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

Theorem metustblOLD 21168
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.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
metustblOLD  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  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 metustblOLD
Dummy variables  r  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 994 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  D )  /\  P  e.  X )  ->  D  e.  ( *Met `  X ) )
2 simp3 996 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  D )  /\  P  e.  X )  ->  P  e.  X )
3 simpr 459 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD
`  D )  /\  P  e.  X )  /\  w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) ) )  /\  w  C_  V )  ->  w  C_  V )
4 vex 3037 . . . . . . . 8  |-  w  e. 
_V
5 eqid 2382 . . . . . . . . 9  |-  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )  =  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) )
65elrnmpt 5162 . . . . . . . 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 724 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD
`  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 3438 . . . . . . . 8  |-  ( w  =  ( `' D " ( 0 [,) r
) )  ->  (
w  C_  V  <->  ( `' D " ( 0 [,) r ) )  C_  V ) )
1110biimpcd 224 . . . . . . 7  |-  ( w 
C_  V  ->  (
w  =  ( `' D " ( 0 [,) r ) )  ->  ( `' D " ( 0 [,) r
) )  C_  V
) )
1211reximdv 2856 . . . . . 6  |-  ( w 
C_  V  ->  ( E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r
) )  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
) )
1312imp 427 . . . . 5  |-  ( ( w  C_  V  /\  E. r  e.  RR+  w  =  ( `' D " ( 0 [,) r
) ) )  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V )
143, 9, 13syl2anc 659 . . . 4  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD
`  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
)
15 ne0i 3717 . . . . . 6  |-  ( P  e.  X  ->  X  =/=  (/) )
162, 15syl 16 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  D )  /\  P  e.  X )  ->  X  =/=  (/) )
17 simp2 995 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  D )  /\  P  e.  X )  ->  V  e.  (metUnifOLD
`  D ) )
18 metuelOLD 21165 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( V  e.  (metUnifOLD `  D )  <->  ( V  C_  ( X  X.  X
)  /\  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) w 
C_  V ) ) )
1918simplbda 622 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  V  e.  (metUnifOLD
`  D ) )  ->  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r ) ) ) w  C_  V
)
2016, 1, 17, 19syl21anc 1225 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  D )  /\  P  e.  X )  ->  E. w  e.  ran  ( r  e.  RR+  |->  ( `' D " ( 0 [,) r
) ) ) w 
C_  V )
2114, 20r19.29a 2924 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  D )  /\  P  e.  X )  ->  E. r  e.  RR+  ( `' D " ( 0 [,) r
) )  C_  V
)
22 imass1 5283 . . . . . 6  |-  ( ( `' D " ( 0 [,) r ) ) 
C_  V  ->  (
( `' D "
( 0 [,) r
) ) " { P } )  C_  ( V " { P }
) )
2322reximi 2850 . . . . 5  |-  ( E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V  ->  E. r  e.  RR+  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) )
24 xmetpsmet 20936 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  D  e.  (PsMet `  X )
)
25 blval2 21163 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  ( P ( ball `  D
) r )  =  ( ( `' D " ( 0 [,) r
) ) " { P } ) )
2624, 25syl3an1 1259 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  ( P ( ball `  D ) r )  =  ( ( `' D " ( 0 [,) r ) )
" { P }
) )
2726sseq1d 3444 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  ( ( P (
ball `  D )
r )  C_  ( V " { P }
)  <->  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) ) )
28273expa 1194 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  r  e.  RR+ )  ->  (
( P ( ball `  D ) r ) 
C_  ( V " { P } )  <->  ( ( `' D " ( 0 [,) r ) )
" { P }
)  C_  ( V " { P } ) ) )
2928rexbidva 2890 . . . . 5  |-  ( ( D  e.  ( *Met `  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 } ) ) )
3023, 29syl5ibr 221 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  ( E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) ) )
3130imp 427 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  E. r  e.  RR+  ( `' D " ( 0 [,) r ) ) 
C_  V )  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  ( V " { P } ) )
321, 2, 21, 31syl21anc 1225 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  D )  /\  P  e.  X )  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  ( V " { P }
) )
33 blssex 21015 . . 3  |-  ( ( D  e.  ( *Met `  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 }
) ) )
34333adant2 1013 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  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 }
) ) )
3532, 34mpbird 232 1  |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   E.wrex 2733   _Vcvv 3034    C_ wss 3389   (/)c0 3711   {csn 3944    |-> cmpt 4425    X. cxp 4911   `'ccnv 4912   ran crn 4914   "cima 4916   ` cfv 5496  (class class class)co 6196   0cc0 9403   RR+crp 11139   [,)cico 11452  PsMetcpsmet 18515   *Metcxmt 18516   ballcbl 18518  metUnifOLDcmetuOLD 18522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ico 11456  df-psmet 18524  df-xmet 18525  df-bl 18527  df-fbas 18529  df-fg 18530  df-metuOLD 18531
This theorem is referenced by:  metutopOLD  21170
  Copyright terms: Public domain W3C validator