MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssblex Structured version   Visualization version   Unicode version
Theorem ssblex 22233
Description: A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
ssblex |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> E. x e. RR+ ( x < R /\ ( P ( ball ` D ) x ) C_ ( P ( ball ` D ) S ) ) )
Distinct variable groups:   x, D   x, R   x, P   x, S   x, X
Proof of Theorem ssblex
StepHypRef Expression
1 simprl 794 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR+ )
21rphalfcld 11884 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR+ )
3 simprr 796 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR+ )
42, 3ifcld 4131 . 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) e. RR+ )
54rpred 11872 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) e. RR )
62rpred 11872 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR )
71rpred 11872 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR )
83rpred 11872 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR )
9 min1 12020 . . . 4 |- ( ( ( R / 2 ) e. RR /\ S e. RR ) -> if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) <_ ( R / 2 ) )
106, 8, 9syl2anc 693 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) <_ ( R / 2 ) )
111rpgt0d 11875 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> 0 < R )
12 halfpos 11262 . . . . 5 |- ( R e. RR -> ( 0 < R <-> ( R / 2 ) < R ) )
137, 12syl 17 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( 0 < R <-> ( R / 2 ) < R ) )
1411, 13mpbid 222 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) < R )
155, 6, 7, 10, 14lelttrd 10195 . 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) < R )
16 simpl 473 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( D e. ( *Met ` X ) /\ P e. X ) )
174rpxrd 11873 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) e. RR* )
183rpxrd 11873 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR* )
19 min2 12021 . . . 4 |- ( ( ( R / 2 ) e. RR /\ S e. RR ) -> if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) <_ S )
206, 8, 19syl2anc 693 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) <_ S )
21 ssbl 22228 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) e. RR* /\ S e. RR* ) /\ if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) <_ S ) -> ( P ( ball ` D ) if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) ) C_ ( P ( ball ` D ) S ) )
2216, 17, 18, 20, 21syl121anc 1331 . 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( P ( ball ` D ) if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) ) C_ ( P ( ball ` D ) S ) )
23 breq1 4656 . . . 4 |- ( x = if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) -> ( x < R <-> if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) < R ) )
24 oveq2 6658 . . . . 5 |- ( x = if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D ) if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) ) )
2524sseq1d 3632 . . . 4 |- ( x = if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) -> ( ( P ( ball ` D ) x ) C_ ( P ( ball ` D ) S ) <-> ( P ( ball ` D ) if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) ) C_ ( P ( ball ` D ) S ) ) )
2623, 25anbi12d 747 . . 3 |- ( x = if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) -> ( ( x < R /\ ( P ( ball ` D ) x ) C_ ( P ( ball ` D ) S ) ) <-> ( if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) < R /\ ( P ( ball ` D ) if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) ) C_ ( P ( ball ` D ) S ) ) ) )
2726rspcev 3309 . 2 |- ( ( if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) e. RR+ /\ ( if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) < R /\ ( P ( ball ` D ) if ( ( R / 2 ) <_ S , ( R / 2 ) , S ) ) C_ ( P ( ball ` D ) S ) ) ) -> E. x e. RR+ ( x < R /\ ( P ( ball ` D ) x ) C_ ( P ( ball ` D ) S ) ) )
284, 15, 22, 27syl12anc 1324 1 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> E. x e. RR+ ( x < R /\ ( P ( ball ` D ) x ) C_ ( P ( ball ` D ) S ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   2c2 11070   RR+crp 11832   *Metcxmt 19731   ballcbl 19733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-psmet 19738  df-xmet 19739  df-bl 19741
This theorem is referenced by:  mopni3  22299
  Copyright terms: Public domain W3C validator