Theorem blsscls2 13287

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 1036	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> R < T )
3		xmetcl 13146	. . . . . . . . 9 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ z e. X ) -> ( P D z ) e. RR* )
4	3	3expa 1198	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ z e. X ) -> ( P D z ) e. RR* )
5	4	adantlr 474	. . . . . . 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 1034	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> R e. RR* )
7		simplr2 1035	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ T e. RR* /\ R < T ) ) /\ z e. X ) -> T e. RR* )
8		xrlelttr 9763	. . . . . . . 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 1434	. . . . . . 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 1233	. . . . . 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 993	. . . . . . 7 |- ( ( R e. RR* /\ T e. RR* /\ R < T ) -> T e. RR* )
13		elbl2 13187	. . . . . . . 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 583	. . . . . . 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 402	. . . . . 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 398	. . . . 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 168	. . . 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 2543	. . 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 3224	. . 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 133	. 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 3193	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   {crab 2452   C_ wss 3121   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   RR*cxr 7953   < clt 7954   <_ cle 7955   *Metcxmet 12774   ballcbl 12776   MetOpencmopn 12779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-psmet 12781  df-xmet 12782  df-bl 12784

This theorem is referenced by: (None)