Metamath Proof Explorer

Theorem isbnd2

Description: The predicate "is a bounded metric space". Uses a single point instead of an arbitrary point in the space. (Contributed by Jeff Madsen, 2-Sep-2009)

Ref Expression
Assertion isbnd2 
|- ( ( M e. ( Bnd ` X ) /\ X =/= (/) ) <-> ( M e. ( *Met ` X ) /\ E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )

Proof

Step Hyp Ref Expression
1 isbndx 
 |-  ( M e. ( Bnd ` X ) <-> ( M e. ( *Met ` X ) /\ A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
2 1 anbi1i 
 |-  ( ( M e. ( Bnd ` X ) /\ X =/= (/) ) <-> ( ( M e. ( *Met ` X ) /\ A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) /\ X =/= (/) ) )
3 anass 
 |-  ( ( ( M e. ( *Met ` X ) /\ A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) /\ X =/= (/) ) <-> ( M e. ( *Met ` X ) /\ ( A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) /\ X =/= (/) ) ) )
4 r19.2z 
 |-  ( ( X =/= (/) /\ A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) -> E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) )
5 4 ancoms 
 |-  ( ( A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) /\ X =/= (/) ) -> E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) )
6 oveq1 
 |-  ( x = y -> ( x ( ball ` M ) r ) = ( y ( ball ` M ) r ) )
7 6 eqeq2d 
 |-  ( x = y -> ( X = ( x ( ball ` M ) r ) <-> X = ( y ( ball ` M ) r ) ) )
8 oveq2 
 |-  ( r = s -> ( y ( ball ` M ) r ) = ( y ( ball ` M ) s ) )
9 8 eqeq2d 
 |-  ( r = s -> ( X = ( y ( ball ` M ) r ) <-> X = ( y ( ball ` M ) s ) ) )
10 7 9 cbvrex2vw 
 |-  ( E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) <-> E. y e. X E. s e. RR+ X = ( y ( ball ` M ) s ) )
11 2rp 
 |-  2 e. RR+
12 rpmulcl 
 |-  ( ( 2 e. RR+ /\ s e. RR+ ) -> ( 2 x. s ) e. RR+ )
13 11 12 mpan 
 |-  ( s e. RR+ -> ( 2 x. s ) e. RR+ )
14 13 ad2antll 
 |-  ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) -> ( 2 x. s ) e. RR+ )
15 14 ad2antrr 
 |-  ( ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) /\ X = ( y ( ball ` M ) s ) ) -> ( 2 x. s ) e. RR+ )
16 rpcn 
 |-  ( s e. RR+ -> s e. CC )
17 2cnd 
 |-  ( s e. RR+ -> 2 e. CC )
18 2ne0 
 |-  2 =/= 0
19 18 a1i 
 |-  ( s e. RR+ -> 2 =/= 0 )
20 divcan3 
 |-  ( ( s e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( ( 2 x. s ) / 2 ) = s )
21 20 eqcomd 
 |-  ( ( s e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> s = ( ( 2 x. s ) / 2 ) )
22 16 17 19 21 syl3anc 
 |-  ( s e. RR+ -> s = ( ( 2 x. s ) / 2 ) )
23 22 oveq2d 
 |-  ( s e. RR+ -> ( y ( ball ` M ) s ) = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) )
24 23 eqeq2d 
 |-  ( s e. RR+ -> ( X = ( y ( ball ` M ) s ) <-> X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) )
25 24 biimpd 
 |-  ( s e. RR+ -> ( X = ( y ( ball ` M ) s ) -> X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) )
26 25 ad2antll 
 |-  ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) -> ( X = ( y ( ball ` M ) s ) -> X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) )
27 26 adantr 
 |-  ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) -> ( X = ( y ( ball ` M ) s ) -> X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) )
28 27 imp 
 |-  ( ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) /\ X = ( y ( ball ` M ) s ) ) -> X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) )
29 simpr 
 |-  ( ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) /\ X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) -> X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) )
30 eleq2 
 |-  ( X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) -> ( x e. X <-> x e. ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) )
31 30 biimpac 
 |-  ( ( x e. X /\ X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) -> x e. ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) )
32 2re 
 |-  2 e. RR
33 rpre 
 |-  ( s e. RR+ -> s e. RR )
34 remulcl 
 |-  ( ( 2 e. RR /\ s e. RR ) -> ( 2 x. s ) e. RR )
35 32 33 34 sylancr 
 |-  ( s e. RR+ -> ( 2 x. s ) e. RR )
36 blhalf 
 |-  ( ( ( M e. ( *Met ` X ) /\ y e. X ) /\ ( ( 2 x. s ) e. RR /\ x e. ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) ) -> ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) C_ ( x ( ball ` M ) ( 2 x. s ) ) )
37 36 expr 
 |-  ( ( ( M e. ( *Met ` X ) /\ y e. X ) /\ ( 2 x. s ) e. RR ) -> ( x e. ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) -> ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) C_ ( x ( ball ` M ) ( 2 x. s ) ) ) )
38 35 37 sylan2 
 |-  ( ( ( M e. ( *Met ` X ) /\ y e. X ) /\ s e. RR+ ) -> ( x e. ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) -> ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) C_ ( x ( ball ` M ) ( 2 x. s ) ) ) )
39 38 anasss 
 |-  ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) -> ( x e. ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) -> ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) C_ ( x ( ball ` M ) ( 2 x. s ) ) ) )
40 39 imp 
 |-  ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) -> ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) C_ ( x ( ball ` M ) ( 2 x. s ) ) )
41 31 40 sylan2 
 |-  ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ ( x e. X /\ X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) ) -> ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) C_ ( x ( ball ` M ) ( 2 x. s ) ) )
42 41 anassrs 
 |-  ( ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) /\ X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) -> ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) C_ ( x ( ball ` M ) ( 2 x. s ) ) )
43 29 42 eqsstrd 
 |-  ( ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) /\ X = ( y ( ball ` M ) ( ( 2 x. s ) / 2 ) ) ) -> X C_ ( x ( ball ` M ) ( 2 x. s ) ) )
44 28 43 syldan 
 |-  ( ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) /\ X = ( y ( ball ` M ) s ) ) -> X C_ ( x ( ball ` M ) ( 2 x. s ) ) )
45 13 adantl 
 |-  ( ( y e. X /\ s e. RR+ ) -> ( 2 x. s ) e. RR+ )
46 rpxr 
 |-  ( ( 2 x. s ) e. RR+ -> ( 2 x. s ) e. RR* )
47 blssm 
 |-  ( ( M e. ( *Met ` X ) /\ x e. X /\ ( 2 x. s ) e. RR* ) -> ( x ( ball ` M ) ( 2 x. s ) ) C_ X )
48 46 47 syl3an3 
 |-  ( ( M e. ( *Met ` X ) /\ x e. X /\ ( 2 x. s ) e. RR+ ) -> ( x ( ball ` M ) ( 2 x. s ) ) C_ X )
49 48 3expa 
 |-  ( ( ( M e. ( *Met ` X ) /\ x e. X ) /\ ( 2 x. s ) e. RR+ ) -> ( x ( ball ` M ) ( 2 x. s ) ) C_ X )
50 45 49 sylan2 
 |-  ( ( ( M e. ( *Met ` X ) /\ x e. X ) /\ ( y e. X /\ s e. RR+ ) ) -> ( x ( ball ` M ) ( 2 x. s ) ) C_ X )
51 50 an32s 
 |-  ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) -> ( x ( ball ` M ) ( 2 x. s ) ) C_ X )
52 51 adantr 
 |-  ( ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) /\ X = ( y ( ball ` M ) s ) ) -> ( x ( ball ` M ) ( 2 x. s ) ) C_ X )
53 44 52 eqssd 
 |-  ( ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) /\ X = ( y ( ball ` M ) s ) ) -> X = ( x ( ball ` M ) ( 2 x. s ) ) )
54 oveq2 
 |-  ( r = ( 2 x. s ) -> ( x ( ball ` M ) r ) = ( x ( ball ` M ) ( 2 x. s ) ) )
55 54 rspceeqv 
 |-  ( ( ( 2 x. s ) e. RR+ /\ X = ( x ( ball ` M ) ( 2 x. s ) ) ) -> E. r e. RR+ X = ( x ( ball ` M ) r ) )
56 15 53 55 syl2anc 
 |-  ( ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) /\ X = ( y ( ball ` M ) s ) ) -> E. r e. RR+ X = ( x ( ball ` M ) r ) )
57 56 ex 
 |-  ( ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) /\ x e. X ) -> ( X = ( y ( ball ` M ) s ) -> E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
58 57 ralrimdva 
 |-  ( ( M e. ( *Met ` X ) /\ ( y e. X /\ s e. RR+ ) ) -> ( X = ( y ( ball ` M ) s ) -> A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
59 58 rexlimdvva 
 |-  ( M e. ( *Met ` X ) -> ( E. y e. X E. s e. RR+ X = ( y ( ball ` M ) s ) -> A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
60 10 59 biimtrid 
 |-  ( M e. ( *Met ` X ) -> ( E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) -> A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
61 rexn0 
 |-  ( E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) -> X =/= (/) )
62 61 a1i 
 |-  ( M e. ( *Met ` X ) -> ( E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) -> X =/= (/) ) )
63 60 62 jcad 
 |-  ( M e. ( *Met ` X ) -> ( E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) -> ( A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) /\ X =/= (/) ) ) )
64 5 63 impbid2 
 |-  ( M e. ( *Met ` X ) -> ( ( A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) /\ X =/= (/) ) <-> E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
65 64 pm5.32i 
 |-  ( ( M e. ( *Met ` X ) /\ ( A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) /\ X =/= (/) ) ) <-> ( M e. ( *Met ` X ) /\ E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
66 2 3 65 3bitri 
 |-  ( ( M e. ( Bnd ` X ) /\ X =/= (/) ) <-> ( M e. ( *Met ` X ) /\ E. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )