Theorem blsscls2 14390

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

Hypotheses

Ref	Expression
mopni.1	|- J = ( MetOpen ` D )
blcld.3	|- S = { z e. X | ( P D z ) <_ R }

Assertion

Ref	Expression
blsscls2	|- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) -> S C_ ( P ( ball ` D ) T ) )

Distinct variable groups:   z, D   z, R   z, P   z, T   z, X

Allowed substitution hints:   S( z)   J( z)

Proof of Theorem blsscls2

Step	Hyp	Ref	Expression
1		blcld.3	. 2 |- S = { z e. X | ( P D z ) <_ R }
2		simplr3 1043	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> R < T )
3		xmetcl 14249	. . . . . . . . 9 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ z e. X ) -> ( P D z ) e. RR* )
4	3	3expa 1205	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ z e. X ) -> ( P D z ) e. RR* )
5	4	adantlr 477	. . . . . . 7 |- ( ( ( ( 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 1041	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> R e. RR* )
7		simplr2 1042	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> T e. RR* )
8		xrlelttr 9824	. . . . . . . 8 |- ( ( ( P D z ) e. RR* /\ R e. RR* /\ T e. RR* ) -> ( ( ( P D z ) <_ R /\ R < T ) -> ( P D z ) < T ) )
9	8	expcomd 1452	. . . . . . 7 |- ( ( ( 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 1249	. . . . . 6 |- ( ( ( ( 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	2, 10	mpd 13	. . . . 5 |- ( ( ( ( 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 1000	. . . . . . 7 |- ( ( R e. RR* /\ T e. RR* /\ R < T ) -> T e. RR* )
13		elbl2 14290	. . . . . . . 8 |- ( ( ( 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 588	. . . . . . 7 |- ( ( ( 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 404	. . . . . 6 |- ( ( ( 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 400	. . . . 5 |- ( ( ( ( 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 169	. . . 4 |- ( ( ( ( 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 2563	. . 3 |- ( ( ( 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 3247	. . 3 |- ( { 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 134	. 2 |- ( ( ( 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	1, 20	eqsstrid 3216	1 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) -> S C_ ( P ( ball ` D ) T ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   A.wral 2468   {crab 2472   C_ wss 3144   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   RR*cxr 8009   < clt 8010   <_ cle 8011   *Metcxmet 13810   ballcbl 13812   MetOpencmopn 13815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-map 6668  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-psmet 13817  df-xmet 13818  df-bl 13820

This theorem is referenced by: (None)