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Theorem xrsblre 21182
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 9651 . . 3  |-  ( P  e.  RR  ->  P  e.  RR* )
2 xrsxmet.1 . . . . 5  |-  D  =  ( dist `  RR*s
)
32xrsxmet 21180 . . . 4  |-  D  e.  ( *Met `  RR* )
4 eqid 2467 . . . . 5  |-  ( `' D " RR )  =  ( `' D " RR )
54blssec 20804 . . . 4  |-  ( ( D  e.  ( *Met `  RR* )  /\  P  e.  RR*  /\  R  e.  RR* )  ->  ( P ( ball `  D
) R )  C_  [ P ] ( `' D " RR ) )
63, 5mp3an1 1311 . . 3  |-  ( ( P  e.  RR*  /\  R  e.  RR* )  ->  ( P ( ball `  D
) R )  C_  [ P ] ( `' D " RR ) )
71, 6sylan 471 . 2  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( P ( ball `  D ) R ) 
C_  [ P ]
( `' D " RR ) )
8 vex 3121 . . . . 5  |-  x  e. 
_V
9 simpl 457 . . . . 5  |-  ( ( P  e.  RR  /\  R  e.  RR* )  ->  P  e.  RR )
10 elecg 7362 . . . . 5  |-  ( ( x  e.  _V  /\  P  e.  RR )  ->  ( x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
118, 9, 10sylancr 663 . . . 4  |-  ( ( P  e.  RR  /\  R  e.  RR* )  -> 
( x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
124xmeterval 20801 . . . . . 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 461 . . . . . . . 8  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =  x )  ->  P  =  x )
15 simplll 757 . . . . . . . 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 2556 . . . . . . 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 1040 . . . . . . . . 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 1038 . . . . . . . . . 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 1039 . . . . . . . . . 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 461 . . . . . . . . . 10  |-  ( ( ( ( P  e.  RR  /\  R  e. 
RR* )  /\  ( P  e.  RR*  /\  x  e.  RR*  /\  ( P D x )  e.  RR ) )  /\  P  =/=  x )  ->  P  =/=  x )
212xrsdsreclb 18333 . . . . . . . . . 10  |-  ( ( P  e.  RR*  /\  x  e.  RR*  /\  P  =/=  x )  ->  (
( P D x )  e.  RR  <->  ( P  e.  RR  /\  x  e.  RR ) ) )
2218, 19, 20, 21syl3anc 1228 . . . . . . . . 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 463 . . . . . . 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 2785 . . . . . 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 3515 . 2  |-  ( ( P  e.  RR  /\  R  e.  RR* )  ->  [ P ] ( `' D " RR ) 
C_  RR )
307, 29sstrd 3519 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118    C_ wss 3481   class class class wbr 4453   `'ccnv 5004   "cima 5008   ` cfv 5594  (class class class)co 6295   [cec 7321   RRcr 9503   RR*cxr 9639   distcds 14580   RR*scxrs 14771   *Metcxmt 18271   ballcbl 18273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-ec 7325  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-icc 11548  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-plusg 14584  df-mulr 14585  df-tset 14590  df-ple 14591  df-ds 14593  df-xrs 14773  df-psmet 18279  df-xmet 18280  df-bl 18282
This theorem is referenced by:  xrsmopn  21183
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