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Theorem xrsblre 21606
Description: Any ball of the metric of the extended reals centered on an element of  RR is entirely contained in  RR. (Contributed by Mario Carneiro, 4-Sep-2015.)
Hypothesis
Ref Expression
xrsxmet.1  |-  D  =  ( dist `  RR*s
)
Assertion
Ref Expression
xrsblre  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( P ( ball `  D ) R ) 
C_  RR )

Proof of Theorem xrsblre
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rexr 9668 . . 3  |-  ( P  e.  RR  ->  P  e.  RR* )
2 xrsxmet.1 . . . . 5  |-  D  =  ( dist `  RR*s
)
32xrsxmet 21604 . . . 4  |-  D  e.  ( *Met `  RR* )
4 eqid 2402 . . . . 5  |-  ( `' D " RR )  =  ( `' D " RR )
54blssec 21228 . . . 4  |-  ( ( D  e.  ( *Met `  RR* )  /\  P  e.  RR*  /\  R  e.  RR* )  ->  ( P ( ball `  D
) R )  C_  [ P ] ( `' D " RR ) )
63, 5mp3an1 1313 . . 3  |-  ( ( P  e.  RR*  /\  R  e.  RR* )  ->  ( P ( ball `  D
) R )  C_  [ P ] ( `' D " RR ) )
71, 6sylan 469 . 2  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( P ( ball `  D ) R ) 
C_  [ P ]
( `' D " RR ) )
8 vex 3061 . . . . 5  |-  x  e. 
_V
9 simpl 455 . . . . 5  |-  ( ( P  e.  RR  /\  R  e.  RR* )  ->  P  e.  RR )
10 elecg 7386 . . . . 5  |-  ( ( x  e.  _V  /\  P  e.  RR )  ->  ( x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
118, 9, 10sylancr 661 . . . 4  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
124xmeterval 21225 . . . . . 6  |-  ( D  e.  ( *Met ` 
RR* )  ->  ( P ( `' D " RR ) x  <->  ( P  e.  RR*  /\  x  e. 
RR*  /\  ( P D x )  e.  RR ) ) )
133, 12ax-mp 5 . . . . 5  |-  ( P ( `' D " RR ) x  <->  ( P  e.  RR*  /\  x  e. 
RR*  /\  ( P D x )  e.  RR ) )
14 simpr 459 . . . . . . . 8  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =  x )  ->  P  =  x )
15 simplll 760 . . . . . . . 8  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =  x )  ->  P  e.  RR )
1614, 15eqeltrrd 2491 . . . . . . 7  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =  x )  ->  x  e.  RR )
17 simplr3 1041 . . . . . . . . 9  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =/=  x )  -> 
( P D x )  e.  RR )
18 simplr1 1039 . . . . . . . . . 10  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =/=  x )  ->  P  e.  RR* )
19 simplr2 1040 . . . . . . . . . 10  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =/=  x )  ->  x  e.  RR* )
20 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =/=  x )  ->  P  =/=  x )
212xrsdsreclb 18783 . . . . . . . . . 10  |-  ( ( P  e.  RR*  /\  x  e.  RR*  /\  P  =/=  x )  ->  (
( P D x )  e.  RR  <->  ( P  e.  RR  /\  x  e.  RR ) ) )
2218, 19, 20, 21syl3anc 1230 . . . . . . . . 9  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =/=  x )  -> 
( ( P D x )  e.  RR  <->  ( P  e.  RR  /\  x  e.  RR )
) )
2317, 22mpbid 210 . . . . . . . 8  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =/=  x )  -> 
( P  e.  RR  /\  x  e.  RR ) )
2423simprd 461 . . . . . . 7  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =/=  x )  ->  x  e.  RR )
2516, 24pm2.61dane 2721 . . . . . 6  |-  ( ( ( P  e.  RR  /\  R  e.  RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  ->  x  e.  RR )
2625ex 432 . . . . 5  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( ( P  e. 
RR*  /\  x  e.  RR* 
/\  ( P D x )  e.  RR )  ->  x  e.  RR ) )
2713, 26syl5bi 217 . . . 4  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( P ( `' D " RR ) x  ->  x  e.  RR ) )
2811, 27sylbid 215 . . 3  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( x  e.  [ P ] ( `' D " RR )  ->  x  e.  RR ) )
2928ssrdv 3447 . 2  |-  ( ( P  e.  RR  /\  R  e.  RR* )  ->  [ P ] ( `' D " RR ) 
C_  RR )
307, 29sstrd 3451 1  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( P ( ball `  D ) R ) 
C_  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3058    C_ wss 3413   class class class wbr 4394   `'ccnv 4821   "cima 4825   ` cfv 5568  (class class class)co 6277   [cec 7345   RRcr 9520   RR*cxr 9656   distcds 14916   RR*scxrs 15112   *Metcxmt 18721   ballcbl 18723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-ec 7349  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-icc 11588  df-fz 11725  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-plusg 14920  df-mulr 14921  df-tset 14926  df-ple 14927  df-ds 14929  df-xrs 15114  df-psmet 18729  df-xmet 18730  df-bl 18732
This theorem is referenced by:  xrsmopn  21607
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