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Theorem xrsblre 20347
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 9425 . . 3  |-  ( P  e.  RR  ->  P  e.  RR* )
2 xrsxmet.1 . . . . 5  |-  D  =  ( dist `  RR*s
)
32xrsxmet 20345 . . . 4  |-  D  e.  ( *Met `  RR* )
4 eqid 2441 . . . . 5  |-  ( `' D " RR )  =  ( `' D " RR )
54blssec 19969 . . . 4  |-  ( ( D  e.  ( *Met `  RR* )  /\  P  e.  RR*  /\  R  e.  RR* )  ->  ( P ( ball `  D
) R )  C_  [ P ] ( `' D " RR ) )
63, 5mp3an1 1296 . . 3  |-  ( ( P  e.  RR*  /\  R  e.  RR* )  ->  ( P ( ball `  D
) R )  C_  [ P ] ( `' D " RR ) )
71, 6sylan 468 . 2  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( P ( ball `  D ) R ) 
C_  [ P ]
( `' D " RR ) )
8 vex 2973 . . . . 5  |-  x  e. 
_V
9 simpl 454 . . . . 5  |-  ( ( P  e.  RR  /\  R  e.  RR* )  ->  P  e.  RR )
10 elecg 7135 . . . . 5  |-  ( ( x  e.  _V  /\  P  e.  RR )  ->  ( x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
118, 9, 10sylancr 658 . . . 4  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
124xmeterval 19966 . . . . . 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 458 . . . . . . . 8  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =  x )  ->  P  =  x )
15 simplll 752 . . . . . . . 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 2516 . . . . . . 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 1027 . . . . . . . . 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 1025 . . . . . . . . . 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 1026 . . . . . . . . . 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 458 . . . . . . . . . 10  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =/=  x )  ->  P  =/=  x )
212xrsdsreclb 17819 . . . . . . . . . 10  |-  ( ( P  e.  RR*  /\  x  e.  RR*  /\  P  =/=  x )  ->  (
( P D x )  e.  RR  <->  ( P  e.  RR  /\  x  e.  RR ) ) )
2218, 19, 20, 21syl3anc 1213 . . . . . . . . 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 460 . . . . . . 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 2687 . . . . . 6  |-  ( ( ( P  e.  RR  /\  R  e.  RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  ->  x  e.  RR )
2625ex 434 . . . . 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 3359 . 2  |-  ( ( P  e.  RR  /\  R  e.  RR* )  ->  [ P ] ( `' D " RR ) 
C_  RR )
307, 29sstrd 3363 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 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    C_ wss 3325   class class class wbr 4289   `'ccnv 4835   "cima 4839   ` cfv 5415  (class class class)co 6090   [cec 7095   RRcr 9277   RR*cxr 9413   distcds 14243   RR*scxrs 14434   *Metcxmt 17760   ballcbl 17762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-ec 7099  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-icc 11303  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-tset 14253  df-ple 14254  df-ds 14256  df-xrs 14436  df-psmet 17768  df-xmet 17769  df-bl 17771
This theorem is referenced by:  xrsmopn  20348
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