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

Theorem blssex 18458
Description: Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
blssex  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( E. x  e.  ran  ( ball `  D ) ( P  e.  x  /\  x  C_  A )  <->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  A
) )
Distinct variable groups:    x, r, A    D, r, x    P, r, x    X, r, x

Proof of Theorem blssex
StepHypRef Expression
1 blss 18456 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  ran  ( ball `  D )  /\  P  e.  x
)  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  x
)
2 sstr 3357 . . . . . . . . 9  |-  ( ( ( P ( ball `  D ) r ) 
C_  x  /\  x  C_  A )  ->  ( P ( ball `  D
) r )  C_  A )
32expcom 426 . . . . . . . 8  |-  ( x 
C_  A  ->  (
( P ( ball `  D ) r ) 
C_  x  ->  ( P ( ball `  D
) r )  C_  A ) )
43reximdv 2818 . . . . . . 7  |-  ( x 
C_  A  ->  ( E. r  e.  RR+  ( P ( ball `  D
) r )  C_  x  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  A ) )
51, 4syl5com 29 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  ran  ( ball `  D )  /\  P  e.  x
)  ->  ( x  C_  A  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  A
) )
653expa 1154 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  x  e.  ran  ( ball `  D
) )  /\  P  e.  x )  ->  (
x  C_  A  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  A ) )
76expimpd 588 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  ran  ( ball `  D )
)  ->  ( ( P  e.  x  /\  x  C_  A )  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  A ) )
87adantlr 697 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  x  e.  ran  ( ball `  D
) )  ->  (
( P  e.  x  /\  x  C_  A )  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  A ) )
98rexlimdva 2831 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( E. x  e.  ran  ( ball `  D ) ( P  e.  x  /\  x  C_  A )  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  A
) )
10 simpll 732 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  D  e.  ( * Met `  X ) )
11 simplr 733 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  P  e.  X )
12 rpxr 10620 . . . . . 6  |-  ( r  e.  RR+  ->  r  e. 
RR* )
1312ad2antrl 710 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR* )
14 blelrn 18448 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  r  e.  RR* )  ->  ( P ( ball `  D ) r )  e.  ran  ( ball `  D ) )
1510, 11, 13, 14syl3anc 1185 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
( P ( ball `  D ) r )  e.  ran  ( ball `  D ) )
16 simprl 734 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR+ )
17 blcntr 18444 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  P  e.  ( P ( ball `  D
) r ) )
1810, 11, 16, 17syl3anc 1185 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  P  e.  ( P
( ball `  D )
r ) )
19 simprr 735 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
( P ( ball `  D ) r ) 
C_  A )
20 eleq2 2498 . . . . . 6  |-  ( x  =  ( P (
ball `  D )
r )  ->  ( P  e.  x  <->  P  e.  ( P ( ball `  D
) r ) ) )
21 sseq1 3370 . . . . . 6  |-  ( x  =  ( P (
ball `  D )
r )  ->  (
x  C_  A  <->  ( P
( ball `  D )
r )  C_  A
) )
2220, 21anbi12d 693 . . . . 5  |-  ( x  =  ( P (
ball `  D )
r )  ->  (
( P  e.  x  /\  x  C_  A )  <-> 
( P  e.  ( P ( ball `  D
) r )  /\  ( P ( ball `  D
) r )  C_  A ) ) )
2322rspcev 3053 . . . 4  |-  ( ( ( P ( ball `  D ) r )  e.  ran  ( ball `  D )  /\  ( P  e.  ( P
( ball `  D )
r )  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  E. x  e.  ran  ( ball `  D )
( P  e.  x  /\  x  C_  A ) )
2415, 18, 19, 23syl12anc 1183 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  E. x  e.  ran  ( ball `  D )
( P  e.  x  /\  x  C_  A ) )
2524rexlimdvaa 2832 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( E. r  e.  RR+  ( P ( ball `  D
) r )  C_  A  ->  E. x  e.  ran  ( ball `  D )
( P  e.  x  /\  x  C_  A ) ) )
269, 25impbid 185 1  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( E. x  e.  ran  ( ball `  D ) ( P  e.  x  /\  x  C_  A )  <->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2707    C_ wss 3321   ran crn 4880   ` cfv 5455  (class class class)co 6082   RR*cxr 9120   RR+crp 10613   * Metcxmt 16687   ballcbl 16689
This theorem is referenced by:  blbas  18461  elmopn2  18476  mopni2  18524  metss  18539  metustblOLD  18611  metutopOLD  18613  tgioo  18828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-n0 10223  df-z 10284  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-psmet 16695  df-xmet 16696  df-bl 16698
  Copyright terms: Public domain W3C validator