Metamath Proof Explorer

Theorem metustbl

Description: The "section" image of an entourage at a point P always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017)

Ref Expression
Assertion metustbl 
|- ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> E. a e. ran ( ball ` D ) ( P e. a /\ a C_ ( V " { P } ) ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> D e. ( PsMet ` X ) )
2 simp3 
 |-  ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> P e. X )
3 simpr 
 |-  ( ( ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) /\ w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) ) /\ w C_ V ) -> w C_ V )
4 eqid 
 |-  ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) = ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) )
5 4 elrnmpt 
 |-  ( w e. _V -> ( w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) <-> E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) ) )
6 5 elv 
 |-  ( w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) <-> E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) )
7 6 biimpi 
 |-  ( w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) -> E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) )
8 7 ad2antlr 
 |-  ( ( ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) /\ w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) ) /\ w C_ V ) -> E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) )
9 sseq1 
 |-  ( w = ( `' D " ( 0 [,) r ) ) -> ( w C_ V <-> ( `' D " ( 0 [,) r ) ) C_ V ) )
10 9 biimpcd 
 |-  ( w C_ V -> ( w = ( `' D " ( 0 [,) r ) ) -> ( `' D " ( 0 [,) r ) ) C_ V ) )
11 10 reximdv 
 |-  ( w C_ V -> ( E. r e. RR+ w = ( `' D " ( 0 [,) r ) ) -> E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V ) )
12 3 8 11 sylc 
 |-  ( ( ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) /\ w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) ) /\ w C_ V ) -> E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V )
13 2 ne0d 
 |-  ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> X =/= (/) )
14 simp2 
 |-  ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> V e. ( metUnif ` D ) )
15 metuel 
 |-  ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( V e. ( metUnif ` D ) <-> ( V C_ ( X X. X ) /\ E. w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) w C_ V ) ) )
16 15 simplbda 
 |-  ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ V e. ( metUnif ` D ) ) -> E. w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) w C_ V )
17 13 1 14 16 syl21anc 
 |-  ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> E. w e. ran ( r e. RR+ |-> ( `' D " ( 0 [,) r ) ) ) w C_ V )
18 12 17 r19.29a 
 |-  ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V )
19 imass1 
 |-  ( ( `' D " ( 0 [,) r ) ) C_ V -> ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) )
20 19 reximi 
 |-  ( E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V -> E. r e. RR+ ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) )
21 blval2 
 |-  ( ( D e. ( PsMet ` X ) /\ P e. X /\ r e. RR+ ) -> ( P ( ball ` D ) r ) = ( ( `' D " ( 0 [,) r ) ) " { P } ) )
22 21 sseq1d 
 |-  ( ( D e. ( PsMet ` X ) /\ P e. X /\ r e. RR+ ) -> ( ( P ( ball ` D ) r ) C_ ( V " { P } ) <-> ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) ) )
23 22 3expa 
 |-  ( ( ( D e. ( PsMet ` X ) /\ P e. X ) /\ r e. RR+ ) -> ( ( P ( ball ` D ) r ) C_ ( V " { P } ) <-> ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) ) )
24 23 rexbidva 
 |-  ( ( D e. ( PsMet ` X ) /\ P e. X ) -> ( E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) <-> E. r e. RR+ ( ( `' D " ( 0 [,) r ) ) " { P } ) C_ ( V " { P } ) ) )
25 20 24 syl5ibr 
 |-  ( ( D e. ( PsMet ` X ) /\ P e. X ) -> ( E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V -> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) ) )
26 25 imp 
 |-  ( ( ( D e. ( PsMet ` X ) /\ P e. X ) /\ E. r e. RR+ ( `' D " ( 0 [,) r ) ) C_ V ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) )
27 1 2 18 26 syl21anc 
 |-  ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) )
28 blssexps 
 |-  ( ( D e. ( PsMet ` X ) /\ P e. X ) -> ( E. a e. ran ( ball ` D ) ( P e. a /\ a C_ ( V " { P } ) ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) ) )
29 28 3adant2 
 |-  ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> ( E. a e. ran ( ball ` D ) ( P e. a /\ a C_ ( V " { P } ) ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ ( V " { P } ) ) )
30 27 29 mpbird 
 |-  ( ( D e. ( PsMet ` X ) /\ V e. ( metUnif ` D ) /\ P e. X ) -> E. a e. ran ( ball ` D ) ( P e. a /\ a C_ ( V " { P } ) ) )