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Theorem ssblex 15657
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 535 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR+ )
21rphalfcld 10127 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR+ )
3 simprr 537 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR+ )
4 rpmincl 12028 . . 3 |- ( ( ( R / 2 ) e. RR+ /\ S e. RR+ ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR+ )
52, 3, 4syl2anc 415 . 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR+ )
65rpred 10114 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR )
72rpred 10114 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> ( R / 2 ) e. RR )
81rpred 10114 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> R e. RR )
93rpred 10114 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR )
10 min1inf 12022 . . . 4 |- ( ( ( R / 2 ) e. RR /\ S e. RR ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ ( R / 2 ) )
117, 9, 10syl2anc 415 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ ( R / 2 ) )
121rpgt0d 10117 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> 0 < R )
13 halfpos 9547 . . . . 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 8458 . 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 10115 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) e. RR* )
193rpxrd 10115 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> S e. RR* )
20 min2inf 12023 . . . 4 |- ( ( ( R / 2 ) e. RR /\ S e. RR ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ S )
217, 9, 20syl2anc 415 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR+ /\ S e. RR+ ) ) -> inf ( { ( R / 2 ) , S } , RR , < ) <_ S )
22 ssbl 15652 . . 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 1283 . 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 4134 . . . 4 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( x < R <-> inf ( { ( R / 2 ) , S } , RR , < ) < R ) )
25 oveq2 6096 . . . . 5 |- ( x = inf ( { ( R / 2 ) , S } , RR , < ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D )inf ( { ( R / 2 ) , S } , RR , < ) ) )
2625sseq1d 3277 . . . 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 477 . . 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 2929 . 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 1276 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   {cpr 3710   class class class wbr 4131   ` cfv 5380  (class class class)co 6088  infcinf 7329   RRcr 8184   0cc0 8185   RR*cxr 8365   < clt 8366   <_ cle 8367   / cdiv 9010   2c2 9364   RR+crp 10071   *Metcxmet 14982   ballcbl 14984
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305
This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-map 6928  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-rp 10072  df-xneg 10191  df-xadd 10192  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-psmet 14989  df-xmet 14990  df-bl 14992
This theorem is used by:  mopni3  15710
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