Step |
Hyp |
Ref |
Expression |
1 |
|
crth.1 |
|- S = ( 0 ..^ ( M x. N ) ) |
2 |
|
crth.2 |
|- T = ( ( 0 ..^ M ) X. ( 0 ..^ N ) ) |
3 |
|
crth.3 |
|- F = ( x e. S |-> <. ( x mod M ) , ( x mod N ) >. ) |
4 |
|
crth.4 |
|- ( ph -> ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) |
5 |
|
phimul.4 |
|- U = { y e. ( 0 ..^ M ) | ( y gcd M ) = 1 } |
6 |
|
phimul.5 |
|- V = { y e. ( 0 ..^ N ) | ( y gcd N ) = 1 } |
7 |
|
phimul.6 |
|- W = { y e. S | ( y gcd ( M x. N ) ) = 1 } |
8 |
|
fzofi |
|- ( 0 ..^ M ) e. Fin |
9 |
|
ssrab2 |
|- { y e. ( 0 ..^ M ) | ( y gcd M ) = 1 } C_ ( 0 ..^ M ) |
10 |
|
ssfi |
|- ( ( ( 0 ..^ M ) e. Fin /\ { y e. ( 0 ..^ M ) | ( y gcd M ) = 1 } C_ ( 0 ..^ M ) ) -> { y e. ( 0 ..^ M ) | ( y gcd M ) = 1 } e. Fin ) |
11 |
8 9 10
|
mp2an |
|- { y e. ( 0 ..^ M ) | ( y gcd M ) = 1 } e. Fin |
12 |
5 11
|
eqeltri |
|- U e. Fin |
13 |
|
fzofi |
|- ( 0 ..^ N ) e. Fin |
14 |
|
ssrab2 |
|- { y e. ( 0 ..^ N ) | ( y gcd N ) = 1 } C_ ( 0 ..^ N ) |
15 |
|
ssfi |
|- ( ( ( 0 ..^ N ) e. Fin /\ { y e. ( 0 ..^ N ) | ( y gcd N ) = 1 } C_ ( 0 ..^ N ) ) -> { y e. ( 0 ..^ N ) | ( y gcd N ) = 1 } e. Fin ) |
16 |
13 14 15
|
mp2an |
|- { y e. ( 0 ..^ N ) | ( y gcd N ) = 1 } e. Fin |
17 |
6 16
|
eqeltri |
|- V e. Fin |
18 |
|
hashxp |
|- ( ( U e. Fin /\ V e. Fin ) -> ( # ` ( U X. V ) ) = ( ( # ` U ) x. ( # ` V ) ) ) |
19 |
12 17 18
|
mp2an |
|- ( # ` ( U X. V ) ) = ( ( # ` U ) x. ( # ` V ) ) |
20 |
|
oveq1 |
|- ( y = w -> ( y gcd ( M x. N ) ) = ( w gcd ( M x. N ) ) ) |
21 |
20
|
eqeq1d |
|- ( y = w -> ( ( y gcd ( M x. N ) ) = 1 <-> ( w gcd ( M x. N ) ) = 1 ) ) |
22 |
21 7
|
elrab2 |
|- ( w e. W <-> ( w e. S /\ ( w gcd ( M x. N ) ) = 1 ) ) |
23 |
22
|
simplbi |
|- ( w e. W -> w e. S ) |
24 |
|
oveq1 |
|- ( x = w -> ( x mod M ) = ( w mod M ) ) |
25 |
|
oveq1 |
|- ( x = w -> ( x mod N ) = ( w mod N ) ) |
26 |
24 25
|
opeq12d |
|- ( x = w -> <. ( x mod M ) , ( x mod N ) >. = <. ( w mod M ) , ( w mod N ) >. ) |
27 |
|
opex |
|- <. ( w mod M ) , ( w mod N ) >. e. _V |
28 |
26 3 27
|
fvmpt |
|- ( w e. S -> ( F ` w ) = <. ( w mod M ) , ( w mod N ) >. ) |
29 |
23 28
|
syl |
|- ( w e. W -> ( F ` w ) = <. ( w mod M ) , ( w mod N ) >. ) |
30 |
29
|
adantl |
|- ( ( ph /\ w e. W ) -> ( F ` w ) = <. ( w mod M ) , ( w mod N ) >. ) |
31 |
23 1
|
eleqtrdi |
|- ( w e. W -> w e. ( 0 ..^ ( M x. N ) ) ) |
32 |
31
|
adantl |
|- ( ( ph /\ w e. W ) -> w e. ( 0 ..^ ( M x. N ) ) ) |
33 |
|
elfzoelz |
|- ( w e. ( 0 ..^ ( M x. N ) ) -> w e. ZZ ) |
34 |
32 33
|
syl |
|- ( ( ph /\ w e. W ) -> w e. ZZ ) |
35 |
4
|
simp1d |
|- ( ph -> M e. NN ) |
36 |
35
|
adantr |
|- ( ( ph /\ w e. W ) -> M e. NN ) |
37 |
|
zmodfzo |
|- ( ( w e. ZZ /\ M e. NN ) -> ( w mod M ) e. ( 0 ..^ M ) ) |
38 |
34 36 37
|
syl2anc |
|- ( ( ph /\ w e. W ) -> ( w mod M ) e. ( 0 ..^ M ) ) |
39 |
|
modgcd |
|- ( ( w e. ZZ /\ M e. NN ) -> ( ( w mod M ) gcd M ) = ( w gcd M ) ) |
40 |
34 36 39
|
syl2anc |
|- ( ( ph /\ w e. W ) -> ( ( w mod M ) gcd M ) = ( w gcd M ) ) |
41 |
36
|
nnzd |
|- ( ( ph /\ w e. W ) -> M e. ZZ ) |
42 |
|
gcddvds |
|- ( ( w e. ZZ /\ M e. ZZ ) -> ( ( w gcd M ) || w /\ ( w gcd M ) || M ) ) |
43 |
34 41 42
|
syl2anc |
|- ( ( ph /\ w e. W ) -> ( ( w gcd M ) || w /\ ( w gcd M ) || M ) ) |
44 |
43
|
simpld |
|- ( ( ph /\ w e. W ) -> ( w gcd M ) || w ) |
45 |
|
nnne0 |
|- ( M e. NN -> M =/= 0 ) |
46 |
|
simpr |
|- ( ( w = 0 /\ M = 0 ) -> M = 0 ) |
47 |
46
|
necon3ai |
|- ( M =/= 0 -> -. ( w = 0 /\ M = 0 ) ) |
48 |
36 45 47
|
3syl |
|- ( ( ph /\ w e. W ) -> -. ( w = 0 /\ M = 0 ) ) |
49 |
|
gcdn0cl |
|- ( ( ( w e. ZZ /\ M e. ZZ ) /\ -. ( w = 0 /\ M = 0 ) ) -> ( w gcd M ) e. NN ) |
50 |
34 41 48 49
|
syl21anc |
|- ( ( ph /\ w e. W ) -> ( w gcd M ) e. NN ) |
51 |
50
|
nnzd |
|- ( ( ph /\ w e. W ) -> ( w gcd M ) e. ZZ ) |
52 |
4
|
simp2d |
|- ( ph -> N e. NN ) |
53 |
52
|
adantr |
|- ( ( ph /\ w e. W ) -> N e. NN ) |
54 |
53
|
nnzd |
|- ( ( ph /\ w e. W ) -> N e. ZZ ) |
55 |
43
|
simprd |
|- ( ( ph /\ w e. W ) -> ( w gcd M ) || M ) |
56 |
51 41 54 55
|
dvdsmultr1d |
|- ( ( ph /\ w e. W ) -> ( w gcd M ) || ( M x. N ) ) |
57 |
36 53
|
nnmulcld |
|- ( ( ph /\ w e. W ) -> ( M x. N ) e. NN ) |
58 |
57
|
nnzd |
|- ( ( ph /\ w e. W ) -> ( M x. N ) e. ZZ ) |
59 |
|
nnne0 |
|- ( ( M x. N ) e. NN -> ( M x. N ) =/= 0 ) |
60 |
|
simpr |
|- ( ( w = 0 /\ ( M x. N ) = 0 ) -> ( M x. N ) = 0 ) |
61 |
60
|
necon3ai |
|- ( ( M x. N ) =/= 0 -> -. ( w = 0 /\ ( M x. N ) = 0 ) ) |
62 |
57 59 61
|
3syl |
|- ( ( ph /\ w e. W ) -> -. ( w = 0 /\ ( M x. N ) = 0 ) ) |
63 |
|
dvdslegcd |
|- ( ( ( ( w gcd M ) e. ZZ /\ w e. ZZ /\ ( M x. N ) e. ZZ ) /\ -. ( w = 0 /\ ( M x. N ) = 0 ) ) -> ( ( ( w gcd M ) || w /\ ( w gcd M ) || ( M x. N ) ) -> ( w gcd M ) <_ ( w gcd ( M x. N ) ) ) ) |
64 |
51 34 58 62 63
|
syl31anc |
|- ( ( ph /\ w e. W ) -> ( ( ( w gcd M ) || w /\ ( w gcd M ) || ( M x. N ) ) -> ( w gcd M ) <_ ( w gcd ( M x. N ) ) ) ) |
65 |
44 56 64
|
mp2and |
|- ( ( ph /\ w e. W ) -> ( w gcd M ) <_ ( w gcd ( M x. N ) ) ) |
66 |
22
|
simprbi |
|- ( w e. W -> ( w gcd ( M x. N ) ) = 1 ) |
67 |
66
|
adantl |
|- ( ( ph /\ w e. W ) -> ( w gcd ( M x. N ) ) = 1 ) |
68 |
65 67
|
breqtrd |
|- ( ( ph /\ w e. W ) -> ( w gcd M ) <_ 1 ) |
69 |
|
nnle1eq1 |
|- ( ( w gcd M ) e. NN -> ( ( w gcd M ) <_ 1 <-> ( w gcd M ) = 1 ) ) |
70 |
50 69
|
syl |
|- ( ( ph /\ w e. W ) -> ( ( w gcd M ) <_ 1 <-> ( w gcd M ) = 1 ) ) |
71 |
68 70
|
mpbid |
|- ( ( ph /\ w e. W ) -> ( w gcd M ) = 1 ) |
72 |
40 71
|
eqtrd |
|- ( ( ph /\ w e. W ) -> ( ( w mod M ) gcd M ) = 1 ) |
73 |
|
oveq1 |
|- ( y = ( w mod M ) -> ( y gcd M ) = ( ( w mod M ) gcd M ) ) |
74 |
73
|
eqeq1d |
|- ( y = ( w mod M ) -> ( ( y gcd M ) = 1 <-> ( ( w mod M ) gcd M ) = 1 ) ) |
75 |
74 5
|
elrab2 |
|- ( ( w mod M ) e. U <-> ( ( w mod M ) e. ( 0 ..^ M ) /\ ( ( w mod M ) gcd M ) = 1 ) ) |
76 |
38 72 75
|
sylanbrc |
|- ( ( ph /\ w e. W ) -> ( w mod M ) e. U ) |
77 |
|
zmodfzo |
|- ( ( w e. ZZ /\ N e. NN ) -> ( w mod N ) e. ( 0 ..^ N ) ) |
78 |
34 53 77
|
syl2anc |
|- ( ( ph /\ w e. W ) -> ( w mod N ) e. ( 0 ..^ N ) ) |
79 |
|
modgcd |
|- ( ( w e. ZZ /\ N e. NN ) -> ( ( w mod N ) gcd N ) = ( w gcd N ) ) |
80 |
34 53 79
|
syl2anc |
|- ( ( ph /\ w e. W ) -> ( ( w mod N ) gcd N ) = ( w gcd N ) ) |
81 |
|
gcddvds |
|- ( ( w e. ZZ /\ N e. ZZ ) -> ( ( w gcd N ) || w /\ ( w gcd N ) || N ) ) |
82 |
34 54 81
|
syl2anc |
|- ( ( ph /\ w e. W ) -> ( ( w gcd N ) || w /\ ( w gcd N ) || N ) ) |
83 |
82
|
simpld |
|- ( ( ph /\ w e. W ) -> ( w gcd N ) || w ) |
84 |
|
nnne0 |
|- ( N e. NN -> N =/= 0 ) |
85 |
|
simpr |
|- ( ( w = 0 /\ N = 0 ) -> N = 0 ) |
86 |
85
|
necon3ai |
|- ( N =/= 0 -> -. ( w = 0 /\ N = 0 ) ) |
87 |
53 84 86
|
3syl |
|- ( ( ph /\ w e. W ) -> -. ( w = 0 /\ N = 0 ) ) |
88 |
|
gcdn0cl |
|- ( ( ( w e. ZZ /\ N e. ZZ ) /\ -. ( w = 0 /\ N = 0 ) ) -> ( w gcd N ) e. NN ) |
89 |
34 54 87 88
|
syl21anc |
|- ( ( ph /\ w e. W ) -> ( w gcd N ) e. NN ) |
90 |
89
|
nnzd |
|- ( ( ph /\ w e. W ) -> ( w gcd N ) e. ZZ ) |
91 |
82
|
simprd |
|- ( ( ph /\ w e. W ) -> ( w gcd N ) || N ) |
92 |
|
dvdsmul2 |
|- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M x. N ) ) |
93 |
41 54 92
|
syl2anc |
|- ( ( ph /\ w e. W ) -> N || ( M x. N ) ) |
94 |
90 54 58 91 93
|
dvdstrd |
|- ( ( ph /\ w e. W ) -> ( w gcd N ) || ( M x. N ) ) |
95 |
|
dvdslegcd |
|- ( ( ( ( w gcd N ) e. ZZ /\ w e. ZZ /\ ( M x. N ) e. ZZ ) /\ -. ( w = 0 /\ ( M x. N ) = 0 ) ) -> ( ( ( w gcd N ) || w /\ ( w gcd N ) || ( M x. N ) ) -> ( w gcd N ) <_ ( w gcd ( M x. N ) ) ) ) |
96 |
90 34 58 62 95
|
syl31anc |
|- ( ( ph /\ w e. W ) -> ( ( ( w gcd N ) || w /\ ( w gcd N ) || ( M x. N ) ) -> ( w gcd N ) <_ ( w gcd ( M x. N ) ) ) ) |
97 |
83 94 96
|
mp2and |
|- ( ( ph /\ w e. W ) -> ( w gcd N ) <_ ( w gcd ( M x. N ) ) ) |
98 |
97 67
|
breqtrd |
|- ( ( ph /\ w e. W ) -> ( w gcd N ) <_ 1 ) |
99 |
|
nnle1eq1 |
|- ( ( w gcd N ) e. NN -> ( ( w gcd N ) <_ 1 <-> ( w gcd N ) = 1 ) ) |
100 |
89 99
|
syl |
|- ( ( ph /\ w e. W ) -> ( ( w gcd N ) <_ 1 <-> ( w gcd N ) = 1 ) ) |
101 |
98 100
|
mpbid |
|- ( ( ph /\ w e. W ) -> ( w gcd N ) = 1 ) |
102 |
80 101
|
eqtrd |
|- ( ( ph /\ w e. W ) -> ( ( w mod N ) gcd N ) = 1 ) |
103 |
|
oveq1 |
|- ( y = ( w mod N ) -> ( y gcd N ) = ( ( w mod N ) gcd N ) ) |
104 |
103
|
eqeq1d |
|- ( y = ( w mod N ) -> ( ( y gcd N ) = 1 <-> ( ( w mod N ) gcd N ) = 1 ) ) |
105 |
104 6
|
elrab2 |
|- ( ( w mod N ) e. V <-> ( ( w mod N ) e. ( 0 ..^ N ) /\ ( ( w mod N ) gcd N ) = 1 ) ) |
106 |
78 102 105
|
sylanbrc |
|- ( ( ph /\ w e. W ) -> ( w mod N ) e. V ) |
107 |
76 106
|
opelxpd |
|- ( ( ph /\ w e. W ) -> <. ( w mod M ) , ( w mod N ) >. e. ( U X. V ) ) |
108 |
30 107
|
eqeltrd |
|- ( ( ph /\ w e. W ) -> ( F ` w ) e. ( U X. V ) ) |
109 |
108
|
ralrimiva |
|- ( ph -> A. w e. W ( F ` w ) e. ( U X. V ) ) |
110 |
1 2 3 4
|
crth |
|- ( ph -> F : S -1-1-onto-> T ) |
111 |
|
f1ofn |
|- ( F : S -1-1-onto-> T -> F Fn S ) |
112 |
|
fnfun |
|- ( F Fn S -> Fun F ) |
113 |
110 111 112
|
3syl |
|- ( ph -> Fun F ) |
114 |
7
|
ssrab3 |
|- W C_ S |
115 |
|
fndm |
|- ( F Fn S -> dom F = S ) |
116 |
110 111 115
|
3syl |
|- ( ph -> dom F = S ) |
117 |
114 116
|
sseqtrrid |
|- ( ph -> W C_ dom F ) |
118 |
|
funimass4 |
|- ( ( Fun F /\ W C_ dom F ) -> ( ( F " W ) C_ ( U X. V ) <-> A. w e. W ( F ` w ) e. ( U X. V ) ) ) |
119 |
113 117 118
|
syl2anc |
|- ( ph -> ( ( F " W ) C_ ( U X. V ) <-> A. w e. W ( F ` w ) e. ( U X. V ) ) ) |
120 |
109 119
|
mpbird |
|- ( ph -> ( F " W ) C_ ( U X. V ) ) |
121 |
5
|
ssrab3 |
|- U C_ ( 0 ..^ M ) |
122 |
6
|
ssrab3 |
|- V C_ ( 0 ..^ N ) |
123 |
|
xpss12 |
|- ( ( U C_ ( 0 ..^ M ) /\ V C_ ( 0 ..^ N ) ) -> ( U X. V ) C_ ( ( 0 ..^ M ) X. ( 0 ..^ N ) ) ) |
124 |
121 122 123
|
mp2an |
|- ( U X. V ) C_ ( ( 0 ..^ M ) X. ( 0 ..^ N ) ) |
125 |
124 2
|
sseqtrri |
|- ( U X. V ) C_ T |
126 |
125
|
sseli |
|- ( z e. ( U X. V ) -> z e. T ) |
127 |
|
f1ocnvfv2 |
|- ( ( F : S -1-1-onto-> T /\ z e. T ) -> ( F ` ( `' F ` z ) ) = z ) |
128 |
110 126 127
|
syl2an |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( F ` ( `' F ` z ) ) = z ) |
129 |
|
f1ocnv |
|- ( F : S -1-1-onto-> T -> `' F : T -1-1-onto-> S ) |
130 |
|
f1of |
|- ( `' F : T -1-1-onto-> S -> `' F : T --> S ) |
131 |
110 129 130
|
3syl |
|- ( ph -> `' F : T --> S ) |
132 |
|
ffvelrn |
|- ( ( `' F : T --> S /\ z e. T ) -> ( `' F ` z ) e. S ) |
133 |
131 126 132
|
syl2an |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( `' F ` z ) e. S ) |
134 |
133 1
|
eleqtrdi |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( `' F ` z ) e. ( 0 ..^ ( M x. N ) ) ) |
135 |
|
elfzoelz |
|- ( ( `' F ` z ) e. ( 0 ..^ ( M x. N ) ) -> ( `' F ` z ) e. ZZ ) |
136 |
134 135
|
syl |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( `' F ` z ) e. ZZ ) |
137 |
35
|
adantr |
|- ( ( ph /\ z e. ( U X. V ) ) -> M e. NN ) |
138 |
|
modgcd |
|- ( ( ( `' F ` z ) e. ZZ /\ M e. NN ) -> ( ( ( `' F ` z ) mod M ) gcd M ) = ( ( `' F ` z ) gcd M ) ) |
139 |
136 137 138
|
syl2anc |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( ( `' F ` z ) mod M ) gcd M ) = ( ( `' F ` z ) gcd M ) ) |
140 |
|
oveq1 |
|- ( w = ( `' F ` z ) -> ( w mod M ) = ( ( `' F ` z ) mod M ) ) |
141 |
|
oveq1 |
|- ( w = ( `' F ` z ) -> ( w mod N ) = ( ( `' F ` z ) mod N ) ) |
142 |
140 141
|
opeq12d |
|- ( w = ( `' F ` z ) -> <. ( w mod M ) , ( w mod N ) >. = <. ( ( `' F ` z ) mod M ) , ( ( `' F ` z ) mod N ) >. ) |
143 |
26
|
cbvmptv |
|- ( x e. S |-> <. ( x mod M ) , ( x mod N ) >. ) = ( w e. S |-> <. ( w mod M ) , ( w mod N ) >. ) |
144 |
3 143
|
eqtri |
|- F = ( w e. S |-> <. ( w mod M ) , ( w mod N ) >. ) |
145 |
|
opex |
|- <. ( ( `' F ` z ) mod M ) , ( ( `' F ` z ) mod N ) >. e. _V |
146 |
142 144 145
|
fvmpt |
|- ( ( `' F ` z ) e. S -> ( F ` ( `' F ` z ) ) = <. ( ( `' F ` z ) mod M ) , ( ( `' F ` z ) mod N ) >. ) |
147 |
133 146
|
syl |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( F ` ( `' F ` z ) ) = <. ( ( `' F ` z ) mod M ) , ( ( `' F ` z ) mod N ) >. ) |
148 |
128 147
|
eqtr3d |
|- ( ( ph /\ z e. ( U X. V ) ) -> z = <. ( ( `' F ` z ) mod M ) , ( ( `' F ` z ) mod N ) >. ) |
149 |
|
simpr |
|- ( ( ph /\ z e. ( U X. V ) ) -> z e. ( U X. V ) ) |
150 |
148 149
|
eqeltrrd |
|- ( ( ph /\ z e. ( U X. V ) ) -> <. ( ( `' F ` z ) mod M ) , ( ( `' F ` z ) mod N ) >. e. ( U X. V ) ) |
151 |
|
opelxp |
|- ( <. ( ( `' F ` z ) mod M ) , ( ( `' F ` z ) mod N ) >. e. ( U X. V ) <-> ( ( ( `' F ` z ) mod M ) e. U /\ ( ( `' F ` z ) mod N ) e. V ) ) |
152 |
150 151
|
sylib |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( ( `' F ` z ) mod M ) e. U /\ ( ( `' F ` z ) mod N ) e. V ) ) |
153 |
152
|
simpld |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( `' F ` z ) mod M ) e. U ) |
154 |
|
oveq1 |
|- ( y = ( ( `' F ` z ) mod M ) -> ( y gcd M ) = ( ( ( `' F ` z ) mod M ) gcd M ) ) |
155 |
154
|
eqeq1d |
|- ( y = ( ( `' F ` z ) mod M ) -> ( ( y gcd M ) = 1 <-> ( ( ( `' F ` z ) mod M ) gcd M ) = 1 ) ) |
156 |
155 5
|
elrab2 |
|- ( ( ( `' F ` z ) mod M ) e. U <-> ( ( ( `' F ` z ) mod M ) e. ( 0 ..^ M ) /\ ( ( ( `' F ` z ) mod M ) gcd M ) = 1 ) ) |
157 |
153 156
|
sylib |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( ( `' F ` z ) mod M ) e. ( 0 ..^ M ) /\ ( ( ( `' F ` z ) mod M ) gcd M ) = 1 ) ) |
158 |
157
|
simprd |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( ( `' F ` z ) mod M ) gcd M ) = 1 ) |
159 |
139 158
|
eqtr3d |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( `' F ` z ) gcd M ) = 1 ) |
160 |
52
|
adantr |
|- ( ( ph /\ z e. ( U X. V ) ) -> N e. NN ) |
161 |
|
modgcd |
|- ( ( ( `' F ` z ) e. ZZ /\ N e. NN ) -> ( ( ( `' F ` z ) mod N ) gcd N ) = ( ( `' F ` z ) gcd N ) ) |
162 |
136 160 161
|
syl2anc |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( ( `' F ` z ) mod N ) gcd N ) = ( ( `' F ` z ) gcd N ) ) |
163 |
152
|
simprd |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( `' F ` z ) mod N ) e. V ) |
164 |
|
oveq1 |
|- ( y = ( ( `' F ` z ) mod N ) -> ( y gcd N ) = ( ( ( `' F ` z ) mod N ) gcd N ) ) |
165 |
164
|
eqeq1d |
|- ( y = ( ( `' F ` z ) mod N ) -> ( ( y gcd N ) = 1 <-> ( ( ( `' F ` z ) mod N ) gcd N ) = 1 ) ) |
166 |
165 6
|
elrab2 |
|- ( ( ( `' F ` z ) mod N ) e. V <-> ( ( ( `' F ` z ) mod N ) e. ( 0 ..^ N ) /\ ( ( ( `' F ` z ) mod N ) gcd N ) = 1 ) ) |
167 |
163 166
|
sylib |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( ( `' F ` z ) mod N ) e. ( 0 ..^ N ) /\ ( ( ( `' F ` z ) mod N ) gcd N ) = 1 ) ) |
168 |
167
|
simprd |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( ( `' F ` z ) mod N ) gcd N ) = 1 ) |
169 |
162 168
|
eqtr3d |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( `' F ` z ) gcd N ) = 1 ) |
170 |
35
|
nnzd |
|- ( ph -> M e. ZZ ) |
171 |
170
|
adantr |
|- ( ( ph /\ z e. ( U X. V ) ) -> M e. ZZ ) |
172 |
52
|
nnzd |
|- ( ph -> N e. ZZ ) |
173 |
172
|
adantr |
|- ( ( ph /\ z e. ( U X. V ) ) -> N e. ZZ ) |
174 |
|
rpmul |
|- ( ( ( `' F ` z ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( ( `' F ` z ) gcd M ) = 1 /\ ( ( `' F ` z ) gcd N ) = 1 ) -> ( ( `' F ` z ) gcd ( M x. N ) ) = 1 ) ) |
175 |
136 171 173 174
|
syl3anc |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( ( ( `' F ` z ) gcd M ) = 1 /\ ( ( `' F ` z ) gcd N ) = 1 ) -> ( ( `' F ` z ) gcd ( M x. N ) ) = 1 ) ) |
176 |
159 169 175
|
mp2and |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( ( `' F ` z ) gcd ( M x. N ) ) = 1 ) |
177 |
|
oveq1 |
|- ( y = ( `' F ` z ) -> ( y gcd ( M x. N ) ) = ( ( `' F ` z ) gcd ( M x. N ) ) ) |
178 |
177
|
eqeq1d |
|- ( y = ( `' F ` z ) -> ( ( y gcd ( M x. N ) ) = 1 <-> ( ( `' F ` z ) gcd ( M x. N ) ) = 1 ) ) |
179 |
178 7
|
elrab2 |
|- ( ( `' F ` z ) e. W <-> ( ( `' F ` z ) e. S /\ ( ( `' F ` z ) gcd ( M x. N ) ) = 1 ) ) |
180 |
133 176 179
|
sylanbrc |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( `' F ` z ) e. W ) |
181 |
|
funfvima2 |
|- ( ( Fun F /\ W C_ dom F ) -> ( ( `' F ` z ) e. W -> ( F ` ( `' F ` z ) ) e. ( F " W ) ) ) |
182 |
113 117 181
|
syl2anc |
|- ( ph -> ( ( `' F ` z ) e. W -> ( F ` ( `' F ` z ) ) e. ( F " W ) ) ) |
183 |
182
|
imp |
|- ( ( ph /\ ( `' F ` z ) e. W ) -> ( F ` ( `' F ` z ) ) e. ( F " W ) ) |
184 |
180 183
|
syldan |
|- ( ( ph /\ z e. ( U X. V ) ) -> ( F ` ( `' F ` z ) ) e. ( F " W ) ) |
185 |
128 184
|
eqeltrrd |
|- ( ( ph /\ z e. ( U X. V ) ) -> z e. ( F " W ) ) |
186 |
120 185
|
eqelssd |
|- ( ph -> ( F " W ) = ( U X. V ) ) |
187 |
|
f1of1 |
|- ( F : S -1-1-onto-> T -> F : S -1-1-> T ) |
188 |
110 187
|
syl |
|- ( ph -> F : S -1-1-> T ) |
189 |
|
fzofi |
|- ( 0 ..^ ( M x. N ) ) e. Fin |
190 |
1 189
|
eqeltri |
|- S e. Fin |
191 |
|
ssfi |
|- ( ( S e. Fin /\ W C_ S ) -> W e. Fin ) |
192 |
190 114 191
|
mp2an |
|- W e. Fin |
193 |
192
|
elexi |
|- W e. _V |
194 |
193
|
f1imaen |
|- ( ( F : S -1-1-> T /\ W C_ S ) -> ( F " W ) ~~ W ) |
195 |
188 114 194
|
sylancl |
|- ( ph -> ( F " W ) ~~ W ) |
196 |
186 195
|
eqbrtrrd |
|- ( ph -> ( U X. V ) ~~ W ) |
197 |
|
xpfi |
|- ( ( U e. Fin /\ V e. Fin ) -> ( U X. V ) e. Fin ) |
198 |
12 17 197
|
mp2an |
|- ( U X. V ) e. Fin |
199 |
|
hashen |
|- ( ( ( U X. V ) e. Fin /\ W e. Fin ) -> ( ( # ` ( U X. V ) ) = ( # ` W ) <-> ( U X. V ) ~~ W ) ) |
200 |
198 192 199
|
mp2an |
|- ( ( # ` ( U X. V ) ) = ( # ` W ) <-> ( U X. V ) ~~ W ) |
201 |
196 200
|
sylibr |
|- ( ph -> ( # ` ( U X. V ) ) = ( # ` W ) ) |
202 |
19 201
|
syl5reqr |
|- ( ph -> ( # ` W ) = ( ( # ` U ) x. ( # ` V ) ) ) |
203 |
35 52
|
nnmulcld |
|- ( ph -> ( M x. N ) e. NN ) |
204 |
|
dfphi2 |
|- ( ( M x. N ) e. NN -> ( phi ` ( M x. N ) ) = ( # ` { y e. ( 0 ..^ ( M x. N ) ) | ( y gcd ( M x. N ) ) = 1 } ) ) |
205 |
1
|
rabeqi |
|- { y e. S | ( y gcd ( M x. N ) ) = 1 } = { y e. ( 0 ..^ ( M x. N ) ) | ( y gcd ( M x. N ) ) = 1 } |
206 |
7 205
|
eqtri |
|- W = { y e. ( 0 ..^ ( M x. N ) ) | ( y gcd ( M x. N ) ) = 1 } |
207 |
206
|
fveq2i |
|- ( # ` W ) = ( # ` { y e. ( 0 ..^ ( M x. N ) ) | ( y gcd ( M x. N ) ) = 1 } ) |
208 |
204 207
|
eqtr4di |
|- ( ( M x. N ) e. NN -> ( phi ` ( M x. N ) ) = ( # ` W ) ) |
209 |
203 208
|
syl |
|- ( ph -> ( phi ` ( M x. N ) ) = ( # ` W ) ) |
210 |
|
dfphi2 |
|- ( M e. NN -> ( phi ` M ) = ( # ` { y e. ( 0 ..^ M ) | ( y gcd M ) = 1 } ) ) |
211 |
5
|
fveq2i |
|- ( # ` U ) = ( # ` { y e. ( 0 ..^ M ) | ( y gcd M ) = 1 } ) |
212 |
210 211
|
eqtr4di |
|- ( M e. NN -> ( phi ` M ) = ( # ` U ) ) |
213 |
35 212
|
syl |
|- ( ph -> ( phi ` M ) = ( # ` U ) ) |
214 |
|
dfphi2 |
|- ( N e. NN -> ( phi ` N ) = ( # ` { y e. ( 0 ..^ N ) | ( y gcd N ) = 1 } ) ) |
215 |
6
|
fveq2i |
|- ( # ` V ) = ( # ` { y e. ( 0 ..^ N ) | ( y gcd N ) = 1 } ) |
216 |
214 215
|
eqtr4di |
|- ( N e. NN -> ( phi ` N ) = ( # ` V ) ) |
217 |
52 216
|
syl |
|- ( ph -> ( phi ` N ) = ( # ` V ) ) |
218 |
213 217
|
oveq12d |
|- ( ph -> ( ( phi ` M ) x. ( phi ` N ) ) = ( ( # ` U ) x. ( # ` V ) ) ) |
219 |
202 209 218
|
3eqtr4d |
|- ( ph -> ( phi ` ( M x. N ) ) = ( ( phi ` M ) x. ( phi ` N ) ) ) |