Theorem blsscls2 14729

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 14588	. . . . . . . . 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 9881	. . . . . . . 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 14629	. . . . . . . 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 2570	. . 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 3260	. . 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 3229	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 2167   A.wral 2475   {crab 2479   C_ wss 3157   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   RR*cxr 8060   < clt 8061   <_ cle 8062   *Metcxmet 14092   ballcbl 14094   MetOpencmopn 14097

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-psmet 14099  df-xmet 14100  df-bl 14102

This theorem is referenced by: (None)