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Theorem ssblex 15018
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 9866 . . 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 11664 . . 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 9853 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR )
72rpred 9853 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR )
81rpred 9853 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR )
93rpred 9853 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR )
10 min1inf 11658 . . . 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 9856 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> 0 < R )
13 halfpos 9303 . . . . 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 8232 . 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 9854 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR* )
193rpxrd 9854 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR* )
20 min2inf 11659 . . . 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 15013 . . 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 1255 . 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 4062 . . . 4 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( x < R <-> inf ( { ( R / 2 ) , S } , RR , < ) < R ) )
25 oveq2 5975 . . . . 5 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) )
2625sseq1d 3230 . . . 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 2884 . 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 1248 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 1373   e. wcel 2178   E.wrex 2487   C_ wss 3174   {cpr 3644   class class class wbr 4059   ` cfv 5290  (class class class)co 5967  infcinf 7111   RRcr 7959   0cc0 7960   RR*cxr 8141   < clt 8142   <_ cle 8143   / cdiv 8780   2c2 9122   RR+crp 9810   *Metcxmet 14413   ballcbl 14415
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080
This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-map 6760  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-xneg 9929  df-xadd 9930  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-psmet 14420  df-xmet 14421  df-bl 14423
This theorem is referenced by:  mopni3  15071
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