MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metustbl Structured version   Unicode version

Theorem metustbl 21210
Description: The "section" image of an entourage at a point  P always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.)
Assertion
Ref Expression
metustbl  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. a  e.  ran  ( ball `  D
) ( P  e.  a  /\  a  C_  ( V " { P } ) ) )
Distinct variable groups:    D, a    P, a    V, a    X, a

Proof of Theorem metustbl
Dummy variables  r  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  D  e.  (PsMet `  X ) )
2 simp3 998 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  P  e.  X )
3 simpr 461 . . . . 5  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X )  /\  w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) )  /\  w  C_  V
)  ->  w  C_  V
)
4 vex 3112 . . . . . . . 8  |-  w  e. 
_V
5 eqid 2457 . . . . . . . . 9  |-  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  =  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )
65elrnmpt 5259 . . . . . . . 8  |-  ( w  e.  _V  ->  (
w  e.  ran  (
r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  <->  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r ) ) ) )
74, 6ax-mp 5 . . . . . . 7  |-  ( w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  <->  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r ) ) )
87biimpi 194 . . . . . 6  |-  ( w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  ->  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r ) ) )
98ad2antlr 726 . . . . 5  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X )  /\  w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) )  /\  w  C_  V
)  ->  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r ) ) )
10 sseq1 3520 . . . . . . 7  |-  ( w  =  ( `' D " ( 0 [,) r
) )  ->  (
w  C_  V  <->  ( `' D " ( 0 [,) r ) )  C_  V ) )
1110biimpcd 224 . . . . . 6  |-  ( w 
C_  V  ->  (
w  =  ( `' D " ( 0 [,) r ) )  ->  ( `' D " ( 0 [,) r
) )  C_  V
) )
1211reximdv 2931 . . . . 5  |-  ( w 
C_  V  ->  ( E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r
) )  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
) )
133, 9, 12sylc 60 . . . 4  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X )  /\  w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) )  /\  w  C_  V
)  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
)
14 ne0i 3799 . . . . . 6  |-  ( P  e.  X  ->  X  =/=  (/) )
152, 14syl 16 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  X  =/=  (/) )
16 simp2 997 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  V  e.  (metUnif `  D ) )
17 metuel 21207 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( V  e.  (metUnif `  D )  <->  ( V  C_  ( X  X.  X )  /\  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) ) w  C_  V
) ) )
1817simplbda 624 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  V  e.  (metUnif `  D ) )  ->  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) ) w  C_  V
)
1915, 1, 16, 18syl21anc 1227 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) w 
C_  V )
2013, 19r19.29a 2999 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
)
21 imass1 5381 . . . . . 6  |-  ( ( `' D " ( 0 [,) r ) ) 
C_  V  ->  (
( `' D "
( 0 [,) r
) ) " { P } )  C_  ( V " { P }
) )
2221reximi 2925 . . . . 5  |-  ( E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V  ->  E. r  e.  RR+  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) )
23 blval2 21204 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  ( P ( ball `  D
) r )  =  ( ( `' D " ( 0 [,) r
) ) " { P } ) )
2423sseq1d 3526 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  (
( P ( ball `  D ) r ) 
C_  ( V " { P } )  <->  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) ) )
25243expa 1196 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  r  e.  RR+ )  ->  (
( P ( ball `  D ) r ) 
C_  ( V " { P } )  <->  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) ) )
2625rexbidva 2965 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( E. r  e.  RR+  ( P ( ball `  D
) r )  C_  ( V " { P } )  <->  E. r  e.  RR+  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) ) )
2722, 26syl5ibr 221 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) ) )
2827imp 429 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V )  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  ( V " { P } ) )
291, 2, 20, 28syl21anc 1227 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) )
30 blssexps 21055 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( E. a  e.  ran  ( ball `  D )
( P  e.  a  /\  a  C_  ( V " { P }
) )  <->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) ) )
31303adant2 1015 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  ( E. a  e.  ran  ( ball `  D ) ( P  e.  a  /\  a  C_  ( V " { P } ) )  <->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) ) )
3229, 31mpbird 232 1  |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X
)  ->  E. a  e.  ran  ( ball `  D
) ( P  e.  a  /\  a  C_  ( V " { P } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   _Vcvv 3109    C_ wss 3471   (/)c0 3793   {csn 4032    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   ran crn 5009   "cima 5011   ` cfv 5594  (class class class)co 6296   0cc0 9509   RR+crp 11245   [,)cico 11556  PsMetcpsmet 18529   ballcbl 18532  metUnifcmetu 18537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-psmet 18538  df-bl 18541  df-fbas 18543  df-fg 18544  df-metu 18546
This theorem is referenced by:  psmetutop  21212
  Copyright terms: Public domain W3C validator