 gcd 0 ) ) |
111 |
107 109 110
|
3eqtr4d |
|- ( N e. NN -> ( ( 0 gcd 0 ) ^ N ) = ( ( 0 ^ N ) gcd ( 0 ^ N ) ) ) |
112 |
111
|
3ad2ant3 |
|- ( ( A = 0 /\ B = 0 /\ N e. NN ) -> ( ( 0 gcd 0 ) ^ N ) = ( ( 0 ^ N ) gcd ( 0 ^ N ) ) ) |
113 |
|
simp1 |
|- ( ( A = 0 /\ B = 0 /\ N e. NN ) -> A = 0 ) |
114 |
|
simp2 |
|- ( ( A = 0 /\ B = 0 /\ N e. NN ) -> B = 0 ) |
115 |
113 114
|
oveq12d |
|- ( ( A = 0 /\ B = 0 /\ N e. NN ) -> ( A gcd B ) = ( 0 gcd 0 ) ) |
116 |
115
|
oveq1d |
|- ( ( A = 0 /\ B = 0 /\ N e. NN ) -> ( ( A gcd B ) ^ N ) = ( ( 0 gcd 0 ) ^ N ) ) |
117 |
113
|
oveq1d |
|- ( ( A = 0 /\ B = 0 /\ N e. NN ) -> ( A ^ N ) = ( 0 ^ N ) ) |
118 |
114
|
oveq1d |
|- ( ( A = 0 /\ B = 0 /\ N e. NN ) -> ( B ^ N ) = ( 0 ^ N ) ) |
119 |
117 118
|
oveq12d |
|- ( ( A = 0 /\ B = 0 /\ N e. NN ) -> ( ( A ^ N ) gcd ( B ^ N ) ) = ( ( 0 ^ N ) gcd ( 0 ^ N ) ) ) |
120 |
112 116 119
|
3eqtr4d |
|- ( ( A = 0 /\ B = 0 /\ N e. NN ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) |
121 |
120
|
3expia |
|- ( ( A = 0 /\ B = 0 ) -> ( N e. NN -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
122 |
58 43
|
eqtr4i |
|- ( 0 ^ 0 ) = ( 1 gcd 1 ) |
123 |
106
|
oveq1i |
|- ( ( 0 gcd 0 ) ^ 0 ) = ( 0 ^ 0 ) |
124 |
58 58
|
oveq12i |
|- ( ( 0 ^ 0 ) gcd ( 0 ^ 0 ) ) = ( 1 gcd 1 ) |
125 |
122 123 124
|
3eqtr4i |
|- ( ( 0 gcd 0 ) ^ 0 ) = ( ( 0 ^ 0 ) gcd ( 0 ^ 0 ) ) |
126 |
|
simp1 |
|- ( ( A = 0 /\ B = 0 /\ N = 0 ) -> A = 0 ) |
127 |
|
simp2 |
|- ( ( A = 0 /\ B = 0 /\ N = 0 ) -> B = 0 ) |
128 |
126 127
|
oveq12d |
|- ( ( A = 0 /\ B = 0 /\ N = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) ) |
129 |
|
simp3 |
|- ( ( A = 0 /\ B = 0 /\ N = 0 ) -> N = 0 ) |
130 |
128 129
|
oveq12d |
|- ( ( A = 0 /\ B = 0 /\ N = 0 ) -> ( ( A gcd B ) ^ N ) = ( ( 0 gcd 0 ) ^ 0 ) ) |
131 |
126 129
|
oveq12d |
|- ( ( A = 0 /\ B = 0 /\ N = 0 ) -> ( A ^ N ) = ( 0 ^ 0 ) ) |
132 |
127 129
|
oveq12d |
|- ( ( A = 0 /\ B = 0 /\ N = 0 ) -> ( B ^ N ) = ( 0 ^ 0 ) ) |
133 |
131 132
|
oveq12d |
|- ( ( A = 0 /\ B = 0 /\ N = 0 ) -> ( ( A ^ N ) gcd ( B ^ N ) ) = ( ( 0 ^ 0 ) gcd ( 0 ^ 0 ) ) ) |
134 |
125 130 133
|
3eqtr4a |
|- ( ( A = 0 /\ B = 0 /\ N = 0 ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) |
135 |
134
|
3expia |
|- ( ( A = 0 /\ B = 0 ) -> ( N = 0 -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
136 |
121 135
|
jaod |
|- ( ( A = 0 /\ B = 0 ) -> ( ( N e. NN \/ N = 0 ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
137 |
5 136
|
syl5bi |
|- ( ( A = 0 /\ B = 0 ) -> ( N e. NN0 -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
138 |
4 67 105 137
|
ccase |
|- ( ( ( A e. NN \/ A = 0 ) /\ ( B e. NN \/ B = 0 ) ) -> ( N e. NN0 -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
139 |
1 2 138
|
syl2anb |
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( N e. NN0 -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) ) |
140 |
139
|
3impia |
|- ( ( A e. NN0 /\ B e. NN0 /\ N e. NN0 ) -> ( ( A gcd B ) ^ N ) = ( ( A ^ N ) gcd ( B ^ N ) ) ) |