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

Theorem blbas 21227
Description: The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)
Assertion
Ref Expression
blbas  |-  ( D  e.  ( *Met `  X )  ->  ran  ( ball `  D )  e. 
TopBases )

Proof of Theorem blbas
Dummy variables  x  r  b  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blin2 21226 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  z  e.  ( x  i^i  y
) )  /\  (
x  e.  ran  ( ball `  D )  /\  y  e.  ran  ( ball `  D ) ) )  ->  E. r  e.  RR+  ( z ( ball `  D ) r ) 
C_  ( x  i^i  y ) )
2 simpll 754 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  z  e.  ( x  i^i  y
) )  /\  (
x  e.  ran  ( ball `  D )  /\  y  e.  ran  ( ball `  D ) ) )  ->  D  e.  ( *Met `  X
) )
3 inss1 3661 . . . . . . . . . . 11  |-  ( x  i^i  y )  C_  x
43sseli 3440 . . . . . . . . . 10  |-  ( z  e.  ( x  i^i  y )  ->  z  e.  x )
5 elunii 4198 . . . . . . . . . 10  |-  ( ( z  e.  x  /\  x  e.  ran  ( ball `  D ) )  -> 
z  e.  U. ran  ( ball `  D )
)
64, 5sylan 471 . . . . . . . . 9  |-  ( ( z  e.  ( x  i^i  y )  /\  x  e.  ran  ( ball `  D ) )  -> 
z  e.  U. ran  ( ball `  D )
)
76ad2ant2lr 748 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  z  e.  ( x  i^i  y
) )  /\  (
x  e.  ran  ( ball `  D )  /\  y  e.  ran  ( ball `  D ) ) )  ->  z  e.  U. ran  ( ball `  D
) )
8 unirnbl 21217 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  U. ran  ( ball `  D )  =  X )
98ad2antrr 726 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  z  e.  ( x  i^i  y
) )  /\  (
x  e.  ran  ( ball `  D )  /\  y  e.  ran  ( ball `  D ) ) )  ->  U. ran  ( ball `  D )  =  X )
107, 9eleqtrd 2494 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  z  e.  ( x  i^i  y
) )  /\  (
x  e.  ran  ( ball `  D )  /\  y  e.  ran  ( ball `  D ) ) )  ->  z  e.  X
)
11 blssex 21224 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  z  e.  X
)  ->  ( E. b  e.  ran  ( ball `  D ) ( z  e.  b  /\  b  C_  ( x  i^i  y
) )  <->  E. r  e.  RR+  ( z (
ball `  D )
r )  C_  (
x  i^i  y )
) )
122, 10, 11syl2anc 661 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  z  e.  ( x  i^i  y
) )  /\  (
x  e.  ran  ( ball `  D )  /\  y  e.  ran  ( ball `  D ) ) )  ->  ( E. b  e.  ran  ( ball `  D
) ( z  e.  b  /\  b  C_  ( x  i^i  y
) )  <->  E. r  e.  RR+  ( z (
ball `  D )
r )  C_  (
x  i^i  y )
) )
131, 12mpbird 234 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  z  e.  ( x  i^i  y
) )  /\  (
x  e.  ran  ( ball `  D )  /\  y  e.  ran  ( ball `  D ) ) )  ->  E. b  e.  ran  ( ball `  D )
( z  e.  b  /\  b  C_  (
x  i^i  y )
) )
1413ex 434 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  z  e.  ( x  i^i  y ) )  ->  ( (
x  e.  ran  ( ball `  D )  /\  y  e.  ran  ( ball `  D ) )  ->  E. b  e.  ran  ( ball `  D )
( z  e.  b  /\  b  C_  (
x  i^i  y )
) ) )
1514ralrimdva 2824 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
( x  e.  ran  ( ball `  D )  /\  y  e.  ran  ( ball `  D )
)  ->  A. z  e.  ( x  i^i  y
) E. b  e. 
ran  ( ball `  D
) ( z  e.  b  /\  b  C_  ( x  i^i  y
) ) ) )
1615ralrimivv 2826 . 2  |-  ( D  e.  ( *Met `  X )  ->  A. x  e.  ran  ( ball `  D
) A. y  e. 
ran  ( ball `  D
) A. z  e.  ( x  i^i  y
) E. b  e. 
ran  ( ball `  D
) ( z  e.  b  /\  b  C_  ( x  i^i  y
) ) )
17 fvex 5861 . . . 4  |-  ( ball `  D )  e.  _V
1817rnex 6720 . . 3  |-  ran  ( ball `  D )  e. 
_V
19 isbasis2g 19743 . . 3  |-  ( ran  ( ball `  D
)  e.  _V  ->  ( ran  ( ball `  D
)  e.  TopBases  <->  A. x  e.  ran  ( ball `  D
) A. y  e. 
ran  ( ball `  D
) A. z  e.  ( x  i^i  y
) E. b  e. 
ran  ( ball `  D
) ( z  e.  b  /\  b  C_  ( x  i^i  y
) ) ) )
2018, 19ax-mp 5 . 2  |-  ( ran  ( ball `  D
)  e.  TopBases  <->  A. x  e.  ran  ( ball `  D
) A. y  e. 
ran  ( ball `  D
) A. z  e.  ( x  i^i  y
) E. b  e. 
ran  ( ball `  D
) ( z  e.  b  /\  b  C_  ( x  i^i  y
) ) )
2116, 20sylibr 214 1  |-  ( D  e.  ( *Met `  X )  ->  ran  ( ball `  D )  e. 
TopBases )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   E.wrex 2757   _Vcvv 3061    i^i cin 3415    C_ wss 3416   U.cuni 4193   ran crn 4826   ` cfv 5571  (class class class)co 6280   RR+crp 11267   *Metcxmt 18725   ballcbl 18727   TopBasesctb 19692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-sup 7937  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-n0 10839  df-z 10908  df-uz 11130  df-q 11230  df-rp 11268  df-xneg 11373  df-xadd 11374  df-xmul 11375  df-psmet 18733  df-xmet 18734  df-bl 18736  df-bases 19695
This theorem is referenced by:  mopntopon  21236  elmopn  21239  imasf1oxms  21286  blssopn  21292  metss  21305
  Copyright terms: Public domain W3C validator