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Theorem mopni3 18485
Description: An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
Hypothesis
Ref Expression
mopni.1  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
mopni3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  J  /\  P  e.  A
)  /\  R  e.  RR+ )  ->  E. x  e.  RR+  ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  A
) )
Distinct variable groups:    x, A    x, D    x, J    x, R    x, P    x, X

Proof of Theorem mopni3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mopni.1 . . . 4  |-  J  =  ( MetOpen `  D )
21mopni2 18484 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A
)  ->  E. y  e.  RR+  ( P (
ball `  D )
y )  C_  A
)
32adantr 452 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  J  /\  P  e.  A
)  /\  R  e.  RR+ )  ->  E. y  e.  RR+  ( P (
ball `  D )
y )  C_  A
)
4 simp1 957 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A
)  ->  D  e.  ( * Met `  X
) )
51mopnss 18437 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J
)  ->  A  C_  X
)
65sselda 3316 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  J )  /\  P  e.  A )  ->  P  e.  X )
763impa 1148 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A
)  ->  P  e.  X )
84, 7jca 519 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A
)  ->  ( D  e.  ( * Met `  X
)  /\  P  e.  X ) )
9 ssblex 18419 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  y  e.  RR+ ) )  ->  E. x  e.  RR+  (
x  <  R  /\  ( P ( ball `  D
) x )  C_  ( P ( ball `  D
) y ) ) )
108, 9sylan 458 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  J  /\  P  e.  A
)  /\  ( R  e.  RR+  /\  y  e.  RR+ ) )  ->  E. x  e.  RR+  ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  ( P ( ball `  D
) y ) ) )
1110anassrs 630 . . . 4  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  J  /\  P  e.  A
)  /\  R  e.  RR+ )  /\  y  e.  RR+ )  ->  E. x  e.  RR+  ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  ( P ( ball `  D
) y ) ) )
12 sstr 3324 . . . . . . 7  |-  ( ( ( P ( ball `  D ) x ) 
C_  ( P (
ball `  D )
y )  /\  ( P ( ball `  D
) y )  C_  A )  ->  ( P ( ball `  D
) x )  C_  A )
1312expcom 425 . . . . . 6  |-  ( ( P ( ball `  D
) y )  C_  A  ->  ( ( P ( ball `  D
) x )  C_  ( P ( ball `  D
) y )  -> 
( P ( ball `  D ) x ) 
C_  A ) )
1413anim2d 549 . . . . 5  |-  ( ( P ( ball `  D
) y )  C_  A  ->  ( ( x  <  R  /\  ( P ( ball `  D
) x )  C_  ( P ( ball `  D
) y ) )  ->  ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  A
) ) )
1514reximdv 2785 . . . 4  |-  ( ( P ( ball `  D
) y )  C_  A  ->  ( E. x  e.  RR+  ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  ( P ( ball `  D
) y ) )  ->  E. x  e.  RR+  ( x  <  R  /\  ( P ( ball `  D
) x )  C_  A ) ) )
1611, 15syl5com 28 . . 3  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  J  /\  P  e.  A
)  /\  R  e.  RR+ )  /\  y  e.  RR+ )  ->  ( ( P ( ball `  D
) y )  C_  A  ->  E. x  e.  RR+  ( x  <  R  /\  ( P ( ball `  D
) x )  C_  A ) ) )
1716rexlimdva 2798 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  J  /\  P  e.  A
)  /\  R  e.  RR+ )  ->  ( E. y  e.  RR+  ( P ( ball `  D
) y )  C_  A  ->  E. x  e.  RR+  ( x  <  R  /\  ( P ( ball `  D
) x )  C_  A ) ) )
183, 17mpd 15 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  J  /\  P  e.  A
)  /\  R  e.  RR+ )  ->  E. x  e.  RR+  ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675    C_ wss 3288   class class class wbr 4180   ` cfv 5421  (class class class)co 6048    < clt 9084   RR+crp 10576   * Metcxmt 16649   ballcbl 16651   MetOpencmopn 16654
This theorem is referenced by:  bcthlem5  19242  lhop1lem  19858  ulmdvlem3  20279  efopn  20510  opnrebl2  26222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-topgen 13630  df-psmet 16657  df-xmet 16658  df-bl 16660  df-mopn 16661  df-top 16926  df-bases 16928  df-topon 16929
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