Step |
Hyp |
Ref |
Expression |
1 |
|
reex |
|- RR e. _V |
2 |
|
elssuni |
|- ( A e. ( topGen ` ran (,) ) -> A C_ U. ( topGen ` ran (,) ) ) |
3 |
|
uniretop |
|- RR = U. ( topGen ` ran (,) ) |
4 |
2 3
|
sseqtrrdi |
|- ( A e. ( topGen ` ran (,) ) -> A C_ RR ) |
5 |
|
ssdomg |
|- ( RR e. _V -> ( A C_ RR -> A ~<_ RR ) ) |
6 |
1 4 5
|
mpsyl |
|- ( A e. ( topGen ` ran (,) ) -> A ~<_ RR ) |
7 |
|
rpnnen |
|- RR ~~ ~P NN |
8 |
|
domentr |
|- ( ( A ~<_ RR /\ RR ~~ ~P NN ) -> A ~<_ ~P NN ) |
9 |
6 7 8
|
sylancl |
|- ( A e. ( topGen ` ran (,) ) -> A ~<_ ~P NN ) |
10 |
|
n0 |
|- ( A =/= (/) <-> E. x x e. A ) |
11 |
4
|
sselda |
|- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> x e. RR ) |
12 |
|
rpnnen2 |
|- ~P NN ~<_ ( 0 [,] 1 ) |
13 |
|
rphalfcl |
|- ( y e. RR+ -> ( y / 2 ) e. RR+ ) |
14 |
13
|
rpred |
|- ( y e. RR+ -> ( y / 2 ) e. RR ) |
15 |
|
resubcl |
|- ( ( x e. RR /\ ( y / 2 ) e. RR ) -> ( x - ( y / 2 ) ) e. RR ) |
16 |
14 15
|
sylan2 |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x - ( y / 2 ) ) e. RR ) |
17 |
|
readdcl |
|- ( ( x e. RR /\ ( y / 2 ) e. RR ) -> ( x + ( y / 2 ) ) e. RR ) |
18 |
14 17
|
sylan2 |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x + ( y / 2 ) ) e. RR ) |
19 |
|
simpl |
|- ( ( x e. RR /\ y e. RR+ ) -> x e. RR ) |
20 |
|
ltsubrp |
|- ( ( x e. RR /\ ( y / 2 ) e. RR+ ) -> ( x - ( y / 2 ) ) < x ) |
21 |
13 20
|
sylan2 |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x - ( y / 2 ) ) < x ) |
22 |
|
ltaddrp |
|- ( ( x e. RR /\ ( y / 2 ) e. RR+ ) -> x < ( x + ( y / 2 ) ) ) |
23 |
13 22
|
sylan2 |
|- ( ( x e. RR /\ y e. RR+ ) -> x < ( x + ( y / 2 ) ) ) |
24 |
16 19 18 21 23
|
lttrd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x - ( y / 2 ) ) < ( x + ( y / 2 ) ) ) |
25 |
|
iccen |
|- ( ( ( x - ( y / 2 ) ) e. RR /\ ( x + ( y / 2 ) ) e. RR /\ ( x - ( y / 2 ) ) < ( x + ( y / 2 ) ) ) -> ( 0 [,] 1 ) ~~ ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) ) |
26 |
16 18 24 25
|
syl3anc |
|- ( ( x e. RR /\ y e. RR+ ) -> ( 0 [,] 1 ) ~~ ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) ) |
27 |
|
domentr |
|- ( ( ~P NN ~<_ ( 0 [,] 1 ) /\ ( 0 [,] 1 ) ~~ ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) ) -> ~P NN ~<_ ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) ) |
28 |
12 26 27
|
sylancr |
|- ( ( x e. RR /\ y e. RR+ ) -> ~P NN ~<_ ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) ) |
29 |
|
ovex |
|- ( ( x - y ) (,) ( x + y ) ) e. _V |
30 |
|
rpre |
|- ( y e. RR+ -> y e. RR ) |
31 |
|
resubcl |
|- ( ( x e. RR /\ y e. RR ) -> ( x - y ) e. RR ) |
32 |
30 31
|
sylan2 |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x - y ) e. RR ) |
33 |
32
|
rexrd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x - y ) e. RR* ) |
34 |
|
readdcl |
|- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR ) |
35 |
30 34
|
sylan2 |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x + y ) e. RR ) |
36 |
35
|
rexrd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x + y ) e. RR* ) |
37 |
19
|
recnd |
|- ( ( x e. RR /\ y e. RR+ ) -> x e. CC ) |
38 |
14
|
adantl |
|- ( ( x e. RR /\ y e. RR+ ) -> ( y / 2 ) e. RR ) |
39 |
38
|
recnd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( y / 2 ) e. CC ) |
40 |
37 39 39
|
subsub4d |
|- ( ( x e. RR /\ y e. RR+ ) -> ( ( x - ( y / 2 ) ) - ( y / 2 ) ) = ( x - ( ( y / 2 ) + ( y / 2 ) ) ) ) |
41 |
30
|
adantl |
|- ( ( x e. RR /\ y e. RR+ ) -> y e. RR ) |
42 |
41
|
recnd |
|- ( ( x e. RR /\ y e. RR+ ) -> y e. CC ) |
43 |
42
|
2halvesd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( ( y / 2 ) + ( y / 2 ) ) = y ) |
44 |
43
|
oveq2d |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x - ( ( y / 2 ) + ( y / 2 ) ) ) = ( x - y ) ) |
45 |
40 44
|
eqtrd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( ( x - ( y / 2 ) ) - ( y / 2 ) ) = ( x - y ) ) |
46 |
13
|
adantl |
|- ( ( x e. RR /\ y e. RR+ ) -> ( y / 2 ) e. RR+ ) |
47 |
16 46
|
ltsubrpd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( ( x - ( y / 2 ) ) - ( y / 2 ) ) < ( x - ( y / 2 ) ) ) |
48 |
45 47
|
eqbrtrrd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x - y ) < ( x - ( y / 2 ) ) ) |
49 |
18 46
|
ltaddrpd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x + ( y / 2 ) ) < ( ( x + ( y / 2 ) ) + ( y / 2 ) ) ) |
50 |
37 39 39
|
addassd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( ( x + ( y / 2 ) ) + ( y / 2 ) ) = ( x + ( ( y / 2 ) + ( y / 2 ) ) ) ) |
51 |
43
|
oveq2d |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x + ( ( y / 2 ) + ( y / 2 ) ) ) = ( x + y ) ) |
52 |
50 51
|
eqtrd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( ( x + ( y / 2 ) ) + ( y / 2 ) ) = ( x + y ) ) |
53 |
49 52
|
breqtrd |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x + ( y / 2 ) ) < ( x + y ) ) |
54 |
|
iccssioo |
|- ( ( ( ( x - y ) e. RR* /\ ( x + y ) e. RR* ) /\ ( ( x - y ) < ( x - ( y / 2 ) ) /\ ( x + ( y / 2 ) ) < ( x + y ) ) ) -> ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) C_ ( ( x - y ) (,) ( x + y ) ) ) |
55 |
33 36 48 53 54
|
syl22anc |
|- ( ( x e. RR /\ y e. RR+ ) -> ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) C_ ( ( x - y ) (,) ( x + y ) ) ) |
56 |
|
ssdomg |
|- ( ( ( x - y ) (,) ( x + y ) ) e. _V -> ( ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) C_ ( ( x - y ) (,) ( x + y ) ) -> ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) ~<_ ( ( x - y ) (,) ( x + y ) ) ) ) |
57 |
29 55 56
|
mpsyl |
|- ( ( x e. RR /\ y e. RR+ ) -> ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) ~<_ ( ( x - y ) (,) ( x + y ) ) ) |
58 |
|
domtr |
|- ( ( ~P NN ~<_ ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) /\ ( ( x - ( y / 2 ) ) [,] ( x + ( y / 2 ) ) ) ~<_ ( ( x - y ) (,) ( x + y ) ) ) -> ~P NN ~<_ ( ( x - y ) (,) ( x + y ) ) ) |
59 |
28 57 58
|
syl2anc |
|- ( ( x e. RR /\ y e. RR+ ) -> ~P NN ~<_ ( ( x - y ) (,) ( x + y ) ) ) |
60 |
|
eqid |
|- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) ) |
61 |
60
|
bl2ioo |
|- ( ( x e. RR /\ y e. RR ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) = ( ( x - y ) (,) ( x + y ) ) ) |
62 |
30 61
|
sylan2 |
|- ( ( x e. RR /\ y e. RR+ ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) = ( ( x - y ) (,) ( x + y ) ) ) |
63 |
59 62
|
breqtrrd |
|- ( ( x e. RR /\ y e. RR+ ) -> ~P NN ~<_ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) ) |
64 |
11 63
|
sylan |
|- ( ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) /\ y e. RR+ ) -> ~P NN ~<_ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) ) |
65 |
|
simplll |
|- ( ( ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) /\ y e. RR+ ) /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) C_ A ) -> A e. ( topGen ` ran (,) ) ) |
66 |
|
simpr |
|- ( ( ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) /\ y e. RR+ ) /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) C_ A ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) C_ A ) |
67 |
|
ssdomg |
|- ( A e. ( topGen ` ran (,) ) -> ( ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) C_ A -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) ~<_ A ) ) |
68 |
65 66 67
|
sylc |
|- ( ( ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) /\ y e. RR+ ) /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) C_ A ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) ~<_ A ) |
69 |
|
domtr |
|- ( ( ~P NN ~<_ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) ~<_ A ) -> ~P NN ~<_ A ) |
70 |
64 68 69
|
syl2an2r |
|- ( ( ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) /\ y e. RR+ ) /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) C_ A ) -> ~P NN ~<_ A ) |
71 |
|
eqid |
|- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) |
72 |
60 71
|
tgioo |
|- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) |
73 |
72
|
eleq2i |
|- ( A e. ( topGen ` ran (,) ) <-> A e. ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) ) |
74 |
60
|
rexmet |
|- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) |
75 |
71
|
mopni2 |
|- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ A e. ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) /\ x e. A ) -> E. y e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) C_ A ) |
76 |
74 75
|
mp3an1 |
|- ( ( A e. ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) /\ x e. A ) -> E. y e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) C_ A ) |
77 |
73 76
|
sylanb |
|- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> E. y e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) y ) C_ A ) |
78 |
70 77
|
r19.29a |
|- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> ~P NN ~<_ A ) |
79 |
78
|
ex |
|- ( A e. ( topGen ` ran (,) ) -> ( x e. A -> ~P NN ~<_ A ) ) |
80 |
79
|
exlimdv |
|- ( A e. ( topGen ` ran (,) ) -> ( E. x x e. A -> ~P NN ~<_ A ) ) |
81 |
10 80
|
syl5bi |
|- ( A e. ( topGen ` ran (,) ) -> ( A =/= (/) -> ~P NN ~<_ A ) ) |
82 |
81
|
imp |
|- ( ( A e. ( topGen ` ran (,) ) /\ A =/= (/) ) -> ~P NN ~<_ A ) |
83 |
|
sbth |
|- ( ( A ~<_ ~P NN /\ ~P NN ~<_ A ) -> A ~~ ~P NN ) |
84 |
9 82 83
|
syl2an2r |
|- ( ( A e. ( topGen ` ran (,) ) /\ A =/= (/) ) -> A ~~ ~P NN ) |