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Theorem blssex 21222
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 21220 . . . . . . 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 3450 . . . . . . . . 9  |-  ( ( ( P ( ball `  D ) r ) 
C_  x  /\  x  C_  A )  ->  ( P ( ball `  D
) r )  C_  A )
32expcom 433 . . . . . . . 8  |-  ( x 
C_  A  ->  (
( P ( ball `  D ) r ) 
C_  x  ->  ( P ( ball `  D
) r )  C_  A ) )
43reximdv 2878 . . . . . . 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 28 . . . . . 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 1197 . . . . 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 601 . . . 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 713 . . 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 2896 . 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 752 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  D  e.  ( *Met `  X ) )
11 simplr 754 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  P  e.  X )
12 rpxr 11272 . . . . . 6  |-  ( r  e.  RR+  ->  r  e. 
RR* )
1312ad2antrl 726 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR* )
14 blelrn 21212 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  r  e.  RR* )  ->  ( P ( ball `  D ) r )  e.  ran  ( ball `  D ) )
1510, 11, 13, 14syl3anc 1230 . . . 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 756 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR+ )
17 blcntr 21208 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  P  e.  ( P ( ball `  D
) r ) )
1810, 11, 16, 17syl3anc 1230 . . . 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 758 . . . 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 2475 . . . . . 6  |-  ( x  =  ( P (
ball `  D )
r )  ->  ( P  e.  x  <->  P  e.  ( P ( ball `  D
) r ) ) )
21 sseq1 3463 . . . . . 6  |-  ( x  =  ( P (
ball `  D )
r )  ->  (
x  C_  A  <->  ( P
( ball `  D )
r )  C_  A
) )
2220, 21anbi12d 709 . . . . 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 3160 . . . 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 1228 . . 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 2897 . 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 190 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2755    C_ wss 3414   ran crn 4824   ` cfv 5569  (class class class)co 6278   RR*cxr 9657   RR+crp 11265   *Metcxmt 18723   ballcbl 18725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-psmet 18731  df-xmet 18732  df-bl 18734
This theorem is referenced by:  blbas  21225  elmopn2  21240  mopni2  21288  metss  21303  metustblOLD  21375  metutopOLD  21377  tgioo  21593
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