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Theorem ssblex 15121
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 529 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR+ )
21rphalfcld 9917 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR+ )
3 simprr 531 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR+ )
4 rpmincl 11765 . . 3 |- ( ( ( R / 2 ) e. RR+ /\ S e. RR+ ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR+ )
52, 3, 4syl2anc 411 . 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR+ )
65rpred 9904 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR )
72rpred 9904 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR )
81rpred 9904 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR )
93rpred 9904 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR )
10 min1inf 11759 . . . 4 |- ( ( ( R / 2 ) e. RR /\ S e. RR ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ ( R / 2 ) )
117, 9, 10syl2anc 411 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ ( R / 2 ) )
121rpgt0d 9907 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> 0 < R )
13 halfpos 9353 . . . . 5 |- ( R e. RR -> ( 0 < R <-> ( R / 2 ) < R ) )
148, 13syl 14 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( 0 < R <-> ( R / 2 ) < R ) )
1512, 14mpbid 147 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) < R )
166, 7, 8, 11, 15lelttrd 8282 . 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) < R )
17 simpl 109 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( D e. ( *Met ` X ) /\ P e. X ) )
185rpxrd 9905 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR* )
193rpxrd 9905 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR* )
20 min2inf 11760 . . . 4 |- ( ( ( R / 2 ) e. RR /\ S e. RR ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ S )
217, 9, 20syl2anc 411 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ S )
22 ssbl 15116 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ (inf ( { ( R / 2 ) , S } , RR , < ) e. RR* /\ S e. RR* ) /\ inf ( { ( R / 2 ) , S } , RR , < ) <_ S ) -> ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) C_ ( P ( ball ` D ) S ) )
2317, 18, 19, 21, 22syl121anc 1276 . 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) C_ ( P ( ball ` D ) S ) )
24 breq1 4086 . . . 4 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( x < R <-> inf ( { ( R / 2 ) , S } , RR , < ) < R ) )
25 oveq2 6015 . . . . 5 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) )
2625sseq1d 3253 . . . 4 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( ( P ( ball ` D ) x ) C_ ( P ( ball ` D ) S ) <-> ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) C_ ( P ( ball ` D ) S ) ) )
2724, 26anbi12d 473 . . 3 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( ( x < R /\ ( P ( ball ` D ) x ) C_ ( P ( ball ` D ) S ) ) <-> (inf ( { ( R / 2 ) , S } , RR , < ) < R /\ ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) C_ ( P ( ball ` D ) S ) ) ) )
2827rspcev 2907 . 2 |- ( (inf ( { ( R / 2 ) , S } , RR , < ) e. RR+ /\ (inf ( { ( R / 2 ) , S } , RR , < ) < R /\ ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) C_ ( P ( ball ` D ) S ) ) ) -> E. x e. RR+ ( x < R /\ ( P ( ball ` D ) x ) C_ ( P ( ball ` D ) S ) ) )
295, 16, 23, 28syl12anc 1269 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 set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   {cpr 3667   class class class wbr 4083   ` cfv 5318  (class class class)co 6007  infcinf 7161   RRcr 8009   0cc0 8010   RR*cxr 8191   < clt 8192   <_ cle 8193   / cdiv 8830   2c2 9172   RR+crp 9861   *Metcxmet 14516   ballcbl 14518
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130
This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-map 6805  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-rp 9862  df-xneg 9980  df-xadd 9981  df-seqfrec 10682  df-exp 10773  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-psmet 14523  df-xmet 14524  df-bl 14526
This theorem is referenced by:  mopni3  15174
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