Step |
Hyp |
Ref |
Expression |
1 |
|
poimir.0 |
|- ( ph -> N e. NN ) |
2 |
|
poimir.i |
|- I = ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) |
3 |
|
poimir.r |
|- R = ( Xt_ ` ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ) |
4 |
|
poimir.1 |
|- ( ph -> F e. ( ( R |`t I ) Cn R ) ) |
5 |
|
poimirlem30.x |
|- X = ( ( F ` ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ) ` n ) |
6 |
|
poimirlem30.2 |
|- ( ph -> G : NN --> ( ( NN0 ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
7 |
|
poimirlem30.3 |
|- ( ( ph /\ k e. NN ) -> ran ( 1st ` ( G ` k ) ) C_ ( 0 ..^ k ) ) |
8 |
|
poimirlem30.4 |
|- ( ( ph /\ ( k e. NN /\ n e. ( 1 ... N ) /\ r e. { <_ , `' <_ } ) ) -> E. j e. ( 0 ... N ) 0 r X ) |
9 |
|
fzfi |
|- ( 1 ... N ) e. Fin |
10 |
|
retop |
|- ( topGen ` ran (,) ) e. Top |
11 |
10
|
fconst6 |
|- ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) : ( 1 ... N ) --> Top |
12 |
|
pttop |
|- ( ( ( 1 ... N ) e. Fin /\ ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) : ( 1 ... N ) --> Top ) -> ( Xt_ ` ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ) e. Top ) |
13 |
9 11 12
|
mp2an |
|- ( Xt_ ` ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ) e. Top |
14 |
3 13
|
eqeltri |
|- R e. Top |
15 |
|
ovex |
|- ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) e. _V |
16 |
2 15
|
eqeltri |
|- I e. _V |
17 |
|
elrest |
|- ( ( R e. Top /\ I e. _V ) -> ( v e. ( R |`t I ) <-> E. z e. R v = ( z i^i I ) ) ) |
18 |
14 16 17
|
mp2an |
|- ( v e. ( R |`t I ) <-> E. z e. R v = ( z i^i I ) ) |
19 |
|
eqid |
|- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) ) |
20 |
3 19
|
ptrecube |
|- ( ( z e. R /\ C e. z ) -> E. c e. RR+ X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) C_ z ) |
21 |
20
|
ex |
|- ( z e. R -> ( C e. z -> E. c e. RR+ X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) C_ z ) ) |
22 |
|
inss1 |
|- ( z i^i I ) C_ z |
23 |
|
sseq1 |
|- ( v = ( z i^i I ) -> ( v C_ z <-> ( z i^i I ) C_ z ) ) |
24 |
22 23
|
mpbiri |
|- ( v = ( z i^i I ) -> v C_ z ) |
25 |
24
|
sseld |
|- ( v = ( z i^i I ) -> ( C e. v -> C e. z ) ) |
26 |
|
ssrin |
|- ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) C_ z -> ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ ( z i^i I ) ) |
27 |
|
sseq2 |
|- ( v = ( z i^i I ) -> ( ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v <-> ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ ( z i^i I ) ) ) |
28 |
26 27
|
syl5ibr |
|- ( v = ( z i^i I ) -> ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) C_ z -> ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) ) |
29 |
28
|
reximdv |
|- ( v = ( z i^i I ) -> ( E. c e. RR+ X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) C_ z -> E. c e. RR+ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) ) |
30 |
25 29
|
imim12d |
|- ( v = ( z i^i I ) -> ( ( C e. z -> E. c e. RR+ X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) C_ z ) -> ( C e. v -> E. c e. RR+ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) ) ) |
31 |
21 30
|
syl5com |
|- ( z e. R -> ( v = ( z i^i I ) -> ( C e. v -> E. c e. RR+ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) ) ) |
32 |
31
|
rexlimiv |
|- ( E. z e. R v = ( z i^i I ) -> ( C e. v -> E. c e. RR+ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) ) |
33 |
18 32
|
sylbi |
|- ( v e. ( R |`t I ) -> ( C e. v -> E. c e. RR+ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) ) |
34 |
33
|
imp |
|- ( ( v e. ( R |`t I ) /\ C e. v ) -> E. c e. RR+ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) |
35 |
34
|
adantl |
|- ( ( ph /\ ( v e. ( R |`t I ) /\ C e. v ) ) -> E. c e. RR+ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) |
36 |
|
resttop |
|- ( ( R e. Top /\ I e. _V ) -> ( R |`t I ) e. Top ) |
37 |
14 16 36
|
mp2an |
|- ( R |`t I ) e. Top |
38 |
|
reex |
|- RR e. _V |
39 |
|
unitssre |
|- ( 0 [,] 1 ) C_ RR |
40 |
|
mapss |
|- ( ( RR e. _V /\ ( 0 [,] 1 ) C_ RR ) -> ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) C_ ( RR ^m ( 1 ... N ) ) ) |
41 |
38 39 40
|
mp2an |
|- ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) C_ ( RR ^m ( 1 ... N ) ) |
42 |
2 41
|
eqsstri |
|- I C_ ( RR ^m ( 1 ... N ) ) |
43 |
|
ovex |
|- ( 1 ... N ) e. _V |
44 |
|
uniretop |
|- RR = U. ( topGen ` ran (,) ) |
45 |
3 44
|
ptuniconst |
|- ( ( ( 1 ... N ) e. _V /\ ( topGen ` ran (,) ) e. Top ) -> ( RR ^m ( 1 ... N ) ) = U. R ) |
46 |
43 10 45
|
mp2an |
|- ( RR ^m ( 1 ... N ) ) = U. R |
47 |
46
|
restuni |
|- ( ( R e. Top /\ I C_ ( RR ^m ( 1 ... N ) ) ) -> I = U. ( R |`t I ) ) |
48 |
14 42 47
|
mp2an |
|- I = U. ( R |`t I ) |
49 |
48
|
eltopss |
|- ( ( ( R |`t I ) e. Top /\ v e. ( R |`t I ) ) -> v C_ I ) |
50 |
37 49
|
mpan |
|- ( v e. ( R |`t I ) -> v C_ I ) |
51 |
50
|
sselda |
|- ( ( v e. ( R |`t I ) /\ C e. v ) -> C e. I ) |
52 |
|
2rp |
|- 2 e. RR+ |
53 |
|
rpdivcl |
|- ( ( 2 e. RR+ /\ c e. RR+ ) -> ( 2 / c ) e. RR+ ) |
54 |
52 53
|
mpan |
|- ( c e. RR+ -> ( 2 / c ) e. RR+ ) |
55 |
54
|
rpred |
|- ( c e. RR+ -> ( 2 / c ) e. RR ) |
56 |
|
ceicl |
|- ( ( 2 / c ) e. RR -> -u ( |_ ` -u ( 2 / c ) ) e. ZZ ) |
57 |
55 56
|
syl |
|- ( c e. RR+ -> -u ( |_ ` -u ( 2 / c ) ) e. ZZ ) |
58 |
|
0red |
|- ( c e. RR+ -> 0 e. RR ) |
59 |
57
|
zred |
|- ( c e. RR+ -> -u ( |_ ` -u ( 2 / c ) ) e. RR ) |
60 |
54
|
rpgt0d |
|- ( c e. RR+ -> 0 < ( 2 / c ) ) |
61 |
|
ceige |
|- ( ( 2 / c ) e. RR -> ( 2 / c ) <_ -u ( |_ ` -u ( 2 / c ) ) ) |
62 |
55 61
|
syl |
|- ( c e. RR+ -> ( 2 / c ) <_ -u ( |_ ` -u ( 2 / c ) ) ) |
63 |
58 55 59 60 62
|
ltletrd |
|- ( c e. RR+ -> 0 < -u ( |_ ` -u ( 2 / c ) ) ) |
64 |
|
elnnz |
|- ( -u ( |_ ` -u ( 2 / c ) ) e. NN <-> ( -u ( |_ ` -u ( 2 / c ) ) e. ZZ /\ 0 < -u ( |_ ` -u ( 2 / c ) ) ) ) |
65 |
57 63 64
|
sylanbrc |
|- ( c e. RR+ -> -u ( |_ ` -u ( 2 / c ) ) e. NN ) |
66 |
|
fveq2 |
|- ( i = -u ( |_ ` -u ( 2 / c ) ) -> ( ZZ>= ` i ) = ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) |
67 |
|
oveq2 |
|- ( i = -u ( |_ ` -u ( 2 / c ) ) -> ( 1 / i ) = ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) |
68 |
67
|
oveq2d |
|- ( i = -u ( |_ ` -u ( 2 / c ) ) -> ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) = ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) |
69 |
68
|
eleq2d |
|- ( i = -u ( |_ ` -u ( 2 / c ) ) -> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) <-> ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
70 |
69
|
ralbidv |
|- ( i = -u ( |_ ` -u ( 2 / c ) ) -> ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) <-> A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
71 |
66 70
|
rexeqbidv |
|- ( i = -u ( |_ ` -u ( 2 / c ) ) -> ( E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) <-> E. k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
72 |
71
|
rspcv |
|- ( -u ( |_ ` -u ( 2 / c ) ) e. NN -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> E. k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
73 |
65 72
|
syl |
|- ( c e. RR+ -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> E. k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
74 |
73
|
adantl |
|- ( ( ( ph /\ C e. I ) /\ c e. RR+ ) -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> E. k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
75 |
|
uznnssnn |
|- ( -u ( |_ ` -u ( 2 / c ) ) e. NN -> ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) C_ NN ) |
76 |
65 75
|
syl |
|- ( c e. RR+ -> ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) C_ NN ) |
77 |
76
|
sseld |
|- ( c e. RR+ -> ( k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) -> k e. NN ) ) |
78 |
77
|
adantl |
|- ( ( ( ph /\ C e. I ) /\ c e. RR+ ) -> ( k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) -> k e. NN ) ) |
79 |
78
|
imdistani |
|- ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) ) |
80 |
65
|
nnrpd |
|- ( c e. RR+ -> -u ( |_ ` -u ( 2 / c ) ) e. RR+ ) |
81 |
54 80
|
lerecd |
|- ( c e. RR+ -> ( ( 2 / c ) <_ -u ( |_ ` -u ( 2 / c ) ) <-> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) <_ ( 1 / ( 2 / c ) ) ) ) |
82 |
62 81
|
mpbid |
|- ( c e. RR+ -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) <_ ( 1 / ( 2 / c ) ) ) |
83 |
|
rpcn |
|- ( c e. RR+ -> c e. CC ) |
84 |
|
rpne0 |
|- ( c e. RR+ -> c =/= 0 ) |
85 |
|
2cn |
|- 2 e. CC |
86 |
|
2ne0 |
|- 2 =/= 0 |
87 |
|
recdiv |
|- ( ( ( 2 e. CC /\ 2 =/= 0 ) /\ ( c e. CC /\ c =/= 0 ) ) -> ( 1 / ( 2 / c ) ) = ( c / 2 ) ) |
88 |
85 86 87
|
mpanl12 |
|- ( ( c e. CC /\ c =/= 0 ) -> ( 1 / ( 2 / c ) ) = ( c / 2 ) ) |
89 |
83 84 88
|
syl2anc |
|- ( c e. RR+ -> ( 1 / ( 2 / c ) ) = ( c / 2 ) ) |
90 |
82 89
|
breqtrd |
|- ( c e. RR+ -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) <_ ( c / 2 ) ) |
91 |
90
|
ad4antlr |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) <_ ( c / 2 ) ) |
92 |
|
elmapi |
|- ( C e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> C : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
93 |
92 2
|
eleq2s |
|- ( C e. I -> C : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
94 |
93
|
ad4antlr |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> C : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
95 |
94
|
ffvelrnda |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( C ` m ) e. ( 0 [,] 1 ) ) |
96 |
39 95
|
sseldi |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( C ` m ) e. RR ) |
97 |
|
simp-4l |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ph ) |
98 |
|
simplr |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> k e. NN ) |
99 |
97 98
|
jca |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ph /\ k e. NN ) ) |
100 |
6
|
ffvelrnda |
|- ( ( ph /\ k e. NN ) -> ( G ` k ) e. ( ( NN0 ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
101 |
|
xp1st |
|- ( ( G ` k ) e. ( ( NN0 ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 1st ` ( G ` k ) ) e. ( NN0 ^m ( 1 ... N ) ) ) |
102 |
|
elmapi |
|- ( ( 1st ` ( G ` k ) ) e. ( NN0 ^m ( 1 ... N ) ) -> ( 1st ` ( G ` k ) ) : ( 1 ... N ) --> NN0 ) |
103 |
100 101 102
|
3syl |
|- ( ( ph /\ k e. NN ) -> ( 1st ` ( G ` k ) ) : ( 1 ... N ) --> NN0 ) |
104 |
103
|
ffvelrnda |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( 1st ` ( G ` k ) ) ` m ) e. NN0 ) |
105 |
104
|
nn0red |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( 1st ` ( G ` k ) ) ` m ) e. RR ) |
106 |
|
simplr |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> k e. NN ) |
107 |
105 106
|
nndivred |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) e. RR ) |
108 |
99 107
|
sylan |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) e. RR ) |
109 |
96 108
|
resubcld |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) e. RR ) |
110 |
109
|
recnd |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) e. CC ) |
111 |
110
|
abscld |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) e. RR ) |
112 |
65
|
nnrecred |
|- ( c e. RR+ -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) e. RR ) |
113 |
112
|
ad4antlr |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) e. RR ) |
114 |
|
rphalfcl |
|- ( c e. RR+ -> ( c / 2 ) e. RR+ ) |
115 |
114
|
rpred |
|- ( c e. RR+ -> ( c / 2 ) e. RR ) |
116 |
115
|
ad4antlr |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( c / 2 ) e. RR ) |
117 |
|
ltletr |
|- ( ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) e. RR /\ ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) e. RR /\ ( c / 2 ) e. RR ) -> ( ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) /\ ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) <_ ( c / 2 ) ) -> ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( c / 2 ) ) ) |
118 |
111 113 116 117
|
syl3anc |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) /\ ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) <_ ( c / 2 ) ) -> ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( c / 2 ) ) ) |
119 |
91 118
|
mpan2d |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) -> ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( c / 2 ) ) ) |
120 |
79 119
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) -> ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( c / 2 ) ) ) |
121 |
|
simpl |
|- ( ( ph /\ C e. I ) -> ph ) |
122 |
76
|
sselda |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> k e. NN ) |
123 |
121 122
|
anim12i |
|- ( ( ( ph /\ C e. I ) /\ ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) ) -> ( ph /\ k e. NN ) ) |
124 |
123
|
anassrs |
|- ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ph /\ k e. NN ) ) |
125 |
|
1re |
|- 1 e. RR |
126 |
|
snssi |
|- ( 1 e. RR -> { 1 } C_ RR ) |
127 |
125 126
|
ax-mp |
|- { 1 } C_ RR |
128 |
|
0re |
|- 0 e. RR |
129 |
|
snssi |
|- ( 0 e. RR -> { 0 } C_ RR ) |
130 |
128 129
|
ax-mp |
|- { 0 } C_ RR |
131 |
127 130
|
unssi |
|- ( { 1 } u. { 0 } ) C_ RR |
132 |
|
1ex |
|- 1 e. _V |
133 |
132
|
fconst |
|- ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) : ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) --> { 1 } |
134 |
|
c0ex |
|- 0 e. _V |
135 |
134
|
fconst |
|- ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) : ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) --> { 0 } |
136 |
133 135
|
pm3.2i |
|- ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) : ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) --> { 1 } /\ ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) : ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) --> { 0 } ) |
137 |
|
xp2nd |
|- ( ( G ` k ) e. ( ( NN0 ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 2nd ` ( G ` k ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
138 |
100 137
|
syl |
|- ( ( ph /\ k e. NN ) -> ( 2nd ` ( G ` k ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
139 |
|
fvex |
|- ( 2nd ` ( G ` k ) ) e. _V |
140 |
|
f1oeq1 |
|- ( f = ( 2nd ` ( G ` k ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( G ` k ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) ) |
141 |
139 140
|
elab |
|- ( ( 2nd ` ( G ` k ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( G ` k ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
142 |
138 141
|
sylib |
|- ( ( ph /\ k e. NN ) -> ( 2nd ` ( G ` k ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
143 |
|
dff1o3 |
|- ( ( 2nd ` ( G ` k ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( ( 2nd ` ( G ` k ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) /\ Fun `' ( 2nd ` ( G ` k ) ) ) ) |
144 |
143
|
simprbi |
|- ( ( 2nd ` ( G ` k ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> Fun `' ( 2nd ` ( G ` k ) ) ) |
145 |
|
imain |
|- ( Fun `' ( 2nd ` ( G ` k ) ) -> ( ( 2nd ` ( G ` k ) ) " ( ( 1 ... j ) i^i ( ( j + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) i^i ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) ) ) |
146 |
142 144 145
|
3syl |
|- ( ( ph /\ k e. NN ) -> ( ( 2nd ` ( G ` k ) ) " ( ( 1 ... j ) i^i ( ( j + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) i^i ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) ) ) |
147 |
|
elfznn0 |
|- ( j e. ( 0 ... N ) -> j e. NN0 ) |
148 |
147
|
nn0red |
|- ( j e. ( 0 ... N ) -> j e. RR ) |
149 |
148
|
ltp1d |
|- ( j e. ( 0 ... N ) -> j < ( j + 1 ) ) |
150 |
|
fzdisj |
|- ( j < ( j + 1 ) -> ( ( 1 ... j ) i^i ( ( j + 1 ) ... N ) ) = (/) ) |
151 |
149 150
|
syl |
|- ( j e. ( 0 ... N ) -> ( ( 1 ... j ) i^i ( ( j + 1 ) ... N ) ) = (/) ) |
152 |
151
|
imaeq2d |
|- ( j e. ( 0 ... N ) -> ( ( 2nd ` ( G ` k ) ) " ( ( 1 ... j ) i^i ( ( j + 1 ) ... N ) ) ) = ( ( 2nd ` ( G ` k ) ) " (/) ) ) |
153 |
|
ima0 |
|- ( ( 2nd ` ( G ` k ) ) " (/) ) = (/) |
154 |
152 153
|
eqtrdi |
|- ( j e. ( 0 ... N ) -> ( ( 2nd ` ( G ` k ) ) " ( ( 1 ... j ) i^i ( ( j + 1 ) ... N ) ) ) = (/) ) |
155 |
146 154
|
sylan9req |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) i^i ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) ) = (/) ) |
156 |
|
fun |
|- ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) : ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) --> { 1 } /\ ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) : ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) --> { 0 } ) /\ ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) i^i ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) ) = (/) ) -> ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) : ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) u. ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) ) --> ( { 1 } u. { 0 } ) ) |
157 |
136 155 156
|
sylancr |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) : ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) u. ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) ) --> ( { 1 } u. { 0 } ) ) |
158 |
|
imaundi |
|- ( ( 2nd ` ( G ` k ) ) " ( ( 1 ... j ) u. ( ( j + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) u. ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) ) |
159 |
|
nn0p1nn |
|- ( j e. NN0 -> ( j + 1 ) e. NN ) |
160 |
147 159
|
syl |
|- ( j e. ( 0 ... N ) -> ( j + 1 ) e. NN ) |
161 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
162 |
160 161
|
eleqtrdi |
|- ( j e. ( 0 ... N ) -> ( j + 1 ) e. ( ZZ>= ` 1 ) ) |
163 |
|
elfzuz3 |
|- ( j e. ( 0 ... N ) -> N e. ( ZZ>= ` j ) ) |
164 |
|
fzsplit2 |
|- ( ( ( j + 1 ) e. ( ZZ>= ` 1 ) /\ N e. ( ZZ>= ` j ) ) -> ( 1 ... N ) = ( ( 1 ... j ) u. ( ( j + 1 ) ... N ) ) ) |
165 |
162 163 164
|
syl2anc |
|- ( j e. ( 0 ... N ) -> ( 1 ... N ) = ( ( 1 ... j ) u. ( ( j + 1 ) ... N ) ) ) |
166 |
165
|
imaeq2d |
|- ( j e. ( 0 ... N ) -> ( ( 2nd ` ( G ` k ) ) " ( 1 ... N ) ) = ( ( 2nd ` ( G ` k ) ) " ( ( 1 ... j ) u. ( ( j + 1 ) ... N ) ) ) ) |
167 |
|
f1ofo |
|- ( ( 2nd ` ( G ` k ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( G ` k ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) ) |
168 |
|
foima |
|- ( ( 2nd ` ( G ` k ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) -> ( ( 2nd ` ( G ` k ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
169 |
142 167 168
|
3syl |
|- ( ( ph /\ k e. NN ) -> ( ( 2nd ` ( G ` k ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
170 |
166 169
|
sylan9req |
|- ( ( j e. ( 0 ... N ) /\ ( ph /\ k e. NN ) ) -> ( ( 2nd ` ( G ` k ) ) " ( ( 1 ... j ) u. ( ( j + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
171 |
170
|
ancoms |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( 2nd ` ( G ` k ) ) " ( ( 1 ... j ) u. ( ( j + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
172 |
158 171
|
eqtr3id |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) u. ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
173 |
172
|
feq2d |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) : ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) u. ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) ) --> ( { 1 } u. { 0 } ) <-> ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) : ( 1 ... N ) --> ( { 1 } u. { 0 } ) ) ) |
174 |
157 173
|
mpbid |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) : ( 1 ... N ) --> ( { 1 } u. { 0 } ) ) |
175 |
174
|
ffvelrnda |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. ( { 1 } u. { 0 } ) ) |
176 |
131 175
|
sseldi |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. RR ) |
177 |
|
simpllr |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> k e. NN ) |
178 |
176 177
|
nndivred |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) e. RR ) |
179 |
178
|
recnd |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) e. CC ) |
180 |
179
|
absnegd |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) = ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) |
181 |
124 180
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) = ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) |
182 |
124 175
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. ( { 1 } u. { 0 } ) ) |
183 |
|
elun |
|- ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. ( { 1 } u. { 0 } ) <-> ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 1 } \/ ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 0 } ) ) |
184 |
182 183
|
sylib |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 1 } \/ ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 0 } ) ) |
185 |
|
nnrecre |
|- ( k e. NN -> ( 1 / k ) e. RR ) |
186 |
|
nnrp |
|- ( k e. NN -> k e. RR+ ) |
187 |
186
|
rpreccld |
|- ( k e. NN -> ( 1 / k ) e. RR+ ) |
188 |
187
|
rpge0d |
|- ( k e. NN -> 0 <_ ( 1 / k ) ) |
189 |
185 188
|
absidd |
|- ( k e. NN -> ( abs ` ( 1 / k ) ) = ( 1 / k ) ) |
190 |
122 189
|
syl |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( abs ` ( 1 / k ) ) = ( 1 / k ) ) |
191 |
122
|
nnrecred |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( 1 / k ) e. RR ) |
192 |
112
|
adantr |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) e. RR ) |
193 |
115
|
adantr |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( c / 2 ) e. RR ) |
194 |
|
eluzle |
|- ( k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) -> -u ( |_ ` -u ( 2 / c ) ) <_ k ) |
195 |
194
|
adantl |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> -u ( |_ ` -u ( 2 / c ) ) <_ k ) |
196 |
65
|
adantr |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> -u ( |_ ` -u ( 2 / c ) ) e. NN ) |
197 |
196
|
nnrpd |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> -u ( |_ ` -u ( 2 / c ) ) e. RR+ ) |
198 |
122
|
nnrpd |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> k e. RR+ ) |
199 |
197 198
|
lerecd |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( -u ( |_ ` -u ( 2 / c ) ) <_ k <-> ( 1 / k ) <_ ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) |
200 |
195 199
|
mpbid |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( 1 / k ) <_ ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) |
201 |
90
|
adantr |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) <_ ( c / 2 ) ) |
202 |
191 192 193 200 201
|
letrd |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( 1 / k ) <_ ( c / 2 ) ) |
203 |
190 202
|
eqbrtrd |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( abs ` ( 1 / k ) ) <_ ( c / 2 ) ) |
204 |
|
elsni |
|- ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 1 } -> ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) = 1 ) |
205 |
204
|
fvoveq1d |
|- ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 1 } -> ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) = ( abs ` ( 1 / k ) ) ) |
206 |
205
|
breq1d |
|- ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 1 } -> ( ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) <-> ( abs ` ( 1 / k ) ) <_ ( c / 2 ) ) ) |
207 |
203 206
|
syl5ibrcom |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 1 } -> ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) ) ) |
208 |
114
|
rpge0d |
|- ( c e. RR+ -> 0 <_ ( c / 2 ) ) |
209 |
|
nncn |
|- ( k e. NN -> k e. CC ) |
210 |
|
nnne0 |
|- ( k e. NN -> k =/= 0 ) |
211 |
209 210
|
div0d |
|- ( k e. NN -> ( 0 / k ) = 0 ) |
212 |
211
|
abs00bd |
|- ( k e. NN -> ( abs ` ( 0 / k ) ) = 0 ) |
213 |
212
|
breq1d |
|- ( k e. NN -> ( ( abs ` ( 0 / k ) ) <_ ( c / 2 ) <-> 0 <_ ( c / 2 ) ) ) |
214 |
213
|
biimparc |
|- ( ( 0 <_ ( c / 2 ) /\ k e. NN ) -> ( abs ` ( 0 / k ) ) <_ ( c / 2 ) ) |
215 |
208 214
|
sylan |
|- ( ( c e. RR+ /\ k e. NN ) -> ( abs ` ( 0 / k ) ) <_ ( c / 2 ) ) |
216 |
122 215
|
syldan |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( abs ` ( 0 / k ) ) <_ ( c / 2 ) ) |
217 |
|
elsni |
|- ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 0 } -> ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) = 0 ) |
218 |
217
|
fvoveq1d |
|- ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 0 } -> ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) = ( abs ` ( 0 / k ) ) ) |
219 |
218
|
breq1d |
|- ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 0 } -> ( ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) <-> ( abs ` ( 0 / k ) ) <_ ( c / 2 ) ) ) |
220 |
216 219
|
syl5ibrcom |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 0 } -> ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) ) ) |
221 |
207 220
|
jaod |
|- ( ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 1 } \/ ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 0 } ) -> ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) ) ) |
222 |
221
|
adantll |
|- ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 1 } \/ ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 0 } ) -> ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) ) ) |
223 |
222
|
ad2antrr |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 1 } \/ ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. { 0 } ) -> ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) ) ) |
224 |
184 223
|
mpd |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) ) |
225 |
181 224
|
eqbrtrd |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) ) |
226 |
79 111
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) e. RR ) |
227 |
|
simpll |
|- ( ( ( ph /\ C e. I ) /\ c e. RR+ ) -> ph ) |
228 |
227
|
anim1i |
|- ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) -> ( ph /\ k e. NN ) ) |
229 |
178
|
renegcld |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) e. RR ) |
230 |
228 229
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) e. RR ) |
231 |
230
|
recnd |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) e. CC ) |
232 |
231
|
abscld |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) e. RR ) |
233 |
79 232
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) e. RR ) |
234 |
115 115
|
jca |
|- ( c e. RR+ -> ( ( c / 2 ) e. RR /\ ( c / 2 ) e. RR ) ) |
235 |
234
|
ad4antlr |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( c / 2 ) e. RR /\ ( c / 2 ) e. RR ) ) |
236 |
|
ltleadd |
|- ( ( ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) e. RR /\ ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) e. RR ) /\ ( ( c / 2 ) e. RR /\ ( c / 2 ) e. RR ) ) -> ( ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( c / 2 ) /\ ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) ) -> ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) |
237 |
226 233 235 236
|
syl21anc |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( c / 2 ) /\ ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) <_ ( c / 2 ) ) -> ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) |
238 |
225 237
|
mpan2d |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( c / 2 ) -> ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) |
239 |
110 231
|
abstrid |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) <_ ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) ) |
240 |
109 230
|
readdcld |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) e. RR ) |
241 |
240
|
recnd |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) e. CC ) |
242 |
241
|
abscld |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) e. RR ) |
243 |
111 232
|
readdcld |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) e. RR ) |
244 |
115 115
|
readdcld |
|- ( c e. RR+ -> ( ( c / 2 ) + ( c / 2 ) ) e. RR ) |
245 |
244
|
ad4antlr |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( c / 2 ) + ( c / 2 ) ) e. RR ) |
246 |
|
lelttr |
|- ( ( ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) e. RR /\ ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) e. RR /\ ( ( c / 2 ) + ( c / 2 ) ) e. RR ) -> ( ( ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) <_ ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) /\ ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) -> ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) |
247 |
242 243 245 246
|
syl3anc |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) <_ ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) /\ ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) -> ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) |
248 |
239 247
|
mpand |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) -> ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) |
249 |
79 248
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) + ( abs ` -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) -> ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) |
250 |
120 238 249
|
3syld |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) -> ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) |
251 |
105
|
adantlr |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( 1st ` ( G ` k ) ) ` m ) e. RR ) |
252 |
251 176
|
readdcld |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) e. RR ) |
253 |
252 177
|
nndivred |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR ) |
254 |
124 253
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR ) |
255 |
250 254
|
jctild |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR /\ ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) ) |
256 |
255
|
adantld |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR /\ ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) ) |
257 |
79
|
adantr |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) -> ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) ) |
258 |
93
|
ad3antlr |
|- ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) -> C : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
259 |
258
|
ffvelrnda |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( C ` m ) e. ( 0 [,] 1 ) ) |
260 |
39 259
|
sseldi |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( C ` m ) e. RR ) |
261 |
80
|
rpreccld |
|- ( c e. RR+ -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) e. RR+ ) |
262 |
261
|
rpxrd |
|- ( c e. RR+ -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) e. RR* ) |
263 |
262
|
ad3antlr |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) e. RR* ) |
264 |
19
|
rexmet |
|- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) |
265 |
|
elbl |
|- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ ( C ` m ) e. RR /\ ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) e. RR* ) -> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) <-> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
266 |
264 265
|
mp3an1 |
|- ( ( ( C ` m ) e. RR /\ ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) e. RR* ) -> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) <-> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
267 |
260 263 266
|
syl2anc |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) <-> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
268 |
|
elmapfn |
|- ( ( 1st ` ( G ` k ) ) e. ( NN0 ^m ( 1 ... N ) ) -> ( 1st ` ( G ` k ) ) Fn ( 1 ... N ) ) |
269 |
100 101 268
|
3syl |
|- ( ( ph /\ k e. NN ) -> ( 1st ` ( G ` k ) ) Fn ( 1 ... N ) ) |
270 |
|
vex |
|- k e. _V |
271 |
|
fnconstg |
|- ( k e. _V -> ( ( 1 ... N ) X. { k } ) Fn ( 1 ... N ) ) |
272 |
270 271
|
mp1i |
|- ( ( ph /\ k e. NN ) -> ( ( 1 ... N ) X. { k } ) Fn ( 1 ... N ) ) |
273 |
|
fzfid |
|- ( ( ph /\ k e. NN ) -> ( 1 ... N ) e. Fin ) |
274 |
|
inidm |
|- ( ( 1 ... N ) i^i ( 1 ... N ) ) = ( 1 ... N ) |
275 |
|
eqidd |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( 1st ` ( G ` k ) ) ` m ) = ( ( 1st ` ( G ` k ) ) ` m ) ) |
276 |
270
|
fvconst2 |
|- ( m e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { k } ) ` m ) = k ) |
277 |
276
|
adantl |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1 ... N ) X. { k } ) ` m ) = k ) |
278 |
269 272 273 273 274 275 277
|
ofval |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) = ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) |
279 |
278
|
oveq2d |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) = ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) |
280 |
227 279
|
sylanl1 |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) = ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) |
281 |
227 107
|
sylanl1 |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) e. RR ) |
282 |
19
|
remetdval |
|- ( ( ( C ` m ) e. RR /\ ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) e. RR ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) = ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) ) |
283 |
260 281 282
|
syl2anc |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) = ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) ) |
284 |
280 283
|
eqtrd |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) = ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) ) |
285 |
284
|
breq1d |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) <-> ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) |
286 |
285
|
anbi2d |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) <-> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
287 |
267 286
|
bitrd |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) <-> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
288 |
257 287
|
sylan |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) <-> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( abs ` ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) ) < ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) ) ) |
289 |
|
rpxr |
|- ( c e. RR+ -> c e. RR* ) |
290 |
289
|
ad4antlr |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> c e. RR* ) |
291 |
|
elbl |
|- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ ( C ` m ) e. RR /\ c e. RR* ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) <-> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < c ) ) ) |
292 |
264 291
|
mp3an1 |
|- ( ( ( C ` m ) e. RR /\ c e. RR* ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) <-> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < c ) ) ) |
293 |
96 290 292
|
syl2anc |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) <-> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < c ) ) ) |
294 |
|
elun |
|- ( z e. ( { 1 } u. { 0 } ) <-> ( z e. { 1 } \/ z e. { 0 } ) ) |
295 |
|
fzofzp1 |
|- ( v e. ( 0 ..^ k ) -> ( v + 1 ) e. ( 0 ... k ) ) |
296 |
|
elsni |
|- ( z e. { 1 } -> z = 1 ) |
297 |
296
|
oveq2d |
|- ( z e. { 1 } -> ( v + z ) = ( v + 1 ) ) |
298 |
297
|
eleq1d |
|- ( z e. { 1 } -> ( ( v + z ) e. ( 0 ... k ) <-> ( v + 1 ) e. ( 0 ... k ) ) ) |
299 |
295 298
|
syl5ibrcom |
|- ( v e. ( 0 ..^ k ) -> ( z e. { 1 } -> ( v + z ) e. ( 0 ... k ) ) ) |
300 |
|
elfzonn0 |
|- ( v e. ( 0 ..^ k ) -> v e. NN0 ) |
301 |
300
|
nn0cnd |
|- ( v e. ( 0 ..^ k ) -> v e. CC ) |
302 |
301
|
addid1d |
|- ( v e. ( 0 ..^ k ) -> ( v + 0 ) = v ) |
303 |
|
elfzofz |
|- ( v e. ( 0 ..^ k ) -> v e. ( 0 ... k ) ) |
304 |
302 303
|
eqeltrd |
|- ( v e. ( 0 ..^ k ) -> ( v + 0 ) e. ( 0 ... k ) ) |
305 |
|
elsni |
|- ( z e. { 0 } -> z = 0 ) |
306 |
305
|
oveq2d |
|- ( z e. { 0 } -> ( v + z ) = ( v + 0 ) ) |
307 |
306
|
eleq1d |
|- ( z e. { 0 } -> ( ( v + z ) e. ( 0 ... k ) <-> ( v + 0 ) e. ( 0 ... k ) ) ) |
308 |
304 307
|
syl5ibrcom |
|- ( v e. ( 0 ..^ k ) -> ( z e. { 0 } -> ( v + z ) e. ( 0 ... k ) ) ) |
309 |
299 308
|
jaod |
|- ( v e. ( 0 ..^ k ) -> ( ( z e. { 1 } \/ z e. { 0 } ) -> ( v + z ) e. ( 0 ... k ) ) ) |
310 |
294 309
|
syl5bi |
|- ( v e. ( 0 ..^ k ) -> ( z e. ( { 1 } u. { 0 } ) -> ( v + z ) e. ( 0 ... k ) ) ) |
311 |
310
|
imp |
|- ( ( v e. ( 0 ..^ k ) /\ z e. ( { 1 } u. { 0 } ) ) -> ( v + z ) e. ( 0 ... k ) ) |
312 |
311
|
adantl |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ ( v e. ( 0 ..^ k ) /\ z e. ( { 1 } u. { 0 } ) ) ) -> ( v + z ) e. ( 0 ... k ) ) |
313 |
|
dffn3 |
|- ( ( 1st ` ( G ` k ) ) Fn ( 1 ... N ) <-> ( 1st ` ( G ` k ) ) : ( 1 ... N ) --> ran ( 1st ` ( G ` k ) ) ) |
314 |
269 313
|
sylib |
|- ( ( ph /\ k e. NN ) -> ( 1st ` ( G ` k ) ) : ( 1 ... N ) --> ran ( 1st ` ( G ` k ) ) ) |
315 |
314 7
|
fssd |
|- ( ( ph /\ k e. NN ) -> ( 1st ` ( G ` k ) ) : ( 1 ... N ) --> ( 0 ..^ k ) ) |
316 |
315
|
adantr |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( 1st ` ( G ` k ) ) : ( 1 ... N ) --> ( 0 ..^ k ) ) |
317 |
|
fzfid |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( 1 ... N ) e. Fin ) |
318 |
312 316 174 317 317 274
|
off |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) : ( 1 ... N ) --> ( 0 ... k ) ) |
319 |
318
|
ffnd |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) Fn ( 1 ... N ) ) |
320 |
270 271
|
mp1i |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( 1 ... N ) X. { k } ) Fn ( 1 ... N ) ) |
321 |
269
|
adantr |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( 1st ` ( G ` k ) ) Fn ( 1 ... N ) ) |
322 |
174
|
ffnd |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) |
323 |
|
eqidd |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( 1st ` ( G ` k ) ) ` m ) = ( ( 1st ` ( G ` k ) ) ` m ) ) |
324 |
|
eqidd |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) = ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) |
325 |
321 322 317 317 274 323 324
|
ofval |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ` m ) = ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) ) |
326 |
276
|
adantl |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1 ... N ) X. { k } ) ` m ) = k ) |
327 |
319 320 317 317 274 325 326
|
ofval |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) = ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) |
328 |
327
|
eleq1d |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR <-> ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR ) ) |
329 |
228 328
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR <-> ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR ) ) |
330 |
327
|
adantl3r |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) = ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) |
331 |
330
|
oveq2d |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) = ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) ) |
332 |
93
|
ad3antlr |
|- ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> C : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
333 |
332
|
ffvelrnda |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( C ` m ) e. ( 0 [,] 1 ) ) |
334 |
39 333
|
sseldi |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( C ` m ) e. RR ) |
335 |
253
|
adantl3r |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR ) |
336 |
19
|
remetdval |
|- ( ( ( C ` m ) e. RR /\ ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) = ( abs ` ( ( C ` m ) - ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) ) ) |
337 |
334 335 336
|
syl2anc |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) = ( abs ` ( ( C ` m ) - ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) ) ) |
338 |
251
|
recnd |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( 1st ` ( G ` k ) ) ` m ) e. CC ) |
339 |
176
|
recnd |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) e. CC ) |
340 |
209
|
ad3antlr |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> k e. CC ) |
341 |
210
|
ad3antlr |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> k =/= 0 ) |
342 |
338 339 340 341
|
divdird |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) = ( ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) + ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) |
343 |
107
|
recnd |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) e. CC ) |
344 |
343
|
adantlr |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) e. CC ) |
345 |
344 179
|
subnegd |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) - -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) = ( ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) + ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) |
346 |
342 345
|
eqtr4d |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) = ( ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) - -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) |
347 |
346
|
oveq2d |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) - ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) = ( ( C ` m ) - ( ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) - -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) ) |
348 |
347
|
adantl3r |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) - ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) = ( ( C ` m ) - ( ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) - -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) ) |
349 |
334
|
recnd |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( C ` m ) e. CC ) |
350 |
107
|
adantllr |
|- ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) e. RR ) |
351 |
350
|
adantlr |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) e. RR ) |
352 |
351
|
recnd |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) e. CC ) |
353 |
179
|
adantl3r |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) e. CC ) |
354 |
353
|
negcld |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) e. CC ) |
355 |
349 352 354
|
subsubd |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) - ( ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) - -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) = ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) |
356 |
348 355
|
eqtrd |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) - ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) = ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) |
357 |
356
|
fveq2d |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( abs ` ( ( C ` m ) - ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) ) ) = ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) ) |
358 |
331 337 357
|
3eqtrd |
|- ( ( ( ( ( ph /\ C e. I ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) = ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) ) |
359 |
358
|
adantl3r |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) = ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) ) |
360 |
83
|
2halvesd |
|- ( c e. RR+ -> ( ( c / 2 ) + ( c / 2 ) ) = c ) |
361 |
360
|
eqcomd |
|- ( c e. RR+ -> c = ( ( c / 2 ) + ( c / 2 ) ) ) |
362 |
361
|
ad4antlr |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> c = ( ( c / 2 ) + ( c / 2 ) ) ) |
363 |
359 362
|
breq12d |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < c <-> ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) |
364 |
329 363
|
anbi12d |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. RR /\ ( ( C ` m ) ( ( abs o. - ) |` ( RR X. RR ) ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) ) < c ) <-> ( ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR /\ ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) ) |
365 |
293 364
|
bitrd |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) <-> ( ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR /\ ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) ) |
366 |
79 365
|
sylanl1 |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) <-> ( ( ( ( ( 1st ` ( G ` k ) ) ` m ) + ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) ) / k ) e. RR /\ ( abs ` ( ( ( C ` m ) - ( ( ( 1st ` ( G ` k ) ) ` m ) / k ) ) + -u ( ( ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ` m ) / k ) ) ) < ( ( c / 2 ) + ( c / 2 ) ) ) ) ) |
367 |
256 288 366
|
3imtr4d |
|- ( ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) /\ m e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) ) ) |
368 |
367
|
ralimdva |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) -> ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> A. m e. ( 1 ... N ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) ) ) |
369 |
|
simplr |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> k e. NN ) |
370 |
|
elfznn0 |
|- ( v e. ( 0 ... k ) -> v e. NN0 ) |
371 |
370
|
nn0red |
|- ( v e. ( 0 ... k ) -> v e. RR ) |
372 |
|
nndivre |
|- ( ( v e. RR /\ k e. NN ) -> ( v / k ) e. RR ) |
373 |
371 372
|
sylan |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> ( v / k ) e. RR ) |
374 |
|
elfzle1 |
|- ( v e. ( 0 ... k ) -> 0 <_ v ) |
375 |
371 374
|
jca |
|- ( v e. ( 0 ... k ) -> ( v e. RR /\ 0 <_ v ) ) |
376 |
186
|
rpregt0d |
|- ( k e. NN -> ( k e. RR /\ 0 < k ) ) |
377 |
|
divge0 |
|- ( ( ( v e. RR /\ 0 <_ v ) /\ ( k e. RR /\ 0 < k ) ) -> 0 <_ ( v / k ) ) |
378 |
375 376 377
|
syl2an |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> 0 <_ ( v / k ) ) |
379 |
|
elfzle2 |
|- ( v e. ( 0 ... k ) -> v <_ k ) |
380 |
379
|
adantr |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> v <_ k ) |
381 |
371
|
adantr |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> v e. RR ) |
382 |
|
1red |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> 1 e. RR ) |
383 |
186
|
adantl |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> k e. RR+ ) |
384 |
381 382 383
|
ledivmuld |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> ( ( v / k ) <_ 1 <-> v <_ ( k x. 1 ) ) ) |
385 |
209
|
mulid1d |
|- ( k e. NN -> ( k x. 1 ) = k ) |
386 |
385
|
breq2d |
|- ( k e. NN -> ( v <_ ( k x. 1 ) <-> v <_ k ) ) |
387 |
386
|
adantl |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> ( v <_ ( k x. 1 ) <-> v <_ k ) ) |
388 |
384 387
|
bitrd |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> ( ( v / k ) <_ 1 <-> v <_ k ) ) |
389 |
380 388
|
mpbird |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> ( v / k ) <_ 1 ) |
390 |
|
elicc01 |
|- ( ( v / k ) e. ( 0 [,] 1 ) <-> ( ( v / k ) e. RR /\ 0 <_ ( v / k ) /\ ( v / k ) <_ 1 ) ) |
391 |
373 378 389 390
|
syl3anbrc |
|- ( ( v e. ( 0 ... k ) /\ k e. NN ) -> ( v / k ) e. ( 0 [,] 1 ) ) |
392 |
391
|
ancoms |
|- ( ( k e. NN /\ v e. ( 0 ... k ) ) -> ( v / k ) e. ( 0 [,] 1 ) ) |
393 |
|
elsni |
|- ( z e. { k } -> z = k ) |
394 |
393
|
oveq2d |
|- ( z e. { k } -> ( v / z ) = ( v / k ) ) |
395 |
394
|
eleq1d |
|- ( z e. { k } -> ( ( v / z ) e. ( 0 [,] 1 ) <-> ( v / k ) e. ( 0 [,] 1 ) ) ) |
396 |
392 395
|
syl5ibrcom |
|- ( ( k e. NN /\ v e. ( 0 ... k ) ) -> ( z e. { k } -> ( v / z ) e. ( 0 [,] 1 ) ) ) |
397 |
396
|
impr |
|- ( ( k e. NN /\ ( v e. ( 0 ... k ) /\ z e. { k } ) ) -> ( v / z ) e. ( 0 [,] 1 ) ) |
398 |
369 397
|
sylan |
|- ( ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) /\ ( v e. ( 0 ... k ) /\ z e. { k } ) ) -> ( v / z ) e. ( 0 [,] 1 ) ) |
399 |
270
|
fconst |
|- ( ( 1 ... N ) X. { k } ) : ( 1 ... N ) --> { k } |
400 |
399
|
a1i |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( 1 ... N ) X. { k } ) : ( 1 ... N ) --> { k } ) |
401 |
398 318 400 317 317 274
|
off |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
402 |
401
|
ffnd |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) Fn ( 1 ... N ) ) |
403 |
124 402
|
sylan |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) Fn ( 1 ... N ) ) |
404 |
368 403
|
jctild |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) -> ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) Fn ( 1 ... N ) /\ A. m e. ( 1 ... N ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) ) ) ) |
405 |
2
|
eleq2i |
|- ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. I <-> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) ) |
406 |
|
ovex |
|- ( 0 [,] 1 ) e. _V |
407 |
406 43
|
elmap |
|- ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) <-> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
408 |
405 407
|
bitri |
|- ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. I <-> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
409 |
401 408
|
sylibr |
|- ( ( ( ph /\ k e. NN ) /\ j e. ( 0 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. I ) |
410 |
124 409
|
sylan |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. I ) |
411 |
404 410
|
jctird |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) -> ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) Fn ( 1 ... N ) /\ A. m e. ( 1 ... N ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) ) /\ ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. I ) ) ) |
412 |
|
elin |
|- ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) <-> ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) /\ ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. I ) ) |
413 |
|
ovex |
|- ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. _V |
414 |
413
|
elixp |
|- ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) <-> ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) Fn ( 1 ... N ) /\ A. m e. ( 1 ... N ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) ) ) |
415 |
414
|
anbi1i |
|- ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) /\ ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. I ) <-> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) Fn ( 1 ... N ) /\ A. m e. ( 1 ... N ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) ) /\ ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. I ) ) |
416 |
412 415
|
bitri |
|- ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) <-> ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) Fn ( 1 ... N ) /\ A. m e. ( 1 ... N ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) ) /\ ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. I ) ) |
417 |
411 416
|
syl6ibr |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) -> ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) ) ) |
418 |
|
ssel |
|- ( ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v -> ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v ) ) |
419 |
418
|
com12 |
|- ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) -> ( ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v ) ) |
420 |
417 419
|
syl6 |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) -> ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v ) ) ) |
421 |
420
|
impd |
|- ( ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) /\ j e. ( 0 ... N ) ) -> ( ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) /\ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) -> ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v ) ) |
422 |
421
|
ralrimdva |
|- ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) /\ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) -> A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v ) ) |
423 |
422
|
expd |
|- ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v -> A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v ) ) ) |
424 |
8
|
3exp2 |
|- ( ph -> ( k e. NN -> ( n e. ( 1 ... N ) -> ( r e. { <_ , `' <_ } -> E. j e. ( 0 ... N ) 0 r X ) ) ) ) |
425 |
424
|
imp43 |
|- ( ( ( ph /\ k e. NN ) /\ ( n e. ( 1 ... N ) /\ r e. { <_ , `' <_ } ) ) -> E. j e. ( 0 ... N ) 0 r X ) |
426 |
|
r19.29 |
|- ( ( A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v /\ E. j e. ( 0 ... N ) 0 r X ) -> E. j e. ( 0 ... N ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v /\ 0 r X ) ) |
427 |
|
fveq2 |
|- ( z = ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) -> ( F ` z ) = ( F ` ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ) ) |
428 |
427
|
fveq1d |
|- ( z = ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) -> ( ( F ` z ) ` n ) = ( ( F ` ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) ) ` n ) ) |
429 |
428 5
|
eqtr4di |
|- ( z = ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) -> ( ( F ` z ) ` n ) = X ) |
430 |
429
|
breq2d |
|- ( z = ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) -> ( 0 r ( ( F ` z ) ` n ) <-> 0 r X ) ) |
431 |
430
|
rspcev |
|- ( ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v /\ 0 r X ) -> E. z e. v 0 r ( ( F ` z ) ` n ) ) |
432 |
431
|
rexlimivw |
|- ( E. j e. ( 0 ... N ) ( ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v /\ 0 r X ) -> E. z e. v 0 r ( ( F ` z ) ` n ) ) |
433 |
426 432
|
syl |
|- ( ( A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v /\ E. j e. ( 0 ... N ) 0 r X ) -> E. z e. v 0 r ( ( F ` z ) ` n ) ) |
434 |
433
|
expcom |
|- ( E. j e. ( 0 ... N ) 0 r X -> ( A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v -> E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
435 |
425 434
|
syl |
|- ( ( ( ph /\ k e. NN ) /\ ( n e. ( 1 ... N ) /\ r e. { <_ , `' <_ } ) ) -> ( A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v -> E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
436 |
435
|
ralrimdvva |
|- ( ( ph /\ k e. NN ) -> ( A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
437 |
122 436
|
sylan2 |
|- ( ( ph /\ ( c e. RR+ /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) ) -> ( A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
438 |
437
|
anassrs |
|- ( ( ( ph /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
439 |
438
|
adantllr |
|- ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( A. j e. ( 0 ... N ) ( ( ( 1st ` ( G ` k ) ) oF + ( ( ( ( 2nd ` ( G ` k ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( G ` k ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) oF / ( ( 1 ... N ) X. { k } ) ) e. v -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
440 |
423 439
|
syl6d |
|- ( ( ( ( ph /\ C e. I ) /\ c e. RR+ ) /\ k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) ) |
441 |
440
|
rexlimdva |
|- ( ( ( ph /\ C e. I ) /\ c e. RR+ ) -> ( E. k e. ( ZZ>= ` -u ( |_ ` -u ( 2 / c ) ) ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / -u ( |_ ` -u ( 2 / c ) ) ) ) -> ( ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) ) |
442 |
74 441
|
syld |
|- ( ( ( ph /\ C e. I ) /\ c e. RR+ ) -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> ( ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) ) |
443 |
442
|
com23 |
|- ( ( ( ph /\ C e. I ) /\ c e. RR+ ) -> ( ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) ) |
444 |
443
|
impr |
|- ( ( ( ph /\ C e. I ) /\ ( c e. RR+ /\ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) ) -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
445 |
51 444
|
sylanl2 |
|- ( ( ( ph /\ ( v e. ( R |`t I ) /\ C e. v ) ) /\ ( c e. RR+ /\ ( X_ m e. ( 1 ... N ) ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) c ) i^i I ) C_ v ) ) -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
446 |
35 445
|
rexlimddv |
|- ( ( ph /\ ( v e. ( R |`t I ) /\ C e. v ) ) -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
447 |
446
|
expr |
|- ( ( ph /\ v e. ( R |`t I ) ) -> ( C e. v -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) ) |
448 |
447
|
com23 |
|- ( ( ph /\ v e. ( R |`t I ) ) -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> ( C e. v -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) ) |
449 |
|
r19.21v |
|- ( A. n e. ( 1 ... N ) ( C e. v -> A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) <-> ( C e. v -> A. n e. ( 1 ... N ) A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
450 |
448 449
|
syl6ibr |
|- ( ( ph /\ v e. ( R |`t I ) ) -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> A. n e. ( 1 ... N ) ( C e. v -> A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) ) |
451 |
450
|
ralrimdva |
|- ( ph -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> A. v e. ( R |`t I ) A. n e. ( 1 ... N ) ( C e. v -> A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) ) |
452 |
|
ralcom |
|- ( A. v e. ( R |`t I ) A. n e. ( 1 ... N ) ( C e. v -> A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) <-> A. n e. ( 1 ... N ) A. v e. ( R |`t I ) ( C e. v -> A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) |
453 |
451 452
|
syl6ib |
|- ( ph -> ( A. i e. NN E. k e. ( ZZ>= ` i ) A. m e. ( 1 ... N ) ( ( ( 1st ` ( G ` k ) ) oF / ( ( 1 ... N ) X. { k } ) ) ` m ) e. ( ( C ` m ) ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ( 1 / i ) ) -> A. n e. ( 1 ... N ) A. v e. ( R |`t I ) ( C e. v -> A. r e. { <_ , `' <_ } E. z e. v 0 r ( ( F ` z ) ` n ) ) ) ) |