Step |
Hyp |
Ref |
Expression |
1 |
|
metust.1 |
|- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) |
2 |
1
|
metustel |
|- ( D e. ( PsMet ` X ) -> ( x e. F <-> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) ) |
3 |
|
simpr |
|- ( ( D e. ( PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x = ( `' D " ( 0 [,) a ) ) ) |
4 |
|
cnvimass |
|- ( `' D " ( 0 [,) a ) ) C_ dom D |
5 |
|
psmetf |
|- ( D e. ( PsMet ` X ) -> D : ( X X. X ) --> RR* ) |
6 |
5
|
fdmd |
|- ( D e. ( PsMet ` X ) -> dom D = ( X X. X ) ) |
7 |
6
|
adantr |
|- ( ( D e. ( PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> dom D = ( X X. X ) ) |
8 |
4 7
|
sseqtrid |
|- ( ( D e. ( PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> ( `' D " ( 0 [,) a ) ) C_ ( X X. X ) ) |
9 |
3 8
|
eqsstrd |
|- ( ( D e. ( PsMet ` X ) /\ x = ( `' D " ( 0 [,) a ) ) ) -> x C_ ( X X. X ) ) |
10 |
9
|
ex |
|- ( D e. ( PsMet ` X ) -> ( x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) ) |
11 |
10
|
rexlimdvw |
|- ( D e. ( PsMet ` X ) -> ( E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) -> x C_ ( X X. X ) ) ) |
12 |
2 11
|
sylbid |
|- ( D e. ( PsMet ` X ) -> ( x e. F -> x C_ ( X X. X ) ) ) |
13 |
12
|
ralrimiv |
|- ( D e. ( PsMet ` X ) -> A. x e. F x C_ ( X X. X ) ) |
14 |
|
pwssb |
|- ( F C_ ~P ( X X. X ) <-> A. x e. F x C_ ( X X. X ) ) |
15 |
13 14
|
sylibr |
|- ( D e. ( PsMet ` X ) -> F C_ ~P ( X X. X ) ) |
16 |
15
|
adantl |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> F C_ ~P ( X X. X ) ) |
17 |
|
cnvexg |
|- ( D e. ( PsMet ` X ) -> `' D e. _V ) |
18 |
|
imaexg |
|- ( `' D e. _V -> ( `' D " ( 0 [,) 1 ) ) e. _V ) |
19 |
|
elisset |
|- ( ( `' D " ( 0 [,) 1 ) ) e. _V -> E. x x = ( `' D " ( 0 [,) 1 ) ) ) |
20 |
|
1rp |
|- 1 e. RR+ |
21 |
|
oveq2 |
|- ( a = 1 -> ( 0 [,) a ) = ( 0 [,) 1 ) ) |
22 |
21
|
imaeq2d |
|- ( a = 1 -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) 1 ) ) ) |
23 |
22
|
rspceeqv |
|- ( ( 1 e. RR+ /\ x = ( `' D " ( 0 [,) 1 ) ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) |
24 |
20 23
|
mpan |
|- ( x = ( `' D " ( 0 [,) 1 ) ) -> E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) |
25 |
24
|
eximi |
|- ( E. x x = ( `' D " ( 0 [,) 1 ) ) -> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) |
26 |
17 18 19 25
|
4syl |
|- ( D e. ( PsMet ` X ) -> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) |
27 |
2
|
exbidv |
|- ( D e. ( PsMet ` X ) -> ( E. x x e. F <-> E. x E. a e. RR+ x = ( `' D " ( 0 [,) a ) ) ) ) |
28 |
26 27
|
mpbird |
|- ( D e. ( PsMet ` X ) -> E. x x e. F ) |
29 |
28
|
adantl |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> E. x x e. F ) |
30 |
|
n0 |
|- ( F =/= (/) <-> E. x x e. F ) |
31 |
29 30
|
sylibr |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> F =/= (/) ) |
32 |
1
|
metustid |
|- ( ( D e. ( PsMet ` X ) /\ x e. F ) -> ( _I |` X ) C_ x ) |
33 |
32
|
adantll |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) C_ x ) |
34 |
|
n0 |
|- ( X =/= (/) <-> E. p p e. X ) |
35 |
34
|
biimpi |
|- ( X =/= (/) -> E. p p e. X ) |
36 |
35
|
adantr |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> E. p p e. X ) |
37 |
|
opelidres |
|- ( p e. X -> ( <. p , p >. e. ( _I |` X ) <-> p e. X ) ) |
38 |
37
|
ibir |
|- ( p e. X -> <. p , p >. e. ( _I |` X ) ) |
39 |
38
|
ne0d |
|- ( p e. X -> ( _I |` X ) =/= (/) ) |
40 |
39
|
exlimiv |
|- ( E. p p e. X -> ( _I |` X ) =/= (/) ) |
41 |
36 40
|
syl |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( _I |` X ) =/= (/) ) |
42 |
41
|
adantr |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ x e. F ) -> ( _I |` X ) =/= (/) ) |
43 |
|
ssn0 |
|- ( ( ( _I |` X ) C_ x /\ ( _I |` X ) =/= (/) ) -> x =/= (/) ) |
44 |
33 42 43
|
syl2anc |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ x e. F ) -> x =/= (/) ) |
45 |
44
|
nelrdva |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> -. (/) e. F ) |
46 |
|
df-nel |
|- ( (/) e/ F <-> -. (/) e. F ) |
47 |
45 46
|
sylibr |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> (/) e/ F ) |
48 |
|
df-ss |
|- ( x C_ y <-> ( x i^i y ) = x ) |
49 |
48
|
biimpi |
|- ( x C_ y -> ( x i^i y ) = x ) |
50 |
49
|
adantl |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) = x ) |
51 |
|
simplrl |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> x e. F ) |
52 |
50 51
|
eqeltrd |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ x C_ y ) -> ( x i^i y ) e. F ) |
53 |
|
sseqin2 |
|- ( y C_ x <-> ( x i^i y ) = y ) |
54 |
53
|
biimpi |
|- ( y C_ x -> ( x i^i y ) = y ) |
55 |
54
|
adantl |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) = y ) |
56 |
|
simplrr |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> y e. F ) |
57 |
55 56
|
eqeltrd |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) /\ y C_ x ) -> ( x i^i y ) e. F ) |
58 |
|
simplr |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> D e. ( PsMet ` X ) ) |
59 |
|
simprl |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> x e. F ) |
60 |
|
simprr |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> y e. F ) |
61 |
1
|
metustto |
|- ( ( D e. ( PsMet ` X ) /\ x e. F /\ y e. F ) -> ( x C_ y \/ y C_ x ) ) |
62 |
58 59 60 61
|
syl3anc |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x C_ y \/ y C_ x ) ) |
63 |
52 57 62
|
mpjaodan |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) e. F ) |
64 |
|
ssidd |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> ( x i^i y ) C_ ( x i^i y ) ) |
65 |
|
sseq1 |
|- ( z = ( x i^i y ) -> ( z C_ ( x i^i y ) <-> ( x i^i y ) C_ ( x i^i y ) ) ) |
66 |
65
|
rspcev |
|- ( ( ( x i^i y ) e. F /\ ( x i^i y ) C_ ( x i^i y ) ) -> E. z e. F z C_ ( x i^i y ) ) |
67 |
63 64 66
|
syl2anc |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( x e. F /\ y e. F ) ) -> E. z e. F z C_ ( x i^i y ) ) |
68 |
67
|
ralrimivva |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> A. x e. F A. y e. F E. z e. F z C_ ( x i^i y ) ) |
69 |
31 47 68
|
3jca |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F E. z e. F z C_ ( x i^i y ) ) ) |
70 |
|
elfvex |
|- ( D e. ( PsMet ` X ) -> X e. _V ) |
71 |
70
|
adantl |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> X e. _V ) |
72 |
71 71
|
xpexd |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( X X. X ) e. _V ) |
73 |
|
isfbas2 |
|- ( ( X X. X ) e. _V -> ( F e. ( fBas ` ( X X. X ) ) <-> ( F C_ ~P ( X X. X ) /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F E. z e. F z C_ ( x i^i y ) ) ) ) ) |
74 |
72 73
|
syl |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( F e. ( fBas ` ( X X. X ) ) <-> ( F C_ ~P ( X X. X ) /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F E. z e. F z C_ ( x i^i y ) ) ) ) ) |
75 |
16 69 74
|
mpbir2and |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> F e. ( fBas ` ( X X. X ) ) ) |