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 |
|
broucube.1 |
|- ( ph -> F e. ( ( R |`t I ) Cn ( R |`t I ) ) ) |
5 |
|
elmapfn |
|- ( x e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> x Fn ( 1 ... N ) ) |
6 |
5 2
|
eleq2s |
|- ( x e. I -> x Fn ( 1 ... N ) ) |
7 |
6
|
adantl |
|- ( ( ph /\ x e. I ) -> x Fn ( 1 ... N ) ) |
8 |
|
ovex |
|- ( 1 ... N ) e. _V |
9 |
|
retopon |
|- ( topGen ` ran (,) ) e. ( TopOn ` RR ) |
10 |
3
|
pttoponconst |
|- ( ( ( 1 ... N ) e. _V /\ ( topGen ` ran (,) ) e. ( TopOn ` RR ) ) -> R e. ( TopOn ` ( RR ^m ( 1 ... N ) ) ) ) |
11 |
8 9 10
|
mp2an |
|- R e. ( TopOn ` ( RR ^m ( 1 ... N ) ) ) |
12 |
|
reex |
|- RR e. _V |
13 |
|
unitssre |
|- ( 0 [,] 1 ) C_ RR |
14 |
|
mapss |
|- ( ( RR e. _V /\ ( 0 [,] 1 ) C_ RR ) -> ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) C_ ( RR ^m ( 1 ... N ) ) ) |
15 |
12 13 14
|
mp2an |
|- ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) C_ ( RR ^m ( 1 ... N ) ) |
16 |
2 15
|
eqsstri |
|- I C_ ( RR ^m ( 1 ... N ) ) |
17 |
|
resttopon |
|- ( ( R e. ( TopOn ` ( RR ^m ( 1 ... N ) ) ) /\ I C_ ( RR ^m ( 1 ... N ) ) ) -> ( R |`t I ) e. ( TopOn ` I ) ) |
18 |
11 16 17
|
mp2an |
|- ( R |`t I ) e. ( TopOn ` I ) |
19 |
18
|
toponunii |
|- I = U. ( R |`t I ) |
20 |
19 19
|
cnf |
|- ( F e. ( ( R |`t I ) Cn ( R |`t I ) ) -> F : I --> I ) |
21 |
4 20
|
syl |
|- ( ph -> F : I --> I ) |
22 |
21
|
ffvelrnda |
|- ( ( ph /\ x e. I ) -> ( F ` x ) e. I ) |
23 |
|
elmapfn |
|- ( ( F ` x ) e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> ( F ` x ) Fn ( 1 ... N ) ) |
24 |
23 2
|
eleq2s |
|- ( ( F ` x ) e. I -> ( F ` x ) Fn ( 1 ... N ) ) |
25 |
22 24
|
syl |
|- ( ( ph /\ x e. I ) -> ( F ` x ) Fn ( 1 ... N ) ) |
26 |
|
ovexd |
|- ( ( ph /\ x e. I ) -> ( 1 ... N ) e. _V ) |
27 |
|
inidm |
|- ( ( 1 ... N ) i^i ( 1 ... N ) ) = ( 1 ... N ) |
28 |
|
eqidd |
|- ( ( ( ph /\ x e. I ) /\ n e. ( 1 ... N ) ) -> ( x ` n ) = ( x ` n ) ) |
29 |
|
eqidd |
|- ( ( ( ph /\ x e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` x ) ` n ) = ( ( F ` x ) ` n ) ) |
30 |
7 25 26 26 27 28 29
|
offval |
|- ( ( ph /\ x e. I ) -> ( x oF - ( F ` x ) ) = ( n e. ( 1 ... N ) |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) ) |
31 |
30
|
mpteq2dva |
|- ( ph -> ( x e. I |-> ( x oF - ( F ` x ) ) ) = ( x e. I |-> ( n e. ( 1 ... N ) |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) ) ) |
32 |
18
|
a1i |
|- ( ph -> ( R |`t I ) e. ( TopOn ` I ) ) |
33 |
|
ovexd |
|- ( ph -> ( 1 ... N ) e. _V ) |
34 |
|
retop |
|- ( topGen ` ran (,) ) e. Top |
35 |
34
|
fconst6 |
|- ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) : ( 1 ... N ) --> Top |
36 |
35
|
a1i |
|- ( ph -> ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) : ( 1 ... N ) --> Top ) |
37 |
18
|
a1i |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( R |`t I ) e. ( TopOn ` I ) ) |
38 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
39 |
38
|
cnfldtop |
|- ( TopOpen ` CCfld ) e. Top |
40 |
|
cnrest2r |
|- ( ( TopOpen ` CCfld ) e. Top -> ( ( R |`t I ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) C_ ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) ) |
41 |
39 40
|
ax-mp |
|- ( ( R |`t I ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) C_ ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) |
42 |
|
resmpt |
|- ( I C_ ( RR ^m ( 1 ... N ) ) -> ( ( x e. ( RR ^m ( 1 ... N ) ) |-> ( x ` n ) ) |` I ) = ( x e. I |-> ( x ` n ) ) ) |
43 |
16 42
|
ax-mp |
|- ( ( x e. ( RR ^m ( 1 ... N ) ) |-> ( x ` n ) ) |` I ) = ( x e. I |-> ( x ` n ) ) |
44 |
11
|
toponunii |
|- ( RR ^m ( 1 ... N ) ) = U. R |
45 |
44 3
|
ptpjcn |
|- ( ( ( 1 ... N ) e. _V /\ ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) : ( 1 ... N ) --> Top /\ n e. ( 1 ... N ) ) -> ( x e. ( RR ^m ( 1 ... N ) ) |-> ( x ` n ) ) e. ( R Cn ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) ) ) |
46 |
8 35 45
|
mp3an12 |
|- ( n e. ( 1 ... N ) -> ( x e. ( RR ^m ( 1 ... N ) ) |-> ( x ` n ) ) e. ( R Cn ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) ) ) |
47 |
44
|
cnrest |
|- ( ( ( x e. ( RR ^m ( 1 ... N ) ) |-> ( x ` n ) ) e. ( R Cn ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) ) /\ I C_ ( RR ^m ( 1 ... N ) ) ) -> ( ( x e. ( RR ^m ( 1 ... N ) ) |-> ( x ` n ) ) |` I ) e. ( ( R |`t I ) Cn ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) ) ) |
48 |
46 16 47
|
sylancl |
|- ( n e. ( 1 ... N ) -> ( ( x e. ( RR ^m ( 1 ... N ) ) |-> ( x ` n ) ) |` I ) e. ( ( R |`t I ) Cn ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) ) ) |
49 |
43 48
|
eqeltrrid |
|- ( n e. ( 1 ... N ) -> ( x e. I |-> ( x ` n ) ) e. ( ( R |`t I ) Cn ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) ) ) |
50 |
|
fvex |
|- ( topGen ` ran (,) ) e. _V |
51 |
50
|
fvconst2 |
|- ( n e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) = ( topGen ` ran (,) ) ) |
52 |
38
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
53 |
51 52
|
eqtrdi |
|- ( n e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) = ( ( TopOpen ` CCfld ) |`t RR ) ) |
54 |
53
|
oveq2d |
|- ( n e. ( 1 ... N ) -> ( ( R |`t I ) Cn ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) ) = ( ( R |`t I ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) |
55 |
49 54
|
eleqtrd |
|- ( n e. ( 1 ... N ) -> ( x e. I |-> ( x ` n ) ) e. ( ( R |`t I ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) |
56 |
41 55
|
sselid |
|- ( n e. ( 1 ... N ) -> ( x e. I |-> ( x ` n ) ) e. ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) ) |
57 |
56
|
adantl |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( x e. I |-> ( x ` n ) ) e. ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) ) |
58 |
21
|
feqmptd |
|- ( ph -> F = ( x e. I |-> ( F ` x ) ) ) |
59 |
58 4
|
eqeltrrd |
|- ( ph -> ( x e. I |-> ( F ` x ) ) e. ( ( R |`t I ) Cn ( R |`t I ) ) ) |
60 |
59
|
adantr |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( x e. I |-> ( F ` x ) ) e. ( ( R |`t I ) Cn ( R |`t I ) ) ) |
61 |
|
fveq1 |
|- ( x = z -> ( x ` n ) = ( z ` n ) ) |
62 |
61
|
cbvmptv |
|- ( x e. I |-> ( x ` n ) ) = ( z e. I |-> ( z ` n ) ) |
63 |
62 57
|
eqeltrrid |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( z e. I |-> ( z ` n ) ) e. ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) ) |
64 |
|
fveq1 |
|- ( z = ( F ` x ) -> ( z ` n ) = ( ( F ` x ) ` n ) ) |
65 |
37 60 37 63 64
|
cnmpt11 |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( x e. I |-> ( ( F ` x ) ` n ) ) e. ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) ) |
66 |
38
|
subcn |
|- - e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) |
67 |
66
|
a1i |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> - e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) ) |
68 |
37 57 65 67
|
cnmpt12f |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) e. ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) ) |
69 |
|
elmapi |
|- ( x e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> x : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
70 |
69 2
|
eleq2s |
|- ( x e. I -> x : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
71 |
70
|
ffvelrnda |
|- ( ( x e. I /\ n e. ( 1 ... N ) ) -> ( x ` n ) e. ( 0 [,] 1 ) ) |
72 |
13 71
|
sselid |
|- ( ( x e. I /\ n e. ( 1 ... N ) ) -> ( x ` n ) e. RR ) |
73 |
72
|
adantll |
|- ( ( ( ph /\ x e. I ) /\ n e. ( 1 ... N ) ) -> ( x ` n ) e. RR ) |
74 |
|
elmapi |
|- ( ( F ` x ) e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> ( F ` x ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
75 |
74 2
|
eleq2s |
|- ( ( F ` x ) e. I -> ( F ` x ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
76 |
22 75
|
syl |
|- ( ( ph /\ x e. I ) -> ( F ` x ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
77 |
76
|
ffvelrnda |
|- ( ( ( ph /\ x e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` x ) ` n ) e. ( 0 [,] 1 ) ) |
78 |
13 77
|
sselid |
|- ( ( ( ph /\ x e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` x ) ` n ) e. RR ) |
79 |
73 78
|
resubcld |
|- ( ( ( ph /\ x e. I ) /\ n e. ( 1 ... N ) ) -> ( ( x ` n ) - ( ( F ` x ) ` n ) ) e. RR ) |
80 |
79
|
an32s |
|- ( ( ( ph /\ n e. ( 1 ... N ) ) /\ x e. I ) -> ( ( x ` n ) - ( ( F ` x ) ` n ) ) e. RR ) |
81 |
80
|
fmpttd |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) : I --> RR ) |
82 |
|
frn |
|- ( ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) : I --> RR -> ran ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) C_ RR ) |
83 |
38
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
84 |
|
ax-resscn |
|- RR C_ CC |
85 |
|
cnrest2 |
|- ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ ran ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) C_ RR /\ RR C_ CC ) -> ( ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) e. ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) <-> ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) e. ( ( R |`t I ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) ) |
86 |
83 84 85
|
mp3an13 |
|- ( ran ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) C_ RR -> ( ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) e. ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) <-> ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) e. ( ( R |`t I ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) ) |
87 |
81 82 86
|
3syl |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) e. ( ( R |`t I ) Cn ( TopOpen ` CCfld ) ) <-> ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) e. ( ( R |`t I ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) ) |
88 |
68 87
|
mpbid |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) e. ( ( R |`t I ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) |
89 |
54
|
adantl |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( R |`t I ) Cn ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) ) = ( ( R |`t I ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) |
90 |
88 89
|
eleqtrrd |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> ( x e. I |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) e. ( ( R |`t I ) Cn ( ( ( 1 ... N ) X. { ( topGen ` ran (,) ) } ) ` n ) ) ) |
91 |
3 32 33 36 90
|
ptcn |
|- ( ph -> ( x e. I |-> ( n e. ( 1 ... N ) |-> ( ( x ` n ) - ( ( F ` x ) ` n ) ) ) ) e. ( ( R |`t I ) Cn R ) ) |
92 |
31 91
|
eqeltrd |
|- ( ph -> ( x e. I |-> ( x oF - ( F ` x ) ) ) e. ( ( R |`t I ) Cn R ) ) |
93 |
|
simpr2 |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 0 ) ) -> z e. I ) |
94 |
|
id |
|- ( x = z -> x = z ) |
95 |
|
fveq2 |
|- ( x = z -> ( F ` x ) = ( F ` z ) ) |
96 |
94 95
|
oveq12d |
|- ( x = z -> ( x oF - ( F ` x ) ) = ( z oF - ( F ` z ) ) ) |
97 |
|
eqid |
|- ( x e. I |-> ( x oF - ( F ` x ) ) ) = ( x e. I |-> ( x oF - ( F ` x ) ) ) |
98 |
|
ovex |
|- ( z oF - ( F ` z ) ) e. _V |
99 |
96 97 98
|
fvmpt |
|- ( z e. I -> ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` z ) = ( z oF - ( F ` z ) ) ) |
100 |
99
|
fveq1d |
|- ( z e. I -> ( ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` z ) ` n ) = ( ( z oF - ( F ` z ) ) ` n ) ) |
101 |
93 100
|
syl |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 0 ) ) -> ( ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` z ) ` n ) = ( ( z oF - ( F ` z ) ) ` n ) ) |
102 |
|
elmapfn |
|- ( z e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> z Fn ( 1 ... N ) ) |
103 |
102 2
|
eleq2s |
|- ( z e. I -> z Fn ( 1 ... N ) ) |
104 |
103
|
adantl |
|- ( ( ( ph /\ ( z ` n ) = 0 ) /\ z e. I ) -> z Fn ( 1 ... N ) ) |
105 |
21
|
ffvelrnda |
|- ( ( ph /\ z e. I ) -> ( F ` z ) e. I ) |
106 |
|
elmapfn |
|- ( ( F ` z ) e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> ( F ` z ) Fn ( 1 ... N ) ) |
107 |
106 2
|
eleq2s |
|- ( ( F ` z ) e. I -> ( F ` z ) Fn ( 1 ... N ) ) |
108 |
105 107
|
syl |
|- ( ( ph /\ z e. I ) -> ( F ` z ) Fn ( 1 ... N ) ) |
109 |
108
|
adantlr |
|- ( ( ( ph /\ ( z ` n ) = 0 ) /\ z e. I ) -> ( F ` z ) Fn ( 1 ... N ) ) |
110 |
|
ovexd |
|- ( ( ( ph /\ ( z ` n ) = 0 ) /\ z e. I ) -> ( 1 ... N ) e. _V ) |
111 |
|
simpllr |
|- ( ( ( ( ph /\ ( z ` n ) = 0 ) /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( z ` n ) = 0 ) |
112 |
|
eqidd |
|- ( ( ( ( ph /\ ( z ` n ) = 0 ) /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` z ) ` n ) = ( ( F ` z ) ` n ) ) |
113 |
104 109 110 110 27 111 112
|
ofval |
|- ( ( ( ( ph /\ ( z ` n ) = 0 ) /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( ( z oF - ( F ` z ) ) ` n ) = ( 0 - ( ( F ` z ) ` n ) ) ) |
114 |
|
df-neg |
|- -u ( ( F ` z ) ` n ) = ( 0 - ( ( F ` z ) ` n ) ) |
115 |
113 114
|
eqtr4di |
|- ( ( ( ( ph /\ ( z ` n ) = 0 ) /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( ( z oF - ( F ` z ) ) ` n ) = -u ( ( F ` z ) ` n ) ) |
116 |
115
|
exp41 |
|- ( ph -> ( ( z ` n ) = 0 -> ( z e. I -> ( n e. ( 1 ... N ) -> ( ( z oF - ( F ` z ) ) ` n ) = -u ( ( F ` z ) ` n ) ) ) ) ) |
117 |
116
|
com24 |
|- ( ph -> ( n e. ( 1 ... N ) -> ( z e. I -> ( ( z ` n ) = 0 -> ( ( z oF - ( F ` z ) ) ` n ) = -u ( ( F ` z ) ` n ) ) ) ) ) |
118 |
117
|
3imp2 |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 0 ) ) -> ( ( z oF - ( F ` z ) ) ` n ) = -u ( ( F ` z ) ` n ) ) |
119 |
101 118
|
eqtrd |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 0 ) ) -> ( ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` z ) ` n ) = -u ( ( F ` z ) ` n ) ) |
120 |
|
elmapi |
|- ( ( F ` z ) e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> ( F ` z ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
121 |
120 2
|
eleq2s |
|- ( ( F ` z ) e. I -> ( F ` z ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
122 |
105 121
|
syl |
|- ( ( ph /\ z e. I ) -> ( F ` z ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
123 |
122
|
ffvelrnda |
|- ( ( ( ph /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` z ) ` n ) e. ( 0 [,] 1 ) ) |
124 |
|
0xr |
|- 0 e. RR* |
125 |
|
1xr |
|- 1 e. RR* |
126 |
|
iccgelb |
|- ( ( 0 e. RR* /\ 1 e. RR* /\ ( ( F ` z ) ` n ) e. ( 0 [,] 1 ) ) -> 0 <_ ( ( F ` z ) ` n ) ) |
127 |
124 125 126
|
mp3an12 |
|- ( ( ( F ` z ) ` n ) e. ( 0 [,] 1 ) -> 0 <_ ( ( F ` z ) ` n ) ) |
128 |
123 127
|
syl |
|- ( ( ( ph /\ z e. I ) /\ n e. ( 1 ... N ) ) -> 0 <_ ( ( F ` z ) ` n ) ) |
129 |
13 123
|
sselid |
|- ( ( ( ph /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` z ) ` n ) e. RR ) |
130 |
129
|
le0neg2d |
|- ( ( ( ph /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( 0 <_ ( ( F ` z ) ` n ) <-> -u ( ( F ` z ) ` n ) <_ 0 ) ) |
131 |
128 130
|
mpbid |
|- ( ( ( ph /\ z e. I ) /\ n e. ( 1 ... N ) ) -> -u ( ( F ` z ) ` n ) <_ 0 ) |
132 |
131
|
an32s |
|- ( ( ( ph /\ n e. ( 1 ... N ) ) /\ z e. I ) -> -u ( ( F ` z ) ` n ) <_ 0 ) |
133 |
132
|
anasss |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I ) ) -> -u ( ( F ` z ) ` n ) <_ 0 ) |
134 |
133
|
3adantr3 |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 0 ) ) -> -u ( ( F ` z ) ` n ) <_ 0 ) |
135 |
119 134
|
eqbrtrd |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 0 ) ) -> ( ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` z ) ` n ) <_ 0 ) |
136 |
|
iccleub |
|- ( ( 0 e. RR* /\ 1 e. RR* /\ ( ( F ` z ) ` n ) e. ( 0 [,] 1 ) ) -> ( ( F ` z ) ` n ) <_ 1 ) |
137 |
124 125 136
|
mp3an12 |
|- ( ( ( F ` z ) ` n ) e. ( 0 [,] 1 ) -> ( ( F ` z ) ` n ) <_ 1 ) |
138 |
123 137
|
syl |
|- ( ( ( ph /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` z ) ` n ) <_ 1 ) |
139 |
|
1red |
|- ( ( ( ph /\ z e. I ) /\ n e. ( 1 ... N ) ) -> 1 e. RR ) |
140 |
139 129
|
subge0d |
|- ( ( ( ph /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( 0 <_ ( 1 - ( ( F ` z ) ` n ) ) <-> ( ( F ` z ) ` n ) <_ 1 ) ) |
141 |
138 140
|
mpbird |
|- ( ( ( ph /\ z e. I ) /\ n e. ( 1 ... N ) ) -> 0 <_ ( 1 - ( ( F ` z ) ` n ) ) ) |
142 |
141
|
an32s |
|- ( ( ( ph /\ n e. ( 1 ... N ) ) /\ z e. I ) -> 0 <_ ( 1 - ( ( F ` z ) ` n ) ) ) |
143 |
142
|
anasss |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I ) ) -> 0 <_ ( 1 - ( ( F ` z ) ` n ) ) ) |
144 |
143
|
3adantr3 |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 1 ) ) -> 0 <_ ( 1 - ( ( F ` z ) ` n ) ) ) |
145 |
|
simpr2 |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 1 ) ) -> z e. I ) |
146 |
145 100
|
syl |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 1 ) ) -> ( ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` z ) ` n ) = ( ( z oF - ( F ` z ) ) ` n ) ) |
147 |
103
|
adantl |
|- ( ( ( ph /\ ( z ` n ) = 1 ) /\ z e. I ) -> z Fn ( 1 ... N ) ) |
148 |
108
|
adantlr |
|- ( ( ( ph /\ ( z ` n ) = 1 ) /\ z e. I ) -> ( F ` z ) Fn ( 1 ... N ) ) |
149 |
|
ovexd |
|- ( ( ( ph /\ ( z ` n ) = 1 ) /\ z e. I ) -> ( 1 ... N ) e. _V ) |
150 |
|
simpllr |
|- ( ( ( ( ph /\ ( z ` n ) = 1 ) /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( z ` n ) = 1 ) |
151 |
|
eqidd |
|- ( ( ( ( ph /\ ( z ` n ) = 1 ) /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` z ) ` n ) = ( ( F ` z ) ` n ) ) |
152 |
147 148 149 149 27 150 151
|
ofval |
|- ( ( ( ( ph /\ ( z ` n ) = 1 ) /\ z e. I ) /\ n e. ( 1 ... N ) ) -> ( ( z oF - ( F ` z ) ) ` n ) = ( 1 - ( ( F ` z ) ` n ) ) ) |
153 |
152
|
exp41 |
|- ( ph -> ( ( z ` n ) = 1 -> ( z e. I -> ( n e. ( 1 ... N ) -> ( ( z oF - ( F ` z ) ) ` n ) = ( 1 - ( ( F ` z ) ` n ) ) ) ) ) ) |
154 |
153
|
com24 |
|- ( ph -> ( n e. ( 1 ... N ) -> ( z e. I -> ( ( z ` n ) = 1 -> ( ( z oF - ( F ` z ) ) ` n ) = ( 1 - ( ( F ` z ) ` n ) ) ) ) ) ) |
155 |
154
|
3imp2 |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 1 ) ) -> ( ( z oF - ( F ` z ) ) ` n ) = ( 1 - ( ( F ` z ) ` n ) ) ) |
156 |
146 155
|
eqtrd |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 1 ) ) -> ( ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` z ) ` n ) = ( 1 - ( ( F ` z ) ` n ) ) ) |
157 |
144 156
|
breqtrrd |
|- ( ( ph /\ ( n e. ( 1 ... N ) /\ z e. I /\ ( z ` n ) = 1 ) ) -> 0 <_ ( ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` z ) ` n ) ) |
158 |
1 2 3 92 135 157
|
poimir |
|- ( ph -> E. c e. I ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` c ) = ( ( 1 ... N ) X. { 0 } ) ) |
159 |
|
id |
|- ( x = c -> x = c ) |
160 |
|
fveq2 |
|- ( x = c -> ( F ` x ) = ( F ` c ) ) |
161 |
159 160
|
oveq12d |
|- ( x = c -> ( x oF - ( F ` x ) ) = ( c oF - ( F ` c ) ) ) |
162 |
|
ovex |
|- ( c oF - ( F ` c ) ) e. _V |
163 |
161 97 162
|
fvmpt |
|- ( c e. I -> ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` c ) = ( c oF - ( F ` c ) ) ) |
164 |
163
|
adantl |
|- ( ( ph /\ c e. I ) -> ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` c ) = ( c oF - ( F ` c ) ) ) |
165 |
164
|
eqeq1d |
|- ( ( ph /\ c e. I ) -> ( ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` c ) = ( ( 1 ... N ) X. { 0 } ) <-> ( c oF - ( F ` c ) ) = ( ( 1 ... N ) X. { 0 } ) ) ) |
166 |
|
elmapfn |
|- ( c e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> c Fn ( 1 ... N ) ) |
167 |
166 2
|
eleq2s |
|- ( c e. I -> c Fn ( 1 ... N ) ) |
168 |
167
|
adantl |
|- ( ( ph /\ c e. I ) -> c Fn ( 1 ... N ) ) |
169 |
21
|
ffvelrnda |
|- ( ( ph /\ c e. I ) -> ( F ` c ) e. I ) |
170 |
|
elmapfn |
|- ( ( F ` c ) e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> ( F ` c ) Fn ( 1 ... N ) ) |
171 |
170 2
|
eleq2s |
|- ( ( F ` c ) e. I -> ( F ` c ) Fn ( 1 ... N ) ) |
172 |
169 171
|
syl |
|- ( ( ph /\ c e. I ) -> ( F ` c ) Fn ( 1 ... N ) ) |
173 |
|
ovexd |
|- ( ( ph /\ c e. I ) -> ( 1 ... N ) e. _V ) |
174 |
|
eqidd |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( c ` n ) = ( c ` n ) ) |
175 |
|
eqidd |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` c ) ` n ) = ( ( F ` c ) ` n ) ) |
176 |
168 172 173 173 27 174 175
|
ofval |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( ( c oF - ( F ` c ) ) ` n ) = ( ( c ` n ) - ( ( F ` c ) ` n ) ) ) |
177 |
|
c0ex |
|- 0 e. _V |
178 |
177
|
fvconst2 |
|- ( n e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { 0 } ) ` n ) = 0 ) |
179 |
178
|
adantl |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( ( ( 1 ... N ) X. { 0 } ) ` n ) = 0 ) |
180 |
176 179
|
eqeq12d |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( ( ( c oF - ( F ` c ) ) ` n ) = ( ( ( 1 ... N ) X. { 0 } ) ` n ) <-> ( ( c ` n ) - ( ( F ` c ) ` n ) ) = 0 ) ) |
181 |
13 84
|
sstri |
|- ( 0 [,] 1 ) C_ CC |
182 |
|
elmapi |
|- ( c e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> c : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
183 |
182 2
|
eleq2s |
|- ( c e. I -> c : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
184 |
183
|
ffvelrnda |
|- ( ( c e. I /\ n e. ( 1 ... N ) ) -> ( c ` n ) e. ( 0 [,] 1 ) ) |
185 |
181 184
|
sselid |
|- ( ( c e. I /\ n e. ( 1 ... N ) ) -> ( c ` n ) e. CC ) |
186 |
185
|
adantll |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( c ` n ) e. CC ) |
187 |
|
elmapi |
|- ( ( F ` c ) e. ( ( 0 [,] 1 ) ^m ( 1 ... N ) ) -> ( F ` c ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
188 |
187 2
|
eleq2s |
|- ( ( F ` c ) e. I -> ( F ` c ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
189 |
169 188
|
syl |
|- ( ( ph /\ c e. I ) -> ( F ` c ) : ( 1 ... N ) --> ( 0 [,] 1 ) ) |
190 |
189
|
ffvelrnda |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` c ) ` n ) e. ( 0 [,] 1 ) ) |
191 |
181 190
|
sselid |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( ( F ` c ) ` n ) e. CC ) |
192 |
186 191
|
subeq0ad |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( ( ( c ` n ) - ( ( F ` c ) ` n ) ) = 0 <-> ( c ` n ) = ( ( F ` c ) ` n ) ) ) |
193 |
180 192
|
bitrd |
|- ( ( ( ph /\ c e. I ) /\ n e. ( 1 ... N ) ) -> ( ( ( c oF - ( F ` c ) ) ` n ) = ( ( ( 1 ... N ) X. { 0 } ) ` n ) <-> ( c ` n ) = ( ( F ` c ) ` n ) ) ) |
194 |
193
|
ralbidva |
|- ( ( ph /\ c e. I ) -> ( A. n e. ( 1 ... N ) ( ( c oF - ( F ` c ) ) ` n ) = ( ( ( 1 ... N ) X. { 0 } ) ` n ) <-> A. n e. ( 1 ... N ) ( c ` n ) = ( ( F ` c ) ` n ) ) ) |
195 |
168 172 173 173 27
|
offn |
|- ( ( ph /\ c e. I ) -> ( c oF - ( F ` c ) ) Fn ( 1 ... N ) ) |
196 |
|
fnconstg |
|- ( 0 e. _V -> ( ( 1 ... N ) X. { 0 } ) Fn ( 1 ... N ) ) |
197 |
177 196
|
ax-mp |
|- ( ( 1 ... N ) X. { 0 } ) Fn ( 1 ... N ) |
198 |
|
eqfnfv |
|- ( ( ( c oF - ( F ` c ) ) Fn ( 1 ... N ) /\ ( ( 1 ... N ) X. { 0 } ) Fn ( 1 ... N ) ) -> ( ( c oF - ( F ` c ) ) = ( ( 1 ... N ) X. { 0 } ) <-> A. n e. ( 1 ... N ) ( ( c oF - ( F ` c ) ) ` n ) = ( ( ( 1 ... N ) X. { 0 } ) ` n ) ) ) |
199 |
195 197 198
|
sylancl |
|- ( ( ph /\ c e. I ) -> ( ( c oF - ( F ` c ) ) = ( ( 1 ... N ) X. { 0 } ) <-> A. n e. ( 1 ... N ) ( ( c oF - ( F ` c ) ) ` n ) = ( ( ( 1 ... N ) X. { 0 } ) ` n ) ) ) |
200 |
|
eqfnfv |
|- ( ( c Fn ( 1 ... N ) /\ ( F ` c ) Fn ( 1 ... N ) ) -> ( c = ( F ` c ) <-> A. n e. ( 1 ... N ) ( c ` n ) = ( ( F ` c ) ` n ) ) ) |
201 |
168 172 200
|
syl2anc |
|- ( ( ph /\ c e. I ) -> ( c = ( F ` c ) <-> A. n e. ( 1 ... N ) ( c ` n ) = ( ( F ` c ) ` n ) ) ) |
202 |
194 199 201
|
3bitr4d |
|- ( ( ph /\ c e. I ) -> ( ( c oF - ( F ` c ) ) = ( ( 1 ... N ) X. { 0 } ) <-> c = ( F ` c ) ) ) |
203 |
165 202
|
bitrd |
|- ( ( ph /\ c e. I ) -> ( ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` c ) = ( ( 1 ... N ) X. { 0 } ) <-> c = ( F ` c ) ) ) |
204 |
203
|
rexbidva |
|- ( ph -> ( E. c e. I ( ( x e. I |-> ( x oF - ( F ` x ) ) ) ` c ) = ( ( 1 ... N ) X. { 0 } ) <-> E. c e. I c = ( F ` c ) ) ) |
205 |
158 204
|
mpbid |
|- ( ph -> E. c e. I c = ( F ` c ) ) |