Step |
Hyp |
Ref |
Expression |
1 |
|
4sq.1 |
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) } |
2 |
|
4sq.2 |
|- ( ph -> N e. NN ) |
3 |
|
4sq.3 |
|- ( ph -> P = ( ( 2 x. N ) + 1 ) ) |
4 |
|
4sq.4 |
|- ( ph -> P e. Prime ) |
5 |
|
4sq.5 |
|- ( ph -> ( 0 ... ( 2 x. N ) ) C_ S ) |
6 |
|
4sq.6 |
|- T = { i e. NN | ( i x. P ) e. S } |
7 |
|
4sq.7 |
|- M = inf ( T , RR , < ) |
8 |
|
4sq.m |
|- ( ph -> M e. ( ZZ>= ` 2 ) ) |
9 |
|
4sq.a |
|- ( ph -> A e. ZZ ) |
10 |
|
4sq.b |
|- ( ph -> B e. ZZ ) |
11 |
|
4sq.c |
|- ( ph -> C e. ZZ ) |
12 |
|
4sq.d |
|- ( ph -> D e. ZZ ) |
13 |
|
4sq.e |
|- E = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) ) |
14 |
|
4sq.f |
|- F = ( ( ( B + ( M / 2 ) ) mod M ) - ( M / 2 ) ) |
15 |
|
4sq.g |
|- G = ( ( ( C + ( M / 2 ) ) mod M ) - ( M / 2 ) ) |
16 |
|
4sq.h |
|- H = ( ( ( D + ( M / 2 ) ) mod M ) - ( M / 2 ) ) |
17 |
|
4sq.r |
|- R = ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M ) |
18 |
|
4sq.p |
|- ( ph -> ( M x. P ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) |
19 |
|
eluz2nn |
|- ( M e. ( ZZ>= ` 2 ) -> M e. NN ) |
20 |
8 19
|
syl |
|- ( ph -> M e. NN ) |
21 |
9 20 13
|
4sqlem5 |
|- ( ph -> ( E e. ZZ /\ ( ( A - E ) / M ) e. ZZ ) ) |
22 |
21
|
simpld |
|- ( ph -> E e. ZZ ) |
23 |
|
zsqcl |
|- ( E e. ZZ -> ( E ^ 2 ) e. ZZ ) |
24 |
22 23
|
syl |
|- ( ph -> ( E ^ 2 ) e. ZZ ) |
25 |
24
|
zred |
|- ( ph -> ( E ^ 2 ) e. RR ) |
26 |
10 20 14
|
4sqlem5 |
|- ( ph -> ( F e. ZZ /\ ( ( B - F ) / M ) e. ZZ ) ) |
27 |
26
|
simpld |
|- ( ph -> F e. ZZ ) |
28 |
|
zsqcl |
|- ( F e. ZZ -> ( F ^ 2 ) e. ZZ ) |
29 |
27 28
|
syl |
|- ( ph -> ( F ^ 2 ) e. ZZ ) |
30 |
29
|
zred |
|- ( ph -> ( F ^ 2 ) e. RR ) |
31 |
25 30
|
readdcld |
|- ( ph -> ( ( E ^ 2 ) + ( F ^ 2 ) ) e. RR ) |
32 |
11 20 15
|
4sqlem5 |
|- ( ph -> ( G e. ZZ /\ ( ( C - G ) / M ) e. ZZ ) ) |
33 |
32
|
simpld |
|- ( ph -> G e. ZZ ) |
34 |
|
zsqcl |
|- ( G e. ZZ -> ( G ^ 2 ) e. ZZ ) |
35 |
33 34
|
syl |
|- ( ph -> ( G ^ 2 ) e. ZZ ) |
36 |
35
|
zred |
|- ( ph -> ( G ^ 2 ) e. RR ) |
37 |
12 20 16
|
4sqlem5 |
|- ( ph -> ( H e. ZZ /\ ( ( D - H ) / M ) e. ZZ ) ) |
38 |
37
|
simpld |
|- ( ph -> H e. ZZ ) |
39 |
|
zsqcl |
|- ( H e. ZZ -> ( H ^ 2 ) e. ZZ ) |
40 |
38 39
|
syl |
|- ( ph -> ( H ^ 2 ) e. ZZ ) |
41 |
40
|
zred |
|- ( ph -> ( H ^ 2 ) e. RR ) |
42 |
36 41
|
readdcld |
|- ( ph -> ( ( G ^ 2 ) + ( H ^ 2 ) ) e. RR ) |
43 |
20
|
nnred |
|- ( ph -> M e. RR ) |
44 |
43
|
resqcld |
|- ( ph -> ( M ^ 2 ) e. RR ) |
45 |
44
|
rehalfcld |
|- ( ph -> ( ( M ^ 2 ) / 2 ) e. RR ) |
46 |
45
|
rehalfcld |
|- ( ph -> ( ( ( M ^ 2 ) / 2 ) / 2 ) e. RR ) |
47 |
9 20 13
|
4sqlem7 |
|- ( ph -> ( E ^ 2 ) <_ ( ( ( M ^ 2 ) / 2 ) / 2 ) ) |
48 |
10 20 14
|
4sqlem7 |
|- ( ph -> ( F ^ 2 ) <_ ( ( ( M ^ 2 ) / 2 ) / 2 ) ) |
49 |
25 30 46 46 47 48
|
le2addd |
|- ( ph -> ( ( E ^ 2 ) + ( F ^ 2 ) ) <_ ( ( ( ( M ^ 2 ) / 2 ) / 2 ) + ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) |
50 |
45
|
recnd |
|- ( ph -> ( ( M ^ 2 ) / 2 ) e. CC ) |
51 |
50
|
2halvesd |
|- ( ph -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) + ( ( ( M ^ 2 ) / 2 ) / 2 ) ) = ( ( M ^ 2 ) / 2 ) ) |
52 |
49 51
|
breqtrd |
|- ( ph -> ( ( E ^ 2 ) + ( F ^ 2 ) ) <_ ( ( M ^ 2 ) / 2 ) ) |
53 |
11 20 15
|
4sqlem7 |
|- ( ph -> ( G ^ 2 ) <_ ( ( ( M ^ 2 ) / 2 ) / 2 ) ) |
54 |
12 20 16
|
4sqlem7 |
|- ( ph -> ( H ^ 2 ) <_ ( ( ( M ^ 2 ) / 2 ) / 2 ) ) |
55 |
36 41 46 46 53 54
|
le2addd |
|- ( ph -> ( ( G ^ 2 ) + ( H ^ 2 ) ) <_ ( ( ( ( M ^ 2 ) / 2 ) / 2 ) + ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) |
56 |
55 51
|
breqtrd |
|- ( ph -> ( ( G ^ 2 ) + ( H ^ 2 ) ) <_ ( ( M ^ 2 ) / 2 ) ) |
57 |
31 42 45 45 52 56
|
le2addd |
|- ( ph -> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) <_ ( ( ( M ^ 2 ) / 2 ) + ( ( M ^ 2 ) / 2 ) ) ) |
58 |
44
|
recnd |
|- ( ph -> ( M ^ 2 ) e. CC ) |
59 |
58
|
2halvesd |
|- ( ph -> ( ( ( M ^ 2 ) / 2 ) + ( ( M ^ 2 ) / 2 ) ) = ( M ^ 2 ) ) |
60 |
57 59
|
breqtrd |
|- ( ph -> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) <_ ( M ^ 2 ) ) |
61 |
43
|
recnd |
|- ( ph -> M e. CC ) |
62 |
61
|
sqvald |
|- ( ph -> ( M ^ 2 ) = ( M x. M ) ) |
63 |
60 62
|
breqtrd |
|- ( ph -> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) <_ ( M x. M ) ) |
64 |
31 42
|
readdcld |
|- ( ph -> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) e. RR ) |
65 |
20
|
nngt0d |
|- ( ph -> 0 < M ) |
66 |
|
ledivmul |
|- ( ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) e. RR /\ M e. RR /\ ( M e. RR /\ 0 < M ) ) -> ( ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M ) <_ M <-> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) <_ ( M x. M ) ) ) |
67 |
64 43 43 65 66
|
syl112anc |
|- ( ph -> ( ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M ) <_ M <-> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) <_ ( M x. M ) ) ) |
68 |
63 67
|
mpbird |
|- ( ph -> ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M ) <_ M ) |
69 |
17 68
|
eqbrtrid |
|- ( ph -> R <_ M ) |
70 |
|
simpr |
|- ( ( ph /\ R = 0 ) -> R = 0 ) |
71 |
17 70
|
syl5eqr |
|- ( ( ph /\ R = 0 ) -> ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M ) = 0 ) |
72 |
64
|
recnd |
|- ( ph -> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) e. CC ) |
73 |
20
|
nnne0d |
|- ( ph -> M =/= 0 ) |
74 |
72 61 73
|
diveq0ad |
|- ( ph -> ( ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M ) = 0 <-> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) = 0 ) ) |
75 |
|
zsqcl2 |
|- ( E e. ZZ -> ( E ^ 2 ) e. NN0 ) |
76 |
22 75
|
syl |
|- ( ph -> ( E ^ 2 ) e. NN0 ) |
77 |
|
zsqcl2 |
|- ( F e. ZZ -> ( F ^ 2 ) e. NN0 ) |
78 |
27 77
|
syl |
|- ( ph -> ( F ^ 2 ) e. NN0 ) |
79 |
76 78
|
nn0addcld |
|- ( ph -> ( ( E ^ 2 ) + ( F ^ 2 ) ) e. NN0 ) |
80 |
79
|
nn0ge0d |
|- ( ph -> 0 <_ ( ( E ^ 2 ) + ( F ^ 2 ) ) ) |
81 |
|
zsqcl2 |
|- ( G e. ZZ -> ( G ^ 2 ) e. NN0 ) |
82 |
33 81
|
syl |
|- ( ph -> ( G ^ 2 ) e. NN0 ) |
83 |
|
zsqcl2 |
|- ( H e. ZZ -> ( H ^ 2 ) e. NN0 ) |
84 |
38 83
|
syl |
|- ( ph -> ( H ^ 2 ) e. NN0 ) |
85 |
82 84
|
nn0addcld |
|- ( ph -> ( ( G ^ 2 ) + ( H ^ 2 ) ) e. NN0 ) |
86 |
85
|
nn0ge0d |
|- ( ph -> 0 <_ ( ( G ^ 2 ) + ( H ^ 2 ) ) ) |
87 |
|
add20 |
|- ( ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) e. RR /\ 0 <_ ( ( E ^ 2 ) + ( F ^ 2 ) ) ) /\ ( ( ( G ^ 2 ) + ( H ^ 2 ) ) e. RR /\ 0 <_ ( ( G ^ 2 ) + ( H ^ 2 ) ) ) ) -> ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) = 0 <-> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) = 0 /\ ( ( G ^ 2 ) + ( H ^ 2 ) ) = 0 ) ) ) |
88 |
31 80 42 86 87
|
syl22anc |
|- ( ph -> ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) = 0 <-> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) = 0 /\ ( ( G ^ 2 ) + ( H ^ 2 ) ) = 0 ) ) ) |
89 |
74 88
|
bitrd |
|- ( ph -> ( ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M ) = 0 <-> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) = 0 /\ ( ( G ^ 2 ) + ( H ^ 2 ) ) = 0 ) ) ) |
90 |
89
|
biimpa |
|- ( ( ph /\ ( ( ( ( E ^ 2 ) + ( F ^ 2 ) ) + ( ( G ^ 2 ) + ( H ^ 2 ) ) ) / M ) = 0 ) -> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) = 0 /\ ( ( G ^ 2 ) + ( H ^ 2 ) ) = 0 ) ) |
91 |
71 90
|
syldan |
|- ( ( ph /\ R = 0 ) -> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) = 0 /\ ( ( G ^ 2 ) + ( H ^ 2 ) ) = 0 ) ) |
92 |
91
|
simpld |
|- ( ( ph /\ R = 0 ) -> ( ( E ^ 2 ) + ( F ^ 2 ) ) = 0 ) |
93 |
76
|
nn0ge0d |
|- ( ph -> 0 <_ ( E ^ 2 ) ) |
94 |
78
|
nn0ge0d |
|- ( ph -> 0 <_ ( F ^ 2 ) ) |
95 |
|
add20 |
|- ( ( ( ( E ^ 2 ) e. RR /\ 0 <_ ( E ^ 2 ) ) /\ ( ( F ^ 2 ) e. RR /\ 0 <_ ( F ^ 2 ) ) ) -> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) = 0 <-> ( ( E ^ 2 ) = 0 /\ ( F ^ 2 ) = 0 ) ) ) |
96 |
25 93 30 94 95
|
syl22anc |
|- ( ph -> ( ( ( E ^ 2 ) + ( F ^ 2 ) ) = 0 <-> ( ( E ^ 2 ) = 0 /\ ( F ^ 2 ) = 0 ) ) ) |
97 |
96
|
biimpa |
|- ( ( ph /\ ( ( E ^ 2 ) + ( F ^ 2 ) ) = 0 ) -> ( ( E ^ 2 ) = 0 /\ ( F ^ 2 ) = 0 ) ) |
98 |
92 97
|
syldan |
|- ( ( ph /\ R = 0 ) -> ( ( E ^ 2 ) = 0 /\ ( F ^ 2 ) = 0 ) ) |
99 |
98
|
simpld |
|- ( ( ph /\ R = 0 ) -> ( E ^ 2 ) = 0 ) |
100 |
9 20 13 99
|
4sqlem9 |
|- ( ( ph /\ R = 0 ) -> ( M ^ 2 ) || ( A ^ 2 ) ) |
101 |
98
|
simprd |
|- ( ( ph /\ R = 0 ) -> ( F ^ 2 ) = 0 ) |
102 |
10 20 14 101
|
4sqlem9 |
|- ( ( ph /\ R = 0 ) -> ( M ^ 2 ) || ( B ^ 2 ) ) |
103 |
20
|
nnsqcld |
|- ( ph -> ( M ^ 2 ) e. NN ) |
104 |
103
|
nnzd |
|- ( ph -> ( M ^ 2 ) e. ZZ ) |
105 |
|
zsqcl |
|- ( A e. ZZ -> ( A ^ 2 ) e. ZZ ) |
106 |
9 105
|
syl |
|- ( ph -> ( A ^ 2 ) e. ZZ ) |
107 |
|
zsqcl |
|- ( B e. ZZ -> ( B ^ 2 ) e. ZZ ) |
108 |
10 107
|
syl |
|- ( ph -> ( B ^ 2 ) e. ZZ ) |
109 |
|
dvds2add |
|- ( ( ( M ^ 2 ) e. ZZ /\ ( A ^ 2 ) e. ZZ /\ ( B ^ 2 ) e. ZZ ) -> ( ( ( M ^ 2 ) || ( A ^ 2 ) /\ ( M ^ 2 ) || ( B ^ 2 ) ) -> ( M ^ 2 ) || ( ( A ^ 2 ) + ( B ^ 2 ) ) ) ) |
110 |
104 106 108 109
|
syl3anc |
|- ( ph -> ( ( ( M ^ 2 ) || ( A ^ 2 ) /\ ( M ^ 2 ) || ( B ^ 2 ) ) -> ( M ^ 2 ) || ( ( A ^ 2 ) + ( B ^ 2 ) ) ) ) |
111 |
110
|
adantr |
|- ( ( ph /\ R = 0 ) -> ( ( ( M ^ 2 ) || ( A ^ 2 ) /\ ( M ^ 2 ) || ( B ^ 2 ) ) -> ( M ^ 2 ) || ( ( A ^ 2 ) + ( B ^ 2 ) ) ) ) |
112 |
100 102 111
|
mp2and |
|- ( ( ph /\ R = 0 ) -> ( M ^ 2 ) || ( ( A ^ 2 ) + ( B ^ 2 ) ) ) |
113 |
91
|
simprd |
|- ( ( ph /\ R = 0 ) -> ( ( G ^ 2 ) + ( H ^ 2 ) ) = 0 ) |
114 |
82
|
nn0ge0d |
|- ( ph -> 0 <_ ( G ^ 2 ) ) |
115 |
84
|
nn0ge0d |
|- ( ph -> 0 <_ ( H ^ 2 ) ) |
116 |
|
add20 |
|- ( ( ( ( G ^ 2 ) e. RR /\ 0 <_ ( G ^ 2 ) ) /\ ( ( H ^ 2 ) e. RR /\ 0 <_ ( H ^ 2 ) ) ) -> ( ( ( G ^ 2 ) + ( H ^ 2 ) ) = 0 <-> ( ( G ^ 2 ) = 0 /\ ( H ^ 2 ) = 0 ) ) ) |
117 |
36 114 41 115 116
|
syl22anc |
|- ( ph -> ( ( ( G ^ 2 ) + ( H ^ 2 ) ) = 0 <-> ( ( G ^ 2 ) = 0 /\ ( H ^ 2 ) = 0 ) ) ) |
118 |
117
|
biimpa |
|- ( ( ph /\ ( ( G ^ 2 ) + ( H ^ 2 ) ) = 0 ) -> ( ( G ^ 2 ) = 0 /\ ( H ^ 2 ) = 0 ) ) |
119 |
113 118
|
syldan |
|- ( ( ph /\ R = 0 ) -> ( ( G ^ 2 ) = 0 /\ ( H ^ 2 ) = 0 ) ) |
120 |
119
|
simpld |
|- ( ( ph /\ R = 0 ) -> ( G ^ 2 ) = 0 ) |
121 |
11 20 15 120
|
4sqlem9 |
|- ( ( ph /\ R = 0 ) -> ( M ^ 2 ) || ( C ^ 2 ) ) |
122 |
119
|
simprd |
|- ( ( ph /\ R = 0 ) -> ( H ^ 2 ) = 0 ) |
123 |
12 20 16 122
|
4sqlem9 |
|- ( ( ph /\ R = 0 ) -> ( M ^ 2 ) || ( D ^ 2 ) ) |
124 |
|
zsqcl |
|- ( C e. ZZ -> ( C ^ 2 ) e. ZZ ) |
125 |
11 124
|
syl |
|- ( ph -> ( C ^ 2 ) e. ZZ ) |
126 |
|
zsqcl |
|- ( D e. ZZ -> ( D ^ 2 ) e. ZZ ) |
127 |
12 126
|
syl |
|- ( ph -> ( D ^ 2 ) e. ZZ ) |
128 |
|
dvds2add |
|- ( ( ( M ^ 2 ) e. ZZ /\ ( C ^ 2 ) e. ZZ /\ ( D ^ 2 ) e. ZZ ) -> ( ( ( M ^ 2 ) || ( C ^ 2 ) /\ ( M ^ 2 ) || ( D ^ 2 ) ) -> ( M ^ 2 ) || ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) |
129 |
104 125 127 128
|
syl3anc |
|- ( ph -> ( ( ( M ^ 2 ) || ( C ^ 2 ) /\ ( M ^ 2 ) || ( D ^ 2 ) ) -> ( M ^ 2 ) || ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) |
130 |
129
|
adantr |
|- ( ( ph /\ R = 0 ) -> ( ( ( M ^ 2 ) || ( C ^ 2 ) /\ ( M ^ 2 ) || ( D ^ 2 ) ) -> ( M ^ 2 ) || ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) |
131 |
121 123 130
|
mp2and |
|- ( ( ph /\ R = 0 ) -> ( M ^ 2 ) || ( ( C ^ 2 ) + ( D ^ 2 ) ) ) |
132 |
106 108
|
zaddcld |
|- ( ph -> ( ( A ^ 2 ) + ( B ^ 2 ) ) e. ZZ ) |
133 |
125 127
|
zaddcld |
|- ( ph -> ( ( C ^ 2 ) + ( D ^ 2 ) ) e. ZZ ) |
134 |
|
dvds2add |
|- ( ( ( M ^ 2 ) e. ZZ /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) e. ZZ /\ ( ( C ^ 2 ) + ( D ^ 2 ) ) e. ZZ ) -> ( ( ( M ^ 2 ) || ( ( A ^ 2 ) + ( B ^ 2 ) ) /\ ( M ^ 2 ) || ( ( C ^ 2 ) + ( D ^ 2 ) ) ) -> ( M ^ 2 ) || ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) ) |
135 |
104 132 133 134
|
syl3anc |
|- ( ph -> ( ( ( M ^ 2 ) || ( ( A ^ 2 ) + ( B ^ 2 ) ) /\ ( M ^ 2 ) || ( ( C ^ 2 ) + ( D ^ 2 ) ) ) -> ( M ^ 2 ) || ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) ) |
136 |
135
|
adantr |
|- ( ( ph /\ R = 0 ) -> ( ( ( M ^ 2 ) || ( ( A ^ 2 ) + ( B ^ 2 ) ) /\ ( M ^ 2 ) || ( ( C ^ 2 ) + ( D ^ 2 ) ) ) -> ( M ^ 2 ) || ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) ) |
137 |
112 131 136
|
mp2and |
|- ( ( ph /\ R = 0 ) -> ( M ^ 2 ) || ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) |
138 |
104
|
adantr |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) e. ZZ ) |
139 |
132
|
adantr |
|- ( ( ph /\ R = M ) -> ( ( A ^ 2 ) + ( B ^ 2 ) ) e. ZZ ) |
140 |
51
|
adantr |
|- ( ( ph /\ R = M ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) + ( ( ( M ^ 2 ) / 2 ) / 2 ) ) = ( ( M ^ 2 ) / 2 ) ) |
141 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
|
4sqlem15 |
|- ( ( ph /\ R = M ) -> ( ( ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( E ^ 2 ) ) = 0 /\ ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( F ^ 2 ) ) = 0 ) /\ ( ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( G ^ 2 ) ) = 0 /\ ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( H ^ 2 ) ) = 0 ) ) ) |
142 |
141
|
simpld |
|- ( ( ph /\ R = M ) -> ( ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( E ^ 2 ) ) = 0 /\ ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( F ^ 2 ) ) = 0 ) ) |
143 |
142
|
simpld |
|- ( ( ph /\ R = M ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( E ^ 2 ) ) = 0 ) |
144 |
46
|
recnd |
|- ( ph -> ( ( ( M ^ 2 ) / 2 ) / 2 ) e. CC ) |
145 |
24
|
zcnd |
|- ( ph -> ( E ^ 2 ) e. CC ) |
146 |
144 145
|
subeq0ad |
|- ( ph -> ( ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( E ^ 2 ) ) = 0 <-> ( ( ( M ^ 2 ) / 2 ) / 2 ) = ( E ^ 2 ) ) ) |
147 |
146
|
adantr |
|- ( ( ph /\ R = M ) -> ( ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( E ^ 2 ) ) = 0 <-> ( ( ( M ^ 2 ) / 2 ) / 2 ) = ( E ^ 2 ) ) ) |
148 |
143 147
|
mpbid |
|- ( ( ph /\ R = M ) -> ( ( ( M ^ 2 ) / 2 ) / 2 ) = ( E ^ 2 ) ) |
149 |
24
|
adantr |
|- ( ( ph /\ R = M ) -> ( E ^ 2 ) e. ZZ ) |
150 |
148 149
|
eqeltrd |
|- ( ( ph /\ R = M ) -> ( ( ( M ^ 2 ) / 2 ) / 2 ) e. ZZ ) |
151 |
150 150
|
zaddcld |
|- ( ( ph /\ R = M ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) + ( ( ( M ^ 2 ) / 2 ) / 2 ) ) e. ZZ ) |
152 |
140 151
|
eqeltrrd |
|- ( ( ph /\ R = M ) -> ( ( M ^ 2 ) / 2 ) e. ZZ ) |
153 |
139 152
|
zsubcld |
|- ( ( ph /\ R = M ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) e. ZZ ) |
154 |
133
|
adantr |
|- ( ( ph /\ R = M ) -> ( ( C ^ 2 ) + ( D ^ 2 ) ) e. ZZ ) |
155 |
154 152
|
zsubcld |
|- ( ( ph /\ R = M ) -> ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) e. ZZ ) |
156 |
106
|
adantr |
|- ( ( ph /\ R = M ) -> ( A ^ 2 ) e. ZZ ) |
157 |
156 150
|
zsubcld |
|- ( ( ph /\ R = M ) -> ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) e. ZZ ) |
158 |
108
|
adantr |
|- ( ( ph /\ R = M ) -> ( B ^ 2 ) e. ZZ ) |
159 |
158 150
|
zsubcld |
|- ( ( ph /\ R = M ) -> ( ( B ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) e. ZZ ) |
160 |
9 20 13 143
|
4sqlem10 |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) |
161 |
142
|
simprd |
|- ( ( ph /\ R = M ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( F ^ 2 ) ) = 0 ) |
162 |
10 20 14 161
|
4sqlem10 |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( B ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) |
163 |
138 157 159 160 162
|
dvds2addd |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) + ( ( B ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) ) |
164 |
106
|
zcnd |
|- ( ph -> ( A ^ 2 ) e. CC ) |
165 |
108
|
zcnd |
|- ( ph -> ( B ^ 2 ) e. CC ) |
166 |
164 165 144 144
|
addsub4d |
|- ( ph -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( ( ( M ^ 2 ) / 2 ) / 2 ) + ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) = ( ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) + ( ( B ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) ) |
167 |
51
|
oveq2d |
|- ( ph -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( ( ( M ^ 2 ) / 2 ) / 2 ) + ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) |
168 |
166 167
|
eqtr3d |
|- ( ph -> ( ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) + ( ( B ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) |
169 |
168
|
adantr |
|- ( ( ph /\ R = M ) -> ( ( ( A ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) + ( ( B ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) |
170 |
163 169
|
breqtrd |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) |
171 |
125
|
adantr |
|- ( ( ph /\ R = M ) -> ( C ^ 2 ) e. ZZ ) |
172 |
171 150
|
zsubcld |
|- ( ( ph /\ R = M ) -> ( ( C ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) e. ZZ ) |
173 |
127
|
adantr |
|- ( ( ph /\ R = M ) -> ( D ^ 2 ) e. ZZ ) |
174 |
173 150
|
zsubcld |
|- ( ( ph /\ R = M ) -> ( ( D ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) e. ZZ ) |
175 |
141
|
simprd |
|- ( ( ph /\ R = M ) -> ( ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( G ^ 2 ) ) = 0 /\ ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( H ^ 2 ) ) = 0 ) ) |
176 |
175
|
simpld |
|- ( ( ph /\ R = M ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( G ^ 2 ) ) = 0 ) |
177 |
11 20 15 176
|
4sqlem10 |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( C ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) |
178 |
175
|
simprd |
|- ( ( ph /\ R = M ) -> ( ( ( ( M ^ 2 ) / 2 ) / 2 ) - ( H ^ 2 ) ) = 0 ) |
179 |
12 20 16 178
|
4sqlem10 |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( D ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) |
180 |
138 172 174 177 179
|
dvds2addd |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( ( C ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) + ( ( D ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) ) |
181 |
125
|
zcnd |
|- ( ph -> ( C ^ 2 ) e. CC ) |
182 |
127
|
zcnd |
|- ( ph -> ( D ^ 2 ) e. CC ) |
183 |
181 182 144 144
|
addsub4d |
|- ( ph -> ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( ( ( M ^ 2 ) / 2 ) / 2 ) + ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) = ( ( ( C ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) + ( ( D ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) ) |
184 |
51
|
oveq2d |
|- ( ph -> ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( ( ( M ^ 2 ) / 2 ) / 2 ) + ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) |
185 |
183 184
|
eqtr3d |
|- ( ph -> ( ( ( C ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) + ( ( D ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) |
186 |
185
|
adantr |
|- ( ( ph /\ R = M ) -> ( ( ( C ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) + ( ( D ^ 2 ) - ( ( ( M ^ 2 ) / 2 ) / 2 ) ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) |
187 |
180 186
|
breqtrd |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) |
188 |
138 153 155 170 187
|
dvds2addd |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) + ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) ) |
189 |
132
|
zcnd |
|- ( ph -> ( ( A ^ 2 ) + ( B ^ 2 ) ) e. CC ) |
190 |
133
|
zcnd |
|- ( ph -> ( ( C ^ 2 ) + ( D ^ 2 ) ) e. CC ) |
191 |
189 190 50 50
|
addsub4d |
|- ( ph -> ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) - ( ( ( M ^ 2 ) / 2 ) + ( ( M ^ 2 ) / 2 ) ) ) = ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) + ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) ) |
192 |
59
|
oveq2d |
|- ( ph -> ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) - ( ( ( M ^ 2 ) / 2 ) + ( ( M ^ 2 ) / 2 ) ) ) = ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) - ( M ^ 2 ) ) ) |
193 |
191 192
|
eqtr3d |
|- ( ph -> ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) + ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) = ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) - ( M ^ 2 ) ) ) |
194 |
193
|
adantr |
|- ( ( ph /\ R = M ) -> ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) + ( ( ( C ^ 2 ) + ( D ^ 2 ) ) - ( ( M ^ 2 ) / 2 ) ) ) = ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) - ( M ^ 2 ) ) ) |
195 |
188 194
|
breqtrd |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) - ( M ^ 2 ) ) ) |
196 |
132 133
|
zaddcld |
|- ( ph -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) e. ZZ ) |
197 |
196
|
adantr |
|- ( ( ph /\ R = M ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) e. ZZ ) |
198 |
|
dvdssubr |
|- ( ( ( M ^ 2 ) e. ZZ /\ ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) e. ZZ ) -> ( ( M ^ 2 ) || ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) <-> ( M ^ 2 ) || ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) - ( M ^ 2 ) ) ) ) |
199 |
138 197 198
|
syl2anc |
|- ( ( ph /\ R = M ) -> ( ( M ^ 2 ) || ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) <-> ( M ^ 2 ) || ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) - ( M ^ 2 ) ) ) ) |
200 |
195 199
|
mpbird |
|- ( ( ph /\ R = M ) -> ( M ^ 2 ) || ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) |
201 |
137 200
|
jaodan |
|- ( ( ph /\ ( R = 0 \/ R = M ) ) -> ( M ^ 2 ) || ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) |
202 |
18
|
adantr |
|- ( ( ph /\ ( R = 0 \/ R = M ) ) -> ( M x. P ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) + ( ( C ^ 2 ) + ( D ^ 2 ) ) ) ) |
203 |
201 202
|
breqtrrd |
|- ( ( ph /\ ( R = 0 \/ R = M ) ) -> ( M ^ 2 ) || ( M x. P ) ) |
204 |
203
|
ex |
|- ( ph -> ( ( R = 0 \/ R = M ) -> ( M ^ 2 ) || ( M x. P ) ) ) |
205 |
69 204
|
jca |
|- ( ph -> ( R <_ M /\ ( ( R = 0 \/ R = M ) -> ( M ^ 2 ) || ( M x. P ) ) ) ) |