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Theorem neibl 18519
Description: The neighborhoods around a point  P of a metric space are those subsets containing a ball around  P. Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypothesis
Ref Expression
mopni.1  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
neibl  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_  X  /\  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  N
) ) )
Distinct variable groups:    D, r    J, r    N, r    P, r    X, r

Proof of Theorem neibl
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mopni.1 . . . . 5  |-  J  =  ( MetOpen `  D )
21mopntop 18458 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
32adantr 452 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  J  e.  Top )
41mopnuni 18459 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
54eleq2d 2502 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  ( P  e.  X  <->  P  e.  U. J ) )
65biimpa 471 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  P  e.  U. J )
7 eqid 2435 . . . 4  |-  U. J  =  U. J
87isneip 17157 . . 3  |-  ( ( J  e.  Top  /\  P  e.  U. J )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_ 
U. J  /\  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) ) ) )
93, 6, 8syl2anc 643 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_ 
U. J  /\  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) ) ) )
104sseq2d 3368 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  ( N  C_  X  <->  N  C_  U. J
) )
1110adantr 452 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( N  C_  X  <->  N  C_  U. J
) )
1211anbi1d 686 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( ( N  C_  X  /\  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) )  <-> 
( N  C_  U. J  /\  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) ) ) )
131mopni2 18511 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  y  e.  J  /\  P  e.  y
)  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  y
)
14 sstr2 3347 . . . . . . . . . . 11  |-  ( ( P ( ball `  D
) r )  C_  y  ->  ( y  C_  N  ->  ( P (
ball `  D )
r )  C_  N
) )
1514com12 29 . . . . . . . . . 10  |-  ( y 
C_  N  ->  (
( P ( ball `  D ) r ) 
C_  y  ->  ( P ( ball `  D
) r )  C_  N ) )
1615reximdv 2809 . . . . . . . . 9  |-  ( y 
C_  N  ->  ( E. r  e.  RR+  ( P ( ball `  D
) r )  C_  y  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  N ) )
1713, 16syl5com 28 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  y  e.  J  /\  P  e.  y
)  ->  ( y  C_  N  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  N
) )
18173exp 1152 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  (
y  e.  J  -> 
( P  e.  y  ->  ( y  C_  N  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  N ) ) ) )
1918imp4a 573 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  (
y  e.  J  -> 
( ( P  e.  y  /\  y  C_  N )  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  N
) ) )
2019ad2antrr 707 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  N  C_  X )  ->  (
y  e.  J  -> 
( ( P  e.  y  /\  y  C_  N )  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  N
) ) )
2120rexlimdv 2821 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  N  C_  X )  ->  ( E. y  e.  J  ( P  e.  y  /\  y  C_  N )  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  N ) )
22 rpxr 10608 . . . . . . . . 9  |-  ( r  e.  RR+  ->  r  e. 
RR* )
231blopn 18518 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  r  e.  RR* )  ->  ( P ( ball `  D ) r )  e.  J )
2422, 23syl3an3 1219 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  ( P ( ball `  D ) r )  e.  J )
25 blcntr 18431 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  P  e.  ( P ( ball `  D
) r ) )
26 eleq2 2496 . . . . . . . . . . 11  |-  ( y  =  ( P (
ball `  D )
r )  ->  ( P  e.  y  <->  P  e.  ( P ( ball `  D
) r ) ) )
27 sseq1 3361 . . . . . . . . . . 11  |-  ( y  =  ( P (
ball `  D )
r )  ->  (
y  C_  N  <->  ( P
( ball `  D )
r )  C_  N
) )
2826, 27anbi12d 692 . . . . . . . . . 10  |-  ( y  =  ( P (
ball `  D )
r )  ->  (
( P  e.  y  /\  y  C_  N
)  <->  ( P  e.  ( P ( ball `  D ) r )  /\  ( P (
ball `  D )
r )  C_  N
) ) )
2928rspcev 3044 . . . . . . . . 9  |-  ( ( ( P ( ball `  D ) r )  e.  J  /\  ( P  e.  ( P
( ball `  D )
r )  /\  ( P ( ball `  D
) r )  C_  N ) )  ->  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) )
3029expr 599 . . . . . . . 8  |-  ( ( ( P ( ball `  D ) r )  e.  J  /\  P  e.  ( P ( ball `  D ) r ) )  ->  ( ( P ( ball `  D
) r )  C_  N  ->  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) ) )
3124, 25, 30syl2anc 643 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  ( ( P (
ball `  D )
r )  C_  N  ->  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) ) )
32313expia 1155 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( r  e.  RR+  ->  ( ( P ( ball `  D
) r )  C_  N  ->  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) ) ) )
3332rexlimdv 2821 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( E. r  e.  RR+  ( P ( ball `  D
) r )  C_  N  ->  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) ) )
3433adantr 452 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  N  C_  X )  ->  ( E. r  e.  RR+  ( P ( ball `  D
) r )  C_  N  ->  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) ) )
3521, 34impbid 184 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  N  C_  X )  ->  ( E. y  e.  J  ( P  e.  y  /\  y  C_  N )  <->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  N ) )
3635pm5.32da 623 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( ( N  C_  X  /\  E. y  e.  J  ( P  e.  y  /\  y  C_  N ) )  <-> 
( N  C_  X  /\  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  N ) ) )
379, 12, 363bitr2d 273 1  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_  X  /\  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   {csn 3806   U.cuni 4007   ` cfv 5445  (class class class)co 6072   RR*cxr 9108   RR+crp 10601   * Metcxmt 16674   ballcbl 16676   MetOpencmopn 16679   Topctop 16946   neicnei 17149
This theorem is referenced by:  reperflem  18837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-n0 10211  df-z 10272  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-topgen 13655  df-psmet 16682  df-xmet 16683  df-bl 16685  df-mopn 16686  df-top 16951  df-bases 16953  df-topon 16954  df-nei 17150
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