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Theorem ssblex 12603
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 520 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR+ )
21rphalfcld 9499 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR+ )
3 simprr 521 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR+ )
4 rpmincl 11012 . . 3 |- ( ( ( R / 2 ) e. RR+ /\ S e. RR+ ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR+ )
52, 3, 4syl2anc 408 . 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR+ )
65rpred 9486 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR )
72rpred 9486 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR )
81rpred 9486 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR )
93rpred 9486 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR )
10 min1inf 11006 . . . 4 |- ( ( ( R / 2 ) e. RR /\ S e. RR ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ ( R / 2 ) )
117, 9, 10syl2anc 408 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ ( R / 2 ) )
121rpgt0d 9489 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> 0 < R )
13 halfpos 8954 . . . . 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 146 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) < R )
166, 7, 8, 11, 15lelttrd 7890 . 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) < R )
17 simpl 108 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( D e. ( *Met ` X ) /\ P e. X ) )
185rpxrd 9487 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR* )
193rpxrd 9487 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR* )
20 min2inf 11007 . . . 4 |- ( ( ( R / 2 ) e. RR /\ S e. RR ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ S )
217, 9, 20syl2anc 408 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ S )
22 ssbl 12598 . . 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 1221 . 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 3932 . . . 4 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( x < R <-> inf ( { ( R / 2 ) , S } , RR , < ) < R ) )
25 oveq2 5782 . . . . 5 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) )
2625sseq1d 3126 . . . 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 464 . . 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 2789 . 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 1214 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 103   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   C_ wss 3071   {cpr 3528   class class class wbr 3929   ` cfv 5123  (class class class)co 5774  infcinf 6870   RRcr 7622   0cc0 7623   RR*cxr 7802   < clt 7803   <_ cle 7804   / cdiv 8435   2c2 8774   RR+crp 9444   *Metcxmet 12152   ballcbl 12154
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-mulrcl 7722  ax-addcom 7723  ax-mulcom 7724  ax-addass 7725  ax-mulass 7726  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-1rid 7730  ax-0id 7731  ax-rnegex 7732  ax-precex 7733  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-ltwlin 7736  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739  ax-pre-mulgt0 7740  ax-pre-mulext 7741  ax-arch 7742  ax-caucvg 7743
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-sup 6871  df-inf 6872  df-pnf 7805  df-mnf 7806  df-xr 7807  df-ltxr 7808  df-le 7809  df-sub 7938  df-neg 7939  df-reap 8340  df-ap 8347  df-div 8436  df-inn 8724  df-2 8782  df-3 8783  df-4 8784  df-n0 8981  df-z 9058  df-uz 9330  df-rp 9445  df-xneg 9562  df-xadd 9563  df-seqfrec 10222  df-exp 10296  df-cj 10617  df-re 10618  df-im 10619  df-rsqrt 10773  df-abs 10774  df-psmet 12159  df-xmet 12160  df-bl 12162
This theorem is referenced by:  mopni3  12656
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