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Theorem ssblex 14751
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 9801 . . 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 11420 . . 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 9788 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR )
72rpred 9788 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR )
81rpred 9788 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR )
93rpred 9788 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR )
10 min1inf 11414 . . . 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 9791 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> 0 < R )
13 halfpos 9239 . . . . 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 8168 . 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 9789 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR* )
193rpxrd 9789 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR* )
20 min2inf 11415 . . . 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 14746 . . 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 1254 . 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 4037 . . . 4 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( x < R <-> inf ( { ( R / 2 ) , S } , RR , < ) < R ) )
25 oveq2 5933 . . . . 5 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) )
2625sseq1d 3213 . . . 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 2868 . 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 1247 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 1364   e. wcel 2167   E.wrex 2476   C_ wss 3157   {cpr 3624   class class class wbr 4034   ` cfv 5259  (class class class)co 5925  infcinf 7058   RRcr 7895   0cc0 7896   RR*cxr 8077   < clt 8078   <_ cle 8079   / cdiv 8716   2c2 9058   RR+crp 9745   *Metcxmet 14168   ballcbl 14170
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-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  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-reu 2482  df-rmo 2483  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-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-xneg 9864  df-xadd 9865  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-psmet 14175  df-xmet 14176  df-bl 14178
This theorem is referenced by:  mopni3  14804
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