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Metamath Proof Explorer

Theorem metequiv

Description: Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Hypotheses metequiv.3 
|- J = ( MetOpen ` C )
metequiv.4 
|- K = ( MetOpen ` D )
Assertion metequiv 
|- ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( J = K <-> A. x e. X ( A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. a e. RR+ E. b e. RR+ ( x ( ball ` C ) b ) C_ ( x ( ball ` D ) a ) ) ) )

Proof

Step Hyp Ref Expression
1 metequiv.3 
 |-  J = ( MetOpen ` C )
2 metequiv.4 
 |-  K = ( MetOpen ` D )
3 1 2 metss 
 |-  ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( J C_ K <-> A. x e. X A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) ) )
4 2 1 metss 
 |-  ( ( D e. ( *Met ` X ) /\ C e. ( *Met ` X ) ) -> ( K C_ J <-> A. x e. X A. a e. RR+ E. b e. RR+ ( x ( ball ` C ) b ) C_ ( x ( ball ` D ) a ) ) )
5 4 ancoms 
 |-  ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( K C_ J <-> A. x e. X A. a e. RR+ E. b e. RR+ ( x ( ball ` C ) b ) C_ ( x ( ball ` D ) a ) ) )
6 3 5 anbi12d 
 |-  ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( ( J C_ K /\ K C_ J ) <-> ( A. x e. X A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. x e. X A. a e. RR+ E. b e. RR+ ( x ( ball ` C ) b ) C_ ( x ( ball ` D ) a ) ) ) )
7 eqss 
 |-  ( J = K <-> ( J C_ K /\ K C_ J ) )
8 r19.26 
 |-  ( A. x e. X ( A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. a e. RR+ E. b e. RR+ ( x ( ball ` C ) b ) C_ ( x ( ball ` D ) a ) ) <-> ( A. x e. X A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. x e. X A. a e. RR+ E. b e. RR+ ( x ( ball ` C ) b ) C_ ( x ( ball ` D ) a ) ) )
9 6 7 8 3bitr4g 
 |-  ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` X ) ) -> ( J = K <-> A. x e. X ( A. r e. RR+ E. s e. RR+ ( x ( ball ` D ) s ) C_ ( x ( ball ` C ) r ) /\ A. a e. RR+ E. b e. RR+ ( x ( ball ` C ) b ) C_ ( x ( ball ` D ) a ) ) ) )