Step |
Hyp |
Ref |
Expression |
1 |
|
lgseisen.1 |
|- ( ph -> P e. ( Prime \ { 2 } ) ) |
2 |
|
lgseisen.2 |
|- ( ph -> Q e. ( Prime \ { 2 } ) ) |
3 |
|
lgseisen.3 |
|- ( ph -> P =/= Q ) |
4 |
|
lgsquad.4 |
|- M = ( ( P - 1 ) / 2 ) |
5 |
|
lgsquad.5 |
|- N = ( ( Q - 1 ) / 2 ) |
6 |
|
lgsquad.6 |
|- S = { <. x , y >. | ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) } |
7 |
1 2 3
|
lgseisen |
|- ( ph -> ( Q /L P ) = ( -u 1 ^ sum_ u e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) |
8 |
4
|
oveq2i |
|- ( 1 ... M ) = ( 1 ... ( ( P - 1 ) / 2 ) ) |
9 |
8
|
sumeq1i |
|- sum_ u e. ( 1 ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) = sum_ u e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) |
10 |
1 4
|
gausslemma2dlem0b |
|- ( ph -> M e. NN ) |
11 |
10
|
nnred |
|- ( ph -> M e. RR ) |
12 |
11
|
rehalfcld |
|- ( ph -> ( M / 2 ) e. RR ) |
13 |
12
|
flcld |
|- ( ph -> ( |_ ` ( M / 2 ) ) e. ZZ ) |
14 |
13
|
zred |
|- ( ph -> ( |_ ` ( M / 2 ) ) e. RR ) |
15 |
14
|
ltp1d |
|- ( ph -> ( |_ ` ( M / 2 ) ) < ( ( |_ ` ( M / 2 ) ) + 1 ) ) |
16 |
|
fzdisj |
|- ( ( |_ ` ( M / 2 ) ) < ( ( |_ ` ( M / 2 ) ) + 1 ) -> ( ( 1 ... ( |_ ` ( M / 2 ) ) ) i^i ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) = (/) ) |
17 |
15 16
|
syl |
|- ( ph -> ( ( 1 ... ( |_ ` ( M / 2 ) ) ) i^i ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) = (/) ) |
18 |
10
|
nnrpd |
|- ( ph -> M e. RR+ ) |
19 |
18
|
rphalfcld |
|- ( ph -> ( M / 2 ) e. RR+ ) |
20 |
19
|
rpge0d |
|- ( ph -> 0 <_ ( M / 2 ) ) |
21 |
|
flge0nn0 |
|- ( ( ( M / 2 ) e. RR /\ 0 <_ ( M / 2 ) ) -> ( |_ ` ( M / 2 ) ) e. NN0 ) |
22 |
12 20 21
|
syl2anc |
|- ( ph -> ( |_ ` ( M / 2 ) ) e. NN0 ) |
23 |
10
|
nnnn0d |
|- ( ph -> M e. NN0 ) |
24 |
|
rphalflt |
|- ( M e. RR+ -> ( M / 2 ) < M ) |
25 |
18 24
|
syl |
|- ( ph -> ( M / 2 ) < M ) |
26 |
10
|
nnzd |
|- ( ph -> M e. ZZ ) |
27 |
|
fllt |
|- ( ( ( M / 2 ) e. RR /\ M e. ZZ ) -> ( ( M / 2 ) < M <-> ( |_ ` ( M / 2 ) ) < M ) ) |
28 |
12 26 27
|
syl2anc |
|- ( ph -> ( ( M / 2 ) < M <-> ( |_ ` ( M / 2 ) ) < M ) ) |
29 |
25 28
|
mpbid |
|- ( ph -> ( |_ ` ( M / 2 ) ) < M ) |
30 |
14 11 29
|
ltled |
|- ( ph -> ( |_ ` ( M / 2 ) ) <_ M ) |
31 |
|
elfz2nn0 |
|- ( ( |_ ` ( M / 2 ) ) e. ( 0 ... M ) <-> ( ( |_ ` ( M / 2 ) ) e. NN0 /\ M e. NN0 /\ ( |_ ` ( M / 2 ) ) <_ M ) ) |
32 |
22 23 30 31
|
syl3anbrc |
|- ( ph -> ( |_ ` ( M / 2 ) ) e. ( 0 ... M ) ) |
33 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
34 |
23 33
|
eleqtrdi |
|- ( ph -> M e. ( ZZ>= ` 0 ) ) |
35 |
|
elfzp12 |
|- ( M e. ( ZZ>= ` 0 ) -> ( ( |_ ` ( M / 2 ) ) e. ( 0 ... M ) <-> ( ( |_ ` ( M / 2 ) ) = 0 \/ ( |_ ` ( M / 2 ) ) e. ( ( 0 + 1 ) ... M ) ) ) ) |
36 |
34 35
|
syl |
|- ( ph -> ( ( |_ ` ( M / 2 ) ) e. ( 0 ... M ) <-> ( ( |_ ` ( M / 2 ) ) = 0 \/ ( |_ ` ( M / 2 ) ) e. ( ( 0 + 1 ) ... M ) ) ) ) |
37 |
32 36
|
mpbid |
|- ( ph -> ( ( |_ ` ( M / 2 ) ) = 0 \/ ( |_ ` ( M / 2 ) ) e. ( ( 0 + 1 ) ... M ) ) ) |
38 |
|
un0 |
|- ( ( 1 ... M ) u. (/) ) = ( 1 ... M ) |
39 |
|
uncom |
|- ( ( 1 ... M ) u. (/) ) = ( (/) u. ( 1 ... M ) ) |
40 |
38 39
|
eqtr3i |
|- ( 1 ... M ) = ( (/) u. ( 1 ... M ) ) |
41 |
|
oveq2 |
|- ( ( |_ ` ( M / 2 ) ) = 0 -> ( 1 ... ( |_ ` ( M / 2 ) ) ) = ( 1 ... 0 ) ) |
42 |
|
fz10 |
|- ( 1 ... 0 ) = (/) |
43 |
41 42
|
eqtrdi |
|- ( ( |_ ` ( M / 2 ) ) = 0 -> ( 1 ... ( |_ ` ( M / 2 ) ) ) = (/) ) |
44 |
|
oveq1 |
|- ( ( |_ ` ( M / 2 ) ) = 0 -> ( ( |_ ` ( M / 2 ) ) + 1 ) = ( 0 + 1 ) ) |
45 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
46 |
44 45
|
eqtrdi |
|- ( ( |_ ` ( M / 2 ) ) = 0 -> ( ( |_ ` ( M / 2 ) ) + 1 ) = 1 ) |
47 |
46
|
oveq1d |
|- ( ( |_ ` ( M / 2 ) ) = 0 -> ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) = ( 1 ... M ) ) |
48 |
43 47
|
uneq12d |
|- ( ( |_ ` ( M / 2 ) ) = 0 -> ( ( 1 ... ( |_ ` ( M / 2 ) ) ) u. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) = ( (/) u. ( 1 ... M ) ) ) |
49 |
40 48
|
eqtr4id |
|- ( ( |_ ` ( M / 2 ) ) = 0 -> ( 1 ... M ) = ( ( 1 ... ( |_ ` ( M / 2 ) ) ) u. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) ) |
50 |
|
fzsplit |
|- ( ( |_ ` ( M / 2 ) ) e. ( 1 ... M ) -> ( 1 ... M ) = ( ( 1 ... ( |_ ` ( M / 2 ) ) ) u. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) ) |
51 |
45
|
oveq1i |
|- ( ( 0 + 1 ) ... M ) = ( 1 ... M ) |
52 |
50 51
|
eleq2s |
|- ( ( |_ ` ( M / 2 ) ) e. ( ( 0 + 1 ) ... M ) -> ( 1 ... M ) = ( ( 1 ... ( |_ ` ( M / 2 ) ) ) u. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) ) |
53 |
49 52
|
jaoi |
|- ( ( ( |_ ` ( M / 2 ) ) = 0 \/ ( |_ ` ( M / 2 ) ) e. ( ( 0 + 1 ) ... M ) ) -> ( 1 ... M ) = ( ( 1 ... ( |_ ` ( M / 2 ) ) ) u. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) ) |
54 |
37 53
|
syl |
|- ( ph -> ( 1 ... M ) = ( ( 1 ... ( |_ ` ( M / 2 ) ) ) u. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) ) |
55 |
|
fzfid |
|- ( ph -> ( 1 ... M ) e. Fin ) |
56 |
2
|
gausslemma2dlem0a |
|- ( ph -> Q e. NN ) |
57 |
56
|
nnred |
|- ( ph -> Q e. RR ) |
58 |
1
|
gausslemma2dlem0a |
|- ( ph -> P e. NN ) |
59 |
57 58
|
nndivred |
|- ( ph -> ( Q / P ) e. RR ) |
60 |
59
|
adantr |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> ( Q / P ) e. RR ) |
61 |
|
2nn |
|- 2 e. NN |
62 |
|
elfznn |
|- ( u e. ( 1 ... M ) -> u e. NN ) |
63 |
62
|
adantl |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> u e. NN ) |
64 |
|
nnmulcl |
|- ( ( 2 e. NN /\ u e. NN ) -> ( 2 x. u ) e. NN ) |
65 |
61 63 64
|
sylancr |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> ( 2 x. u ) e. NN ) |
66 |
65
|
nnred |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> ( 2 x. u ) e. RR ) |
67 |
60 66
|
remulcld |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> ( ( Q / P ) x. ( 2 x. u ) ) e. RR ) |
68 |
56
|
nnrpd |
|- ( ph -> Q e. RR+ ) |
69 |
58
|
nnrpd |
|- ( ph -> P e. RR+ ) |
70 |
68 69
|
rpdivcld |
|- ( ph -> ( Q / P ) e. RR+ ) |
71 |
70
|
adantr |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> ( Q / P ) e. RR+ ) |
72 |
65
|
nnrpd |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> ( 2 x. u ) e. RR+ ) |
73 |
71 72
|
rpmulcld |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> ( ( Q / P ) x. ( 2 x. u ) ) e. RR+ ) |
74 |
73
|
rpge0d |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> 0 <_ ( ( Q / P ) x. ( 2 x. u ) ) ) |
75 |
|
flge0nn0 |
|- ( ( ( ( Q / P ) x. ( 2 x. u ) ) e. RR /\ 0 <_ ( ( Q / P ) x. ( 2 x. u ) ) ) -> ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. NN0 ) |
76 |
67 74 75
|
syl2anc |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. NN0 ) |
77 |
76
|
nn0cnd |
|- ( ( ph /\ u e. ( 1 ... M ) ) -> ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. CC ) |
78 |
17 54 55 77
|
fsumsplit |
|- ( ph -> sum_ u e. ( 1 ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) = ( sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) + sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) |
79 |
9 78
|
eqtr3id |
|- ( ph -> sum_ u e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) = ( sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) + sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) |
80 |
79
|
oveq2d |
|- ( ph -> ( -u 1 ^ sum_ u e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) = ( -u 1 ^ ( sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) + sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
81 |
|
neg1cn |
|- -u 1 e. CC |
82 |
81
|
a1i |
|- ( ph -> -u 1 e. CC ) |
83 |
|
fzfid |
|- ( ph -> ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) e. Fin ) |
84 |
|
ssun2 |
|- ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) C_ ( ( 1 ... ( |_ ` ( M / 2 ) ) ) u. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) |
85 |
84 54
|
sseqtrrid |
|- ( ph -> ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) C_ ( 1 ... M ) ) |
86 |
85
|
sselda |
|- ( ( ph /\ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) -> u e. ( 1 ... M ) ) |
87 |
86 76
|
syldan |
|- ( ( ph /\ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) -> ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. NN0 ) |
88 |
83 87
|
fsumnn0cl |
|- ( ph -> sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. NN0 ) |
89 |
|
fzfid |
|- ( ph -> ( 1 ... ( |_ ` ( M / 2 ) ) ) e. Fin ) |
90 |
|
ssun1 |
|- ( 1 ... ( |_ ` ( M / 2 ) ) ) C_ ( ( 1 ... ( |_ ` ( M / 2 ) ) ) u. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ) |
91 |
90 54
|
sseqtrrid |
|- ( ph -> ( 1 ... ( |_ ` ( M / 2 ) ) ) C_ ( 1 ... M ) ) |
92 |
91
|
sselda |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> u e. ( 1 ... M ) ) |
93 |
92 76
|
syldan |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. NN0 ) |
94 |
89 93
|
fsumnn0cl |
|- ( ph -> sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. NN0 ) |
95 |
82 88 94
|
expaddd |
|- ( ph -> ( -u 1 ^ ( sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) + sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) = ( ( -u 1 ^ sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) x. ( -u 1 ^ sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
96 |
|
fzfid |
|- ( ph -> ( 1 ... N ) e. Fin ) |
97 |
|
xpfi |
|- ( ( ( 1 ... M ) e. Fin /\ ( 1 ... N ) e. Fin ) -> ( ( 1 ... M ) X. ( 1 ... N ) ) e. Fin ) |
98 |
55 96 97
|
syl2anc |
|- ( ph -> ( ( 1 ... M ) X. ( 1 ... N ) ) e. Fin ) |
99 |
|
opabssxp |
|- { <. x , y >. | ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) } C_ ( ( 1 ... M ) X. ( 1 ... N ) ) |
100 |
6 99
|
eqsstri |
|- S C_ ( ( 1 ... M ) X. ( 1 ... N ) ) |
101 |
|
ssfi |
|- ( ( ( ( 1 ... M ) X. ( 1 ... N ) ) e. Fin /\ S C_ ( ( 1 ... M ) X. ( 1 ... N ) ) ) -> S e. Fin ) |
102 |
98 100 101
|
sylancl |
|- ( ph -> S e. Fin ) |
103 |
|
ssrab2 |
|- { z e. S | -. 2 || ( 1st ` z ) } C_ S |
104 |
|
ssfi |
|- ( ( S e. Fin /\ { z e. S | -. 2 || ( 1st ` z ) } C_ S ) -> { z e. S | -. 2 || ( 1st ` z ) } e. Fin ) |
105 |
102 103 104
|
sylancl |
|- ( ph -> { z e. S | -. 2 || ( 1st ` z ) } e. Fin ) |
106 |
|
hashcl |
|- ( { z e. S | -. 2 || ( 1st ` z ) } e. Fin -> ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) e. NN0 ) |
107 |
105 106
|
syl |
|- ( ph -> ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) e. NN0 ) |
108 |
|
ssrab2 |
|- { z e. S | 2 || ( 1st ` z ) } C_ S |
109 |
|
ssfi |
|- ( ( S e. Fin /\ { z e. S | 2 || ( 1st ` z ) } C_ S ) -> { z e. S | 2 || ( 1st ` z ) } e. Fin ) |
110 |
102 108 109
|
sylancl |
|- ( ph -> { z e. S | 2 || ( 1st ` z ) } e. Fin ) |
111 |
|
hashcl |
|- ( { z e. S | 2 || ( 1st ` z ) } e. Fin -> ( # ` { z e. S | 2 || ( 1st ` z ) } ) e. NN0 ) |
112 |
110 111
|
syl |
|- ( ph -> ( # ` { z e. S | 2 || ( 1st ` z ) } ) e. NN0 ) |
113 |
82 107 112
|
expaddd |
|- ( ph -> ( -u 1 ^ ( ( # ` { z e. S | 2 || ( 1st ` z ) } ) + ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) ) ) = ( ( -u 1 ^ ( # ` { z e. S | 2 || ( 1st ` z ) } ) ) x. ( -u 1 ^ ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) ) ) ) |
114 |
92 65
|
syldan |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( 2 x. u ) e. NN ) |
115 |
|
fzfid |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) e. Fin ) |
116 |
|
xpsnen2g |
|- ( ( ( 2 x. u ) e. NN /\ ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) e. Fin ) -> ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ~~ ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) |
117 |
114 115 116
|
syl2anc |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ~~ ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) |
118 |
|
hasheni |
|- ( ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ~~ ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) -> ( # ` ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) = ( # ` ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
119 |
117 118
|
syl |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( # ` ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) = ( # ` ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
120 |
|
ssrab2 |
|- { z e. S | ( 2 x. u ) = ( 1st ` z ) } C_ S |
121 |
6
|
relopabiv |
|- Rel S |
122 |
|
relss |
|- ( { z e. S | ( 2 x. u ) = ( 1st ` z ) } C_ S -> ( Rel S -> Rel { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) |
123 |
120 121 122
|
mp2 |
|- Rel { z e. S | ( 2 x. u ) = ( 1st ` z ) } |
124 |
|
relxp |
|- Rel ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) |
125 |
6
|
eleq2i |
|- ( <. x , y >. e. S <-> <. x , y >. e. { <. x , y >. | ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) } ) |
126 |
|
opabidw |
|- ( <. x , y >. e. { <. x , y >. | ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) } <-> ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) ) |
127 |
125 126
|
bitri |
|- ( <. x , y >. e. S <-> ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) ) |
128 |
|
anass |
|- ( ( ( y e. NN /\ y <_ N ) /\ ( y x. P ) < ( Q x. ( 2 x. u ) ) ) <-> ( y e. NN /\ ( y <_ N /\ ( y x. P ) < ( Q x. ( 2 x. u ) ) ) ) ) |
129 |
114
|
adantr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( 2 x. u ) e. NN ) |
130 |
129
|
nnred |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( 2 x. u ) e. RR ) |
131 |
58
|
ad2antrr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> P e. NN ) |
132 |
131
|
nnred |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> P e. RR ) |
133 |
132
|
rehalfcld |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( P / 2 ) e. RR ) |
134 |
11
|
adantr |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> M e. RR ) |
135 |
134
|
adantr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> M e. RR ) |
136 |
|
elfzle2 |
|- ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) -> u <_ ( |_ ` ( M / 2 ) ) ) |
137 |
136
|
adantl |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> u <_ ( |_ ` ( M / 2 ) ) ) |
138 |
134
|
rehalfcld |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( M / 2 ) e. RR ) |
139 |
|
elfzelz |
|- ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) -> u e. ZZ ) |
140 |
139
|
adantl |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> u e. ZZ ) |
141 |
|
flge |
|- ( ( ( M / 2 ) e. RR /\ u e. ZZ ) -> ( u <_ ( M / 2 ) <-> u <_ ( |_ ` ( M / 2 ) ) ) ) |
142 |
138 140 141
|
syl2anc |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( u <_ ( M / 2 ) <-> u <_ ( |_ ` ( M / 2 ) ) ) ) |
143 |
137 142
|
mpbird |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> u <_ ( M / 2 ) ) |
144 |
|
elfznn |
|- ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) -> u e. NN ) |
145 |
144
|
adantl |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> u e. NN ) |
146 |
145
|
nnred |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> u e. RR ) |
147 |
|
2re |
|- 2 e. RR |
148 |
147
|
a1i |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> 2 e. RR ) |
149 |
|
2pos |
|- 0 < 2 |
150 |
149
|
a1i |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> 0 < 2 ) |
151 |
|
lemuldiv2 |
|- ( ( u e. RR /\ M e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( 2 x. u ) <_ M <-> u <_ ( M / 2 ) ) ) |
152 |
146 134 148 150 151
|
syl112anc |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( ( 2 x. u ) <_ M <-> u <_ ( M / 2 ) ) ) |
153 |
143 152
|
mpbird |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( 2 x. u ) <_ M ) |
154 |
153
|
adantr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( 2 x. u ) <_ M ) |
155 |
132
|
ltm1d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( P - 1 ) < P ) |
156 |
|
peano2rem |
|- ( P e. RR -> ( P - 1 ) e. RR ) |
157 |
132 156
|
syl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( P - 1 ) e. RR ) |
158 |
147
|
a1i |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> 2 e. RR ) |
159 |
149
|
a1i |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> 0 < 2 ) |
160 |
|
ltdiv1 |
|- ( ( ( P - 1 ) e. RR /\ P e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( P - 1 ) < P <-> ( ( P - 1 ) / 2 ) < ( P / 2 ) ) ) |
161 |
157 132 158 159 160
|
syl112anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( P - 1 ) < P <-> ( ( P - 1 ) / 2 ) < ( P / 2 ) ) ) |
162 |
155 161
|
mpbid |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( P - 1 ) / 2 ) < ( P / 2 ) ) |
163 |
4 162
|
eqbrtrid |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> M < ( P / 2 ) ) |
164 |
130 135 133 154 163
|
lelttrd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( 2 x. u ) < ( P / 2 ) ) |
165 |
131
|
nnrpd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> P e. RR+ ) |
166 |
|
rphalflt |
|- ( P e. RR+ -> ( P / 2 ) < P ) |
167 |
165 166
|
syl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( P / 2 ) < P ) |
168 |
130 133 132 164 167
|
lttrd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( 2 x. u ) < P ) |
169 |
130 132
|
ltnled |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( 2 x. u ) < P <-> -. P <_ ( 2 x. u ) ) ) |
170 |
168 169
|
mpbid |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> -. P <_ ( 2 x. u ) ) |
171 |
1
|
eldifad |
|- ( ph -> P e. Prime ) |
172 |
171
|
ad2antrr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> P e. Prime ) |
173 |
|
prmz |
|- ( P e. Prime -> P e. ZZ ) |
174 |
172 173
|
syl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> P e. ZZ ) |
175 |
|
dvdsle |
|- ( ( P e. ZZ /\ ( 2 x. u ) e. NN ) -> ( P || ( 2 x. u ) -> P <_ ( 2 x. u ) ) ) |
176 |
174 129 175
|
syl2anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( P || ( 2 x. u ) -> P <_ ( 2 x. u ) ) ) |
177 |
170 176
|
mtod |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> -. P || ( 2 x. u ) ) |
178 |
2
|
eldifad |
|- ( ph -> Q e. Prime ) |
179 |
|
prmrp |
|- ( ( P e. Prime /\ Q e. Prime ) -> ( ( P gcd Q ) = 1 <-> P =/= Q ) ) |
180 |
171 178 179
|
syl2anc |
|- ( ph -> ( ( P gcd Q ) = 1 <-> P =/= Q ) ) |
181 |
3 180
|
mpbird |
|- ( ph -> ( P gcd Q ) = 1 ) |
182 |
181
|
ad2antrr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( P gcd Q ) = 1 ) |
183 |
178
|
ad2antrr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> Q e. Prime ) |
184 |
|
prmz |
|- ( Q e. Prime -> Q e. ZZ ) |
185 |
183 184
|
syl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> Q e. ZZ ) |
186 |
129
|
nnzd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( 2 x. u ) e. ZZ ) |
187 |
|
coprmdvds |
|- ( ( P e. ZZ /\ Q e. ZZ /\ ( 2 x. u ) e. ZZ ) -> ( ( P || ( Q x. ( 2 x. u ) ) /\ ( P gcd Q ) = 1 ) -> P || ( 2 x. u ) ) ) |
188 |
174 185 186 187
|
syl3anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( P || ( Q x. ( 2 x. u ) ) /\ ( P gcd Q ) = 1 ) -> P || ( 2 x. u ) ) ) |
189 |
182 188
|
mpan2d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( P || ( Q x. ( 2 x. u ) ) -> P || ( 2 x. u ) ) ) |
190 |
177 189
|
mtod |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> -. P || ( Q x. ( 2 x. u ) ) ) |
191 |
|
nnz |
|- ( y e. NN -> y e. ZZ ) |
192 |
191
|
adantl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> y e. ZZ ) |
193 |
|
dvdsmul2 |
|- ( ( y e. ZZ /\ P e. ZZ ) -> P || ( y x. P ) ) |
194 |
192 174 193
|
syl2anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> P || ( y x. P ) ) |
195 |
|
breq2 |
|- ( ( Q x. ( 2 x. u ) ) = ( y x. P ) -> ( P || ( Q x. ( 2 x. u ) ) <-> P || ( y x. P ) ) ) |
196 |
194 195
|
syl5ibrcom |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( Q x. ( 2 x. u ) ) = ( y x. P ) -> P || ( Q x. ( 2 x. u ) ) ) ) |
197 |
196
|
necon3bd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( -. P || ( Q x. ( 2 x. u ) ) -> ( Q x. ( 2 x. u ) ) =/= ( y x. P ) ) ) |
198 |
190 197
|
mpd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q x. ( 2 x. u ) ) =/= ( y x. P ) ) |
199 |
|
simpr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> y e. NN ) |
200 |
199 131
|
nnmulcld |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( y x. P ) e. NN ) |
201 |
200
|
nnred |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( y x. P ) e. RR ) |
202 |
56
|
adantr |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> Q e. NN ) |
203 |
202 114
|
nnmulcld |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( Q x. ( 2 x. u ) ) e. NN ) |
204 |
203
|
adantr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q x. ( 2 x. u ) ) e. NN ) |
205 |
204
|
nnred |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q x. ( 2 x. u ) ) e. RR ) |
206 |
201 205
|
ltlend |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y x. P ) < ( Q x. ( 2 x. u ) ) <-> ( ( y x. P ) <_ ( Q x. ( 2 x. u ) ) /\ ( Q x. ( 2 x. u ) ) =/= ( y x. P ) ) ) ) |
207 |
198 206
|
mpbiran2d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y x. P ) < ( Q x. ( 2 x. u ) ) <-> ( y x. P ) <_ ( Q x. ( 2 x. u ) ) ) ) |
208 |
|
nnre |
|- ( y e. NN -> y e. RR ) |
209 |
208
|
adantl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> y e. RR ) |
210 |
131
|
nngt0d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> 0 < P ) |
211 |
|
lemuldiv |
|- ( ( y e. RR /\ ( Q x. ( 2 x. u ) ) e. RR /\ ( P e. RR /\ 0 < P ) ) -> ( ( y x. P ) <_ ( Q x. ( 2 x. u ) ) <-> y <_ ( ( Q x. ( 2 x. u ) ) / P ) ) ) |
212 |
209 205 132 210 211
|
syl112anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y x. P ) <_ ( Q x. ( 2 x. u ) ) <-> y <_ ( ( Q x. ( 2 x. u ) ) / P ) ) ) |
213 |
202
|
adantr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> Q e. NN ) |
214 |
213
|
nncnd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> Q e. CC ) |
215 |
129
|
nncnd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( 2 x. u ) e. CC ) |
216 |
131
|
nncnd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> P e. CC ) |
217 |
131
|
nnne0d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> P =/= 0 ) |
218 |
214 215 216 217
|
div23d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( Q x. ( 2 x. u ) ) / P ) = ( ( Q / P ) x. ( 2 x. u ) ) ) |
219 |
218
|
breq2d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( y <_ ( ( Q x. ( 2 x. u ) ) / P ) <-> y <_ ( ( Q / P ) x. ( 2 x. u ) ) ) ) |
220 |
207 212 219
|
3bitrd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y x. P ) < ( Q x. ( 2 x. u ) ) <-> y <_ ( ( Q / P ) x. ( 2 x. u ) ) ) ) |
221 |
213
|
nnred |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> Q e. RR ) |
222 |
213
|
nngt0d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> 0 < Q ) |
223 |
|
ltmul2 |
|- ( ( ( 2 x. u ) e. RR /\ ( P / 2 ) e. RR /\ ( Q e. RR /\ 0 < Q ) ) -> ( ( 2 x. u ) < ( P / 2 ) <-> ( Q x. ( 2 x. u ) ) < ( Q x. ( P / 2 ) ) ) ) |
224 |
130 133 221 222 223
|
syl112anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( 2 x. u ) < ( P / 2 ) <-> ( Q x. ( 2 x. u ) ) < ( Q x. ( P / 2 ) ) ) ) |
225 |
164 224
|
mpbid |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q x. ( 2 x. u ) ) < ( Q x. ( P / 2 ) ) ) |
226 |
|
2cnd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> 2 e. CC ) |
227 |
|
2ne0 |
|- 2 =/= 0 |
228 |
227
|
a1i |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> 2 =/= 0 ) |
229 |
|
divass |
|- ( ( Q e. CC /\ P e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( Q x. P ) / 2 ) = ( Q x. ( P / 2 ) ) ) |
230 |
|
div23 |
|- ( ( Q e. CC /\ P e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( Q x. P ) / 2 ) = ( ( Q / 2 ) x. P ) ) |
231 |
229 230
|
eqtr3d |
|- ( ( Q e. CC /\ P e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( Q x. ( P / 2 ) ) = ( ( Q / 2 ) x. P ) ) |
232 |
214 216 226 228 231
|
syl112anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q x. ( P / 2 ) ) = ( ( Q / 2 ) x. P ) ) |
233 |
225 232
|
breqtrd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q x. ( 2 x. u ) ) < ( ( Q / 2 ) x. P ) ) |
234 |
221
|
rehalfcld |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q / 2 ) e. RR ) |
235 |
234 132
|
remulcld |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( Q / 2 ) x. P ) e. RR ) |
236 |
|
lttr |
|- ( ( ( y x. P ) e. RR /\ ( Q x. ( 2 x. u ) ) e. RR /\ ( ( Q / 2 ) x. P ) e. RR ) -> ( ( ( y x. P ) < ( Q x. ( 2 x. u ) ) /\ ( Q x. ( 2 x. u ) ) < ( ( Q / 2 ) x. P ) ) -> ( y x. P ) < ( ( Q / 2 ) x. P ) ) ) |
237 |
201 205 235 236
|
syl3anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( ( y x. P ) < ( Q x. ( 2 x. u ) ) /\ ( Q x. ( 2 x. u ) ) < ( ( Q / 2 ) x. P ) ) -> ( y x. P ) < ( ( Q / 2 ) x. P ) ) ) |
238 |
233 237
|
mpan2d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y x. P ) < ( Q x. ( 2 x. u ) ) -> ( y x. P ) < ( ( Q / 2 ) x. P ) ) ) |
239 |
|
ltmul1 |
|- ( ( y e. RR /\ ( Q / 2 ) e. RR /\ ( P e. RR /\ 0 < P ) ) -> ( y < ( Q / 2 ) <-> ( y x. P ) < ( ( Q / 2 ) x. P ) ) ) |
240 |
209 234 132 210 239
|
syl112anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( y < ( Q / 2 ) <-> ( y x. P ) < ( ( Q / 2 ) x. P ) ) ) |
241 |
238 240
|
sylibrd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y x. P ) < ( Q x. ( 2 x. u ) ) -> y < ( Q / 2 ) ) ) |
242 |
|
peano2rem |
|- ( Q e. RR -> ( Q - 1 ) e. RR ) |
243 |
221 242
|
syl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q - 1 ) e. RR ) |
244 |
243
|
recnd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q - 1 ) e. CC ) |
245 |
214 244 226 228
|
divsubdird |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( Q - ( Q - 1 ) ) / 2 ) = ( ( Q / 2 ) - ( ( Q - 1 ) / 2 ) ) ) |
246 |
5
|
oveq2i |
|- ( ( Q / 2 ) - N ) = ( ( Q / 2 ) - ( ( Q - 1 ) / 2 ) ) |
247 |
245 246
|
eqtr4di |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( Q - ( Q - 1 ) ) / 2 ) = ( ( Q / 2 ) - N ) ) |
248 |
|
ax-1cn |
|- 1 e. CC |
249 |
|
nncan |
|- ( ( Q e. CC /\ 1 e. CC ) -> ( Q - ( Q - 1 ) ) = 1 ) |
250 |
214 248 249
|
sylancl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q - ( Q - 1 ) ) = 1 ) |
251 |
250
|
oveq1d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( Q - ( Q - 1 ) ) / 2 ) = ( 1 / 2 ) ) |
252 |
|
halflt1 |
|- ( 1 / 2 ) < 1 |
253 |
251 252
|
eqbrtrdi |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( Q - ( Q - 1 ) ) / 2 ) < 1 ) |
254 |
247 253
|
eqbrtrrd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( Q / 2 ) - N ) < 1 ) |
255 |
2 5
|
gausslemma2dlem0b |
|- ( ph -> N e. NN ) |
256 |
255
|
ad2antrr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> N e. NN ) |
257 |
256
|
nnred |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> N e. RR ) |
258 |
|
1red |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> 1 e. RR ) |
259 |
234 257 258
|
ltsubadd2d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( ( Q / 2 ) - N ) < 1 <-> ( Q / 2 ) < ( N + 1 ) ) ) |
260 |
254 259
|
mpbid |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( Q / 2 ) < ( N + 1 ) ) |
261 |
|
peano2re |
|- ( N e. RR -> ( N + 1 ) e. RR ) |
262 |
257 261
|
syl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( N + 1 ) e. RR ) |
263 |
|
lttr |
|- ( ( y e. RR /\ ( Q / 2 ) e. RR /\ ( N + 1 ) e. RR ) -> ( ( y < ( Q / 2 ) /\ ( Q / 2 ) < ( N + 1 ) ) -> y < ( N + 1 ) ) ) |
264 |
209 234 262 263
|
syl3anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y < ( Q / 2 ) /\ ( Q / 2 ) < ( N + 1 ) ) -> y < ( N + 1 ) ) ) |
265 |
260 264
|
mpan2d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( y < ( Q / 2 ) -> y < ( N + 1 ) ) ) |
266 |
241 265
|
syld |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y x. P ) < ( Q x. ( 2 x. u ) ) -> y < ( N + 1 ) ) ) |
267 |
|
nnleltp1 |
|- ( ( y e. NN /\ N e. NN ) -> ( y <_ N <-> y < ( N + 1 ) ) ) |
268 |
199 256 267
|
syl2anc |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( y <_ N <-> y < ( N + 1 ) ) ) |
269 |
266 268
|
sylibrd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y x. P ) < ( Q x. ( 2 x. u ) ) -> y <_ N ) ) |
270 |
269
|
pm4.71rd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y x. P ) < ( Q x. ( 2 x. u ) ) <-> ( y <_ N /\ ( y x. P ) < ( Q x. ( 2 x. u ) ) ) ) ) |
271 |
92 67
|
syldan |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( ( Q / P ) x. ( 2 x. u ) ) e. RR ) |
272 |
|
flge |
|- ( ( ( ( Q / P ) x. ( 2 x. u ) ) e. RR /\ y e. ZZ ) -> ( y <_ ( ( Q / P ) x. ( 2 x. u ) ) <-> y <_ ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) |
273 |
271 191 272
|
syl2an |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( y <_ ( ( Q / P ) x. ( 2 x. u ) ) <-> y <_ ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) |
274 |
220 270 273
|
3bitr3d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ y e. NN ) -> ( ( y <_ N /\ ( y x. P ) < ( Q x. ( 2 x. u ) ) ) <-> y <_ ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) |
275 |
274
|
pm5.32da |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( ( y e. NN /\ ( y <_ N /\ ( y x. P ) < ( Q x. ( 2 x. u ) ) ) ) <-> ( y e. NN /\ y <_ ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
276 |
128 275
|
syl5bb |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( ( ( y e. NN /\ y <_ N ) /\ ( y x. P ) < ( Q x. ( 2 x. u ) ) ) <-> ( y e. NN /\ y <_ ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
277 |
276
|
adantr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( ( ( y e. NN /\ y <_ N ) /\ ( y x. P ) < ( Q x. ( 2 x. u ) ) ) <-> ( y e. NN /\ y <_ ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
278 |
|
simpr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> x = ( 2 x. u ) ) |
279 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
280 |
114 279
|
eleqtrdi |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( 2 x. u ) e. ( ZZ>= ` 1 ) ) |
281 |
26
|
adantr |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> M e. ZZ ) |
282 |
|
elfz5 |
|- ( ( ( 2 x. u ) e. ( ZZ>= ` 1 ) /\ M e. ZZ ) -> ( ( 2 x. u ) e. ( 1 ... M ) <-> ( 2 x. u ) <_ M ) ) |
283 |
280 281 282
|
syl2anc |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( ( 2 x. u ) e. ( 1 ... M ) <-> ( 2 x. u ) <_ M ) ) |
284 |
153 283
|
mpbird |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( 2 x. u ) e. ( 1 ... M ) ) |
285 |
284
|
adantr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( 2 x. u ) e. ( 1 ... M ) ) |
286 |
278 285
|
eqeltrd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> x e. ( 1 ... M ) ) |
287 |
286
|
biantrurd |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( y e. ( 1 ... N ) <-> ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) ) ) |
288 |
255
|
nnzd |
|- ( ph -> N e. ZZ ) |
289 |
288
|
ad2antrr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> N e. ZZ ) |
290 |
|
fznn |
|- ( N e. ZZ -> ( y e. ( 1 ... N ) <-> ( y e. NN /\ y <_ N ) ) ) |
291 |
289 290
|
syl |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( y e. ( 1 ... N ) <-> ( y e. NN /\ y <_ N ) ) ) |
292 |
287 291
|
bitr3d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) <-> ( y e. NN /\ y <_ N ) ) ) |
293 |
|
oveq1 |
|- ( x = ( 2 x. u ) -> ( x x. Q ) = ( ( 2 x. u ) x. Q ) ) |
294 |
114
|
nncnd |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( 2 x. u ) e. CC ) |
295 |
202
|
nncnd |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> Q e. CC ) |
296 |
294 295
|
mulcomd |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( ( 2 x. u ) x. Q ) = ( Q x. ( 2 x. u ) ) ) |
297 |
293 296
|
sylan9eqr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( x x. Q ) = ( Q x. ( 2 x. u ) ) ) |
298 |
297
|
breq2d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( ( y x. P ) < ( x x. Q ) <-> ( y x. P ) < ( Q x. ( 2 x. u ) ) ) ) |
299 |
292 298
|
anbi12d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) <-> ( ( y e. NN /\ y <_ N ) /\ ( y x. P ) < ( Q x. ( 2 x. u ) ) ) ) ) |
300 |
271
|
flcld |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. ZZ ) |
301 |
|
fznn |
|- ( ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. ZZ -> ( y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) <-> ( y e. NN /\ y <_ ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
302 |
300 301
|
syl |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) <-> ( y e. NN /\ y <_ ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
303 |
302
|
adantr |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) <-> ( y e. NN /\ y <_ ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
304 |
277 299 303
|
3bitr4d |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) <-> y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
305 |
127 304
|
syl5bb |
|- ( ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) /\ x = ( 2 x. u ) ) -> ( <. x , y >. e. S <-> y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
306 |
305
|
pm5.32da |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( ( x = ( 2 x. u ) /\ <. x , y >. e. S ) <-> ( x = ( 2 x. u ) /\ y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) ) |
307 |
|
vex |
|- x e. _V |
308 |
|
vex |
|- y e. _V |
309 |
307 308
|
op1std |
|- ( z = <. x , y >. -> ( 1st ` z ) = x ) |
310 |
309
|
eqeq2d |
|- ( z = <. x , y >. -> ( ( 2 x. u ) = ( 1st ` z ) <-> ( 2 x. u ) = x ) ) |
311 |
|
eqcom |
|- ( ( 2 x. u ) = x <-> x = ( 2 x. u ) ) |
312 |
310 311
|
bitrdi |
|- ( z = <. x , y >. -> ( ( 2 x. u ) = ( 1st ` z ) <-> x = ( 2 x. u ) ) ) |
313 |
312
|
elrab |
|- ( <. x , y >. e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } <-> ( <. x , y >. e. S /\ x = ( 2 x. u ) ) ) |
314 |
313
|
biancomi |
|- ( <. x , y >. e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } <-> ( x = ( 2 x. u ) /\ <. x , y >. e. S ) ) |
315 |
|
opelxp |
|- ( <. x , y >. e. ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) <-> ( x e. { ( 2 x. u ) } /\ y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
316 |
|
velsn |
|- ( x e. { ( 2 x. u ) } <-> x = ( 2 x. u ) ) |
317 |
316
|
anbi1i |
|- ( ( x e. { ( 2 x. u ) } /\ y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) <-> ( x = ( 2 x. u ) /\ y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
318 |
315 317
|
bitri |
|- ( <. x , y >. e. ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) <-> ( x = ( 2 x. u ) /\ y e. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
319 |
306 314 318
|
3bitr4g |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( <. x , y >. e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } <-> <. x , y >. e. ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) ) |
320 |
123 124 319
|
eqrelrdv |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> { z e. S | ( 2 x. u ) = ( 1st ` z ) } = ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
321 |
320
|
eqcomd |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) = { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) |
322 |
321
|
fveq2d |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( # ` ( { ( 2 x. u ) } X. ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) = ( # ` { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) |
323 |
|
hashfz1 |
|- ( ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) e. NN0 -> ( # ` ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) = ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) |
324 |
93 323
|
syl |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( # ` ( 1 ... ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) = ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) |
325 |
119 322 324
|
3eqtr3rd |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) = ( # ` { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) |
326 |
325
|
sumeq2dv |
|- ( ph -> sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) = sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( # ` { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) |
327 |
102
|
adantr |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> S e. Fin ) |
328 |
|
ssfi |
|- ( ( S e. Fin /\ { z e. S | ( 2 x. u ) = ( 1st ` z ) } C_ S ) -> { z e. S | ( 2 x. u ) = ( 1st ` z ) } e. Fin ) |
329 |
327 120 328
|
sylancl |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> { z e. S | ( 2 x. u ) = ( 1st ` z ) } e. Fin ) |
330 |
|
fveq2 |
|- ( z = v -> ( 1st ` z ) = ( 1st ` v ) ) |
331 |
330
|
eqeq2d |
|- ( z = v -> ( ( 2 x. u ) = ( 1st ` z ) <-> ( 2 x. u ) = ( 1st ` v ) ) ) |
332 |
331
|
elrab |
|- ( v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } <-> ( v e. S /\ ( 2 x. u ) = ( 1st ` v ) ) ) |
333 |
332
|
simprbi |
|- ( v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } -> ( 2 x. u ) = ( 1st ` v ) ) |
334 |
333
|
ad2antll |
|- ( ( ph /\ ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) -> ( 2 x. u ) = ( 1st ` v ) ) |
335 |
334
|
oveq1d |
|- ( ( ph /\ ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) -> ( ( 2 x. u ) / 2 ) = ( ( 1st ` v ) / 2 ) ) |
336 |
145
|
nncnd |
|- ( ( ph /\ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) -> u e. CC ) |
337 |
336
|
adantrr |
|- ( ( ph /\ ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) -> u e. CC ) |
338 |
|
2cnd |
|- ( ( ph /\ ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) -> 2 e. CC ) |
339 |
227
|
a1i |
|- ( ( ph /\ ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) -> 2 =/= 0 ) |
340 |
337 338 339
|
divcan3d |
|- ( ( ph /\ ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) -> ( ( 2 x. u ) / 2 ) = u ) |
341 |
335 340
|
eqtr3d |
|- ( ( ph /\ ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) -> ( ( 1st ` v ) / 2 ) = u ) |
342 |
341
|
ralrimivva |
|- ( ph -> A. u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) A. v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } ( ( 1st ` v ) / 2 ) = u ) |
343 |
|
invdisj |
|- ( A. u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) A. v e. { z e. S | ( 2 x. u ) = ( 1st ` z ) } ( ( 1st ` v ) / 2 ) = u -> Disj_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) |
344 |
342 343
|
syl |
|- ( ph -> Disj_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) |
345 |
89 329 344
|
hashiun |
|- ( ph -> ( # ` U_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) = sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( # ` { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) ) |
346 |
|
iunrab |
|- U_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) { z e. S | ( 2 x. u ) = ( 1st ` z ) } = { z e. S | E. u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( 2 x. u ) = ( 1st ` z ) } |
347 |
|
2cn |
|- 2 e. CC |
348 |
|
zcn |
|- ( u e. ZZ -> u e. CC ) |
349 |
348
|
adantl |
|- ( ( ( ph /\ z e. S ) /\ u e. ZZ ) -> u e. CC ) |
350 |
|
mulcom |
|- ( ( 2 e. CC /\ u e. CC ) -> ( 2 x. u ) = ( u x. 2 ) ) |
351 |
347 349 350
|
sylancr |
|- ( ( ( ph /\ z e. S ) /\ u e. ZZ ) -> ( 2 x. u ) = ( u x. 2 ) ) |
352 |
351
|
eqeq1d |
|- ( ( ( ph /\ z e. S ) /\ u e. ZZ ) -> ( ( 2 x. u ) = ( 1st ` z ) <-> ( u x. 2 ) = ( 1st ` z ) ) ) |
353 |
352
|
rexbidva |
|- ( ( ph /\ z e. S ) -> ( E. u e. ZZ ( 2 x. u ) = ( 1st ` z ) <-> E. u e. ZZ ( u x. 2 ) = ( 1st ` z ) ) ) |
354 |
139
|
anim1i |
|- ( ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ ( 2 x. u ) = ( 1st ` z ) ) -> ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) |
355 |
354
|
reximi2 |
|- ( E. u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( 2 x. u ) = ( 1st ` z ) -> E. u e. ZZ ( 2 x. u ) = ( 1st ` z ) ) |
356 |
|
simprr |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( 2 x. u ) = ( 1st ` z ) ) |
357 |
|
simpr |
|- ( ( ph /\ z e. S ) -> z e. S ) |
358 |
100 357
|
sselid |
|- ( ( ph /\ z e. S ) -> z e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) |
359 |
|
xp1st |
|- ( z e. ( ( 1 ... M ) X. ( 1 ... N ) ) -> ( 1st ` z ) e. ( 1 ... M ) ) |
360 |
358 359
|
syl |
|- ( ( ph /\ z e. S ) -> ( 1st ` z ) e. ( 1 ... M ) ) |
361 |
360
|
adantr |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( 1st ` z ) e. ( 1 ... M ) ) |
362 |
|
elfzle2 |
|- ( ( 1st ` z ) e. ( 1 ... M ) -> ( 1st ` z ) <_ M ) |
363 |
361 362
|
syl |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( 1st ` z ) <_ M ) |
364 |
356 363
|
eqbrtrd |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( 2 x. u ) <_ M ) |
365 |
|
zre |
|- ( u e. ZZ -> u e. RR ) |
366 |
365
|
ad2antrl |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> u e. RR ) |
367 |
11
|
ad2antrr |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> M e. RR ) |
368 |
147
|
a1i |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> 2 e. RR ) |
369 |
149
|
a1i |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> 0 < 2 ) |
370 |
366 367 368 369 151
|
syl112anc |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( ( 2 x. u ) <_ M <-> u <_ ( M / 2 ) ) ) |
371 |
364 370
|
mpbid |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> u <_ ( M / 2 ) ) |
372 |
12
|
ad2antrr |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( M / 2 ) e. RR ) |
373 |
|
simprl |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> u e. ZZ ) |
374 |
372 373 141
|
syl2anc |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( u <_ ( M / 2 ) <-> u <_ ( |_ ` ( M / 2 ) ) ) ) |
375 |
371 374
|
mpbid |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> u <_ ( |_ ` ( M / 2 ) ) ) |
376 |
|
2t0e0 |
|- ( 2 x. 0 ) = 0 |
377 |
|
elfznn |
|- ( ( 1st ` z ) e. ( 1 ... M ) -> ( 1st ` z ) e. NN ) |
378 |
361 377
|
syl |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( 1st ` z ) e. NN ) |
379 |
356 378
|
eqeltrd |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( 2 x. u ) e. NN ) |
380 |
379
|
nngt0d |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> 0 < ( 2 x. u ) ) |
381 |
376 380
|
eqbrtrid |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( 2 x. 0 ) < ( 2 x. u ) ) |
382 |
|
0red |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> 0 e. RR ) |
383 |
|
ltmul2 |
|- ( ( 0 e. RR /\ u e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( 0 < u <-> ( 2 x. 0 ) < ( 2 x. u ) ) ) |
384 |
382 366 368 369 383
|
syl112anc |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( 0 < u <-> ( 2 x. 0 ) < ( 2 x. u ) ) ) |
385 |
381 384
|
mpbird |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> 0 < u ) |
386 |
|
elnnz |
|- ( u e. NN <-> ( u e. ZZ /\ 0 < u ) ) |
387 |
373 385 386
|
sylanbrc |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> u e. NN ) |
388 |
387 279
|
eleqtrdi |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> u e. ( ZZ>= ` 1 ) ) |
389 |
13
|
ad2antrr |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( |_ ` ( M / 2 ) ) e. ZZ ) |
390 |
|
elfz5 |
|- ( ( u e. ( ZZ>= ` 1 ) /\ ( |_ ` ( M / 2 ) ) e. ZZ ) -> ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) <-> u <_ ( |_ ` ( M / 2 ) ) ) ) |
391 |
388 389 390
|
syl2anc |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) <-> u <_ ( |_ ` ( M / 2 ) ) ) ) |
392 |
375 391
|
mpbird |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ) |
393 |
392 356
|
jca |
|- ( ( ( ph /\ z e. S ) /\ ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) ) -> ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ ( 2 x. u ) = ( 1st ` z ) ) ) |
394 |
393
|
ex |
|- ( ( ph /\ z e. S ) -> ( ( u e. ZZ /\ ( 2 x. u ) = ( 1st ` z ) ) -> ( u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) /\ ( 2 x. u ) = ( 1st ` z ) ) ) ) |
395 |
394
|
reximdv2 |
|- ( ( ph /\ z e. S ) -> ( E. u e. ZZ ( 2 x. u ) = ( 1st ` z ) -> E. u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( 2 x. u ) = ( 1st ` z ) ) ) |
396 |
355 395
|
impbid2 |
|- ( ( ph /\ z e. S ) -> ( E. u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( 2 x. u ) = ( 1st ` z ) <-> E. u e. ZZ ( 2 x. u ) = ( 1st ` z ) ) ) |
397 |
|
2z |
|- 2 e. ZZ |
398 |
360
|
elfzelzd |
|- ( ( ph /\ z e. S ) -> ( 1st ` z ) e. ZZ ) |
399 |
|
divides |
|- ( ( 2 e. ZZ /\ ( 1st ` z ) e. ZZ ) -> ( 2 || ( 1st ` z ) <-> E. u e. ZZ ( u x. 2 ) = ( 1st ` z ) ) ) |
400 |
397 398 399
|
sylancr |
|- ( ( ph /\ z e. S ) -> ( 2 || ( 1st ` z ) <-> E. u e. ZZ ( u x. 2 ) = ( 1st ` z ) ) ) |
401 |
353 396 400
|
3bitr4d |
|- ( ( ph /\ z e. S ) -> ( E. u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( 2 x. u ) = ( 1st ` z ) <-> 2 || ( 1st ` z ) ) ) |
402 |
401
|
rabbidva |
|- ( ph -> { z e. S | E. u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( 2 x. u ) = ( 1st ` z ) } = { z e. S | 2 || ( 1st ` z ) } ) |
403 |
346 402
|
syl5eq |
|- ( ph -> U_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) { z e. S | ( 2 x. u ) = ( 1st ` z ) } = { z e. S | 2 || ( 1st ` z ) } ) |
404 |
403
|
fveq2d |
|- ( ph -> ( # ` U_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) { z e. S | ( 2 x. u ) = ( 1st ` z ) } ) = ( # ` { z e. S | 2 || ( 1st ` z ) } ) ) |
405 |
326 345 404
|
3eqtr2d |
|- ( ph -> sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) = ( # ` { z e. S | 2 || ( 1st ` z ) } ) ) |
406 |
405
|
oveq2d |
|- ( ph -> ( -u 1 ^ sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) = ( -u 1 ^ ( # ` { z e. S | 2 || ( 1st ` z ) } ) ) ) |
407 |
1 2 3 4 5 6
|
lgsquadlem1 |
|- ( ph -> ( -u 1 ^ sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) = ( -u 1 ^ ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) ) ) |
408 |
406 407
|
oveq12d |
|- ( ph -> ( ( -u 1 ^ sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) x. ( -u 1 ^ sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) = ( ( -u 1 ^ ( # ` { z e. S | 2 || ( 1st ` z ) } ) ) x. ( -u 1 ^ ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) ) ) ) |
409 |
113 408
|
eqtr4d |
|- ( ph -> ( -u 1 ^ ( ( # ` { z e. S | 2 || ( 1st ` z ) } ) + ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) ) ) = ( ( -u 1 ^ sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) x. ( -u 1 ^ sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) ) |
410 |
|
inrab |
|- ( { z e. S | 2 || ( 1st ` z ) } i^i { z e. S | -. 2 || ( 1st ` z ) } ) = { z e. S | ( 2 || ( 1st ` z ) /\ -. 2 || ( 1st ` z ) ) } |
411 |
|
pm3.24 |
|- -. ( 2 || ( 1st ` z ) /\ -. 2 || ( 1st ` z ) ) |
412 |
411
|
a1i |
|- ( ph -> -. ( 2 || ( 1st ` z ) /\ -. 2 || ( 1st ` z ) ) ) |
413 |
412
|
ralrimivw |
|- ( ph -> A. z e. S -. ( 2 || ( 1st ` z ) /\ -. 2 || ( 1st ` z ) ) ) |
414 |
|
rabeq0 |
|- ( { z e. S | ( 2 || ( 1st ` z ) /\ -. 2 || ( 1st ` z ) ) } = (/) <-> A. z e. S -. ( 2 || ( 1st ` z ) /\ -. 2 || ( 1st ` z ) ) ) |
415 |
413 414
|
sylibr |
|- ( ph -> { z e. S | ( 2 || ( 1st ` z ) /\ -. 2 || ( 1st ` z ) ) } = (/) ) |
416 |
410 415
|
syl5eq |
|- ( ph -> ( { z e. S | 2 || ( 1st ` z ) } i^i { z e. S | -. 2 || ( 1st ` z ) } ) = (/) ) |
417 |
|
hashun |
|- ( ( { z e. S | 2 || ( 1st ` z ) } e. Fin /\ { z e. S | -. 2 || ( 1st ` z ) } e. Fin /\ ( { z e. S | 2 || ( 1st ` z ) } i^i { z e. S | -. 2 || ( 1st ` z ) } ) = (/) ) -> ( # ` ( { z e. S | 2 || ( 1st ` z ) } u. { z e. S | -. 2 || ( 1st ` z ) } ) ) = ( ( # ` { z e. S | 2 || ( 1st ` z ) } ) + ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) ) ) |
418 |
110 105 416 417
|
syl3anc |
|- ( ph -> ( # ` ( { z e. S | 2 || ( 1st ` z ) } u. { z e. S | -. 2 || ( 1st ` z ) } ) ) = ( ( # ` { z e. S | 2 || ( 1st ` z ) } ) + ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) ) ) |
419 |
|
unrab |
|- ( { z e. S | 2 || ( 1st ` z ) } u. { z e. S | -. 2 || ( 1st ` z ) } ) = { z e. S | ( 2 || ( 1st ` z ) \/ -. 2 || ( 1st ` z ) ) } |
420 |
|
exmid |
|- ( 2 || ( 1st ` z ) \/ -. 2 || ( 1st ` z ) ) |
421 |
420
|
rgenw |
|- A. z e. S ( 2 || ( 1st ` z ) \/ -. 2 || ( 1st ` z ) ) |
422 |
|
rabid2 |
|- ( S = { z e. S | ( 2 || ( 1st ` z ) \/ -. 2 || ( 1st ` z ) ) } <-> A. z e. S ( 2 || ( 1st ` z ) \/ -. 2 || ( 1st ` z ) ) ) |
423 |
421 422
|
mpbir |
|- S = { z e. S | ( 2 || ( 1st ` z ) \/ -. 2 || ( 1st ` z ) ) } |
424 |
419 423
|
eqtr4i |
|- ( { z e. S | 2 || ( 1st ` z ) } u. { z e. S | -. 2 || ( 1st ` z ) } ) = S |
425 |
424
|
fveq2i |
|- ( # ` ( { z e. S | 2 || ( 1st ` z ) } u. { z e. S | -. 2 || ( 1st ` z ) } ) ) = ( # ` S ) |
426 |
418 425
|
eqtr3di |
|- ( ph -> ( ( # ` { z e. S | 2 || ( 1st ` z ) } ) + ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) ) = ( # ` S ) ) |
427 |
426
|
oveq2d |
|- ( ph -> ( -u 1 ^ ( ( # ` { z e. S | 2 || ( 1st ` z ) } ) + ( # ` { z e. S | -. 2 || ( 1st ` z ) } ) ) ) = ( -u 1 ^ ( # ` S ) ) ) |
428 |
95 409 427
|
3eqtr2d |
|- ( ph -> ( -u 1 ^ ( sum_ u e. ( 1 ... ( |_ ` ( M / 2 ) ) ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) + sum_ u e. ( ( ( |_ ` ( M / 2 ) ) + 1 ) ... M ) ( |_ ` ( ( Q / P ) x. ( 2 x. u ) ) ) ) ) = ( -u 1 ^ ( # ` S ) ) ) |
429 |
7 80 428
|
3eqtrd |
|- ( ph -> ( Q /L P ) = ( -u 1 ^ ( # ` S ) ) ) |