Step |
Hyp |
Ref |
Expression |
1 |
|
metust.1 |
|- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) |
2 |
|
relres |
|- Rel ( _I |` X ) |
3 |
2
|
a1i |
|- ( ( D e. ( PsMet ` X ) /\ A e. F ) -> Rel ( _I |` X ) ) |
4 |
|
vex |
|- q e. _V |
5 |
4
|
brresi |
|- ( p ( _I |` X ) q <-> ( p e. X /\ p _I q ) ) |
6 |
|
df-br |
|- ( p ( _I |` X ) q <-> <. p , q >. e. ( _I |` X ) ) |
7 |
4
|
ideq |
|- ( p _I q <-> p = q ) |
8 |
7
|
anbi2i |
|- ( ( p e. X /\ p _I q ) <-> ( p e. X /\ p = q ) ) |
9 |
5 6 8
|
3bitr3i |
|- ( <. p , q >. e. ( _I |` X ) <-> ( p e. X /\ p = q ) ) |
10 |
9
|
biimpi |
|- ( <. p , q >. e. ( _I |` X ) -> ( p e. X /\ p = q ) ) |
11 |
10
|
ad2antlr |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p e. X /\ p = q ) ) |
12 |
11
|
simprd |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> p = q ) |
13 |
|
df-ov |
|- ( p D p ) = ( D ` <. p , p >. ) |
14 |
|
opeq2 |
|- ( p = q -> <. p , p >. = <. p , q >. ) |
15 |
14
|
fveq2d |
|- ( p = q -> ( D ` <. p , p >. ) = ( D ` <. p , q >. ) ) |
16 |
13 15
|
eqtrid |
|- ( p = q -> ( p D p ) = ( D ` <. p , q >. ) ) |
17 |
12 16
|
syl |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p D p ) = ( D ` <. p , q >. ) ) |
18 |
|
simplll |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> D e. ( PsMet ` X ) ) |
19 |
11
|
simpld |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> p e. X ) |
20 |
|
psmet0 |
|- ( ( D e. ( PsMet ` X ) /\ p e. X ) -> ( p D p ) = 0 ) |
21 |
18 19 20
|
syl2anc |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p D p ) = 0 ) |
22 |
17 21
|
eqtr3d |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( D ` <. p , q >. ) = 0 ) |
23 |
|
0xr |
|- 0 e. RR* |
24 |
|
rpxr |
|- ( a e. RR+ -> a e. RR* ) |
25 |
|
rpgt0 |
|- ( a e. RR+ -> 0 < a ) |
26 |
|
lbico1 |
|- ( ( 0 e. RR* /\ a e. RR* /\ 0 < a ) -> 0 e. ( 0 [,) a ) ) |
27 |
23 24 25 26
|
mp3an2i |
|- ( a e. RR+ -> 0 e. ( 0 [,) a ) ) |
28 |
27
|
adantl |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> 0 e. ( 0 [,) a ) ) |
29 |
22 28
|
eqeltrd |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( D ` <. p , q >. ) e. ( 0 [,) a ) ) |
30 |
|
psmetf |
|- ( D e. ( PsMet ` X ) -> D : ( X X. X ) --> RR* ) |
31 |
30
|
ffund |
|- ( D e. ( PsMet ` X ) -> Fun D ) |
32 |
31
|
ad3antrrr |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> Fun D ) |
33 |
12 19
|
eqeltrrd |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> q e. X ) |
34 |
19 33
|
opelxpd |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. ( X X. X ) ) |
35 |
30
|
fdmd |
|- ( D e. ( PsMet ` X ) -> dom D = ( X X. X ) ) |
36 |
35
|
ad3antrrr |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> dom D = ( X X. X ) ) |
37 |
34 36
|
eleqtrrd |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. dom D ) |
38 |
|
fvimacnv |
|- ( ( Fun D /\ <. p , q >. e. dom D ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) ) |
39 |
32 37 38
|
syl2anc |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) ) |
40 |
29 39
|
mpbid |
|- ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) |
41 |
40
|
adantr |
|- ( ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) |
42 |
|
simpr |
|- ( ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> A = ( `' D " ( 0 [,) a ) ) ) |
43 |
41 42
|
eleqtrrd |
|- ( ( ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. p , q >. e. A ) |
44 |
|
simplr |
|- ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> A e. F ) |
45 |
1
|
metustel |
|- ( D e. ( PsMet ` X ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) ) |
46 |
45
|
ad2antrr |
|- ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) ) |
47 |
44 46
|
mpbid |
|- ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) |
48 |
43 47
|
r19.29a |
|- ( ( ( D e. ( PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> <. p , q >. e. A ) |
49 |
48
|
ex |
|- ( ( D e. ( PsMet ` X ) /\ A e. F ) -> ( <. p , q >. e. ( _I |` X ) -> <. p , q >. e. A ) ) |
50 |
3 49
|
relssdv |
|- ( ( D e. ( PsMet ` X ) /\ A e. F ) -> ( _I |` X ) C_ A ) |