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Metamath Proof Explorer

Theorem blsscls2

Description: A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014)

Ref Expression
Hypotheses mopni.1 
|- J = ( MetOpen ` D )
blcld.3 
|- S = { z e. X | ( P D z ) <_ R }
Assertion blsscls2 
|- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) -> S C_ ( P ( ball ` D ) T ) )

Proof

Step Hyp Ref Expression
1 mopni.1 
 |-  J = ( MetOpen ` D )
2 blcld.3 
 |-  S = { z e. X | ( P D z ) <_ R }
3 simplr3 
 |-  ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> R < T )
4 xmetcl 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ z e. X ) -> ( P D z ) e. RR* )
5 4 ad4ant124 
 |-  ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> ( P D z ) e. RR* )
6 simplr1 
 |-  ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> R e. RR* )
7 simplr2 
 |-  ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> T e. RR* )
8 xrlelttr 
 |-  ( ( ( P D z ) e. RR* /\ R e. RR* /\ T e. RR* ) -> ( ( ( P D z ) <_ R /\ R < T ) -> ( P D z ) < T ) )
9 8 expcomd 
 |-  ( ( ( P D z ) e. RR* /\ R e. RR* /\ T e. RR* ) -> ( R < T -> ( ( P D z ) <_ R -> ( P D z ) < T ) ) )
10 5 6 7 9 syl3anc 
 |-  ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> ( R < T -> ( ( P D z ) <_ R -> ( P D z ) < T ) ) )
11 3 10 mpd 
 |-  ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> ( ( P D z ) <_ R -> ( P D z ) < T ) )
12 simp2 
 |-  ( ( R e. RR* /\ T e. RR* /\ R < T ) -> T e. RR* )
13 elbl2 
 |-  ( ( ( D e. ( *Met ` X ) /\ T e. RR* ) /\ ( P e. X /\ z e. X ) ) -> ( z e. ( P ( ball ` D ) T ) <-> ( P D z ) < T ) )
14 13 an4s 
 |-  ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( T e. RR* /\ z e. X ) ) -> ( z e. ( P ( ball ` D ) T ) <-> ( P D z ) < T ) )
15 12 14 sylanr1 
 |-  ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( ( R e. RR* /\ T e. RR* /\ R < T ) /\ z e. X ) ) -> ( z e. ( P ( ball ` D ) T ) <-> ( P D z ) < T ) )
16 15 anassrs 
 |-  ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> ( z e. ( P ( ball ` D ) T ) <-> ( P D z ) < T ) )
17 11 16 sylibrd 
 |-  ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> ( ( P D z ) <_ R -> z e. ( P ( ball ` D ) T ) ) )
18 17 ralrimiva 
 |-  ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) -> A. z e. X ( ( P D z ) <_ R -> z e. ( P ( ball ` D ) T ) ) )
19 rabss 
 |-  ( { z e. X | ( P D z ) <_ R } C_ ( P ( ball ` D ) T ) <-> A. z e. X ( ( P D z ) <_ R -> z e. ( P ( ball ` D ) T ) ) )
20 18 19 sylibr 
 |-  ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) -> { z e. X | ( P D z ) <_ R } C_ ( P ( ball ` D ) T ) )
21 2 20 eqsstrid 
 |-  ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) -> S C_ ( P ( ball ` D ) T ) )