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Theorem xrsblre 20407
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 9448 . . 3  |-  ( P  e.  RR  ->  P  e.  RR* )
2 xrsxmet.1 . . . . 5  |-  D  =  ( dist `  RR*s
)
32xrsxmet 20405 . . . 4  |-  D  e.  ( *Met `  RR* )
4 eqid 2443 . . . . 5  |-  ( `' D " RR )  =  ( `' D " RR )
54blssec 20029 . . . 4  |-  ( ( D  e.  ( *Met `  RR* )  /\  P  e.  RR*  /\  R  e.  RR* )  ->  ( P ( ball `  D
) R )  C_  [ P ] ( `' D " RR ) )
63, 5mp3an1 1301 . . 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 2994 . . . . 5  |-  x  e. 
_V
9 simpl 457 . . . . 5  |-  ( ( P  e.  RR  /\  R  e.  RR* )  ->  P  e.  RR )
10 elecg 7158 . . . . 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 20026 . . . . . 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 2518 . . . . . . 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 1032 . . . . . . . . 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 1030 . . . . . . . . . 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 1031 . . . . . . . . . 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 17879 . . . . . . . . . 10  |-  ( ( P  e.  RR*  /\  x  e.  RR*  /\  P  =/=  x )  ->  (
( P D x )  e.  RR  <->  ( P  e.  RR  /\  x  e.  RR ) ) )
2218, 19, 20, 21syl3anc 1218 . . . . . . . . 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 2711 . . . . . 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 3381 . 2  |-  ( ( P  e.  RR  /\  R  e.  RR* )  ->  [ P ] ( `' D " RR ) 
C_  RR )
307, 29sstrd 3385 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2991    C_ wss 3347   class class class wbr 4311   `'ccnv 4858   "cima 4862   ` cfv 5437  (class class class)co 6110   [cec 7118   RRcr 9300   RR*cxr 9436   distcds 14266   RR*scxrs 14457   *Metcxmt 17820   ballcbl 17822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-ec 7122  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-5 10402  df-6 10403  df-7 10404  df-8 10405  df-9 10406  df-10 10407  df-n0 10599  df-z 10666  df-dec 10775  df-uz 10881  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-icc 11326  df-fz 11457  df-seq 11826  df-exp 11885  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-struct 14195  df-ndx 14196  df-slot 14197  df-base 14198  df-plusg 14270  df-mulr 14271  df-tset 14276  df-ple 14277  df-ds 14279  df-xrs 14459  df-psmet 17828  df-xmet 17829  df-bl 17831
This theorem is referenced by:  xrsmopn  20408
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