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Theorem blssexps 13223
Description: Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blssexps  |-  ( ( D  e.  (PsMet `  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 blssexps
StepHypRef Expression
1 blssps 13221 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  ran  ( ball `  D
)  /\  P  e.  x )  ->  E. r  e.  RR+  ( P (
ball `  D )
r )  C_  x
)
2 sstr 3155 . . . . . . . . 9  |-  ( ( ( P ( ball `  D ) r ) 
C_  x  /\  x  C_  A )  ->  ( P ( ball `  D
) r )  C_  A )
32expcom 115 . . . . . . . 8  |-  ( x 
C_  A  ->  (
( P ( ball `  D ) r ) 
C_  x  ->  ( P ( ball `  D
) r )  C_  A ) )
43reximdv 2571 . . . . . . 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.  (PsMet `  X )  /\  x  e.  ran  ( ball `  D
)  /\  P  e.  x )  ->  (
x  C_  A  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  A ) )
653expa 1198 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  ran  ( ball `  D
) )  /\  P  e.  x )  ->  (
x  C_  A  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  A ) )
76expimpd 361 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  ran  ( ball `  D
) )  ->  (
( P  e.  x  /\  x  C_  A )  ->  E. r  e.  RR+  ( P ( ball `  D
) r )  C_  A ) )
87adantlr 474 . . 3  |-  ( ( ( D  e.  (PsMet `  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 2587 . 2  |-  ( ( D  e.  (PsMet `  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 524 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  D  e.  (PsMet `  X
) )
11 simplr 525 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  P  e.  X )
12 rpxr 9618 . . . . . 6  |-  ( r  e.  RR+  ->  r  e. 
RR* )
1312ad2antrl 487 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR* )
14 blelrnps 13213 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR* )  ->  ( P ( ball `  D
) r )  e. 
ran  ( ball `  D
) )
1510, 11, 13, 14syl3anc 1233 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
( P ( ball `  D ) r )  e.  ran  ( ball `  D ) )
16 simprl 526 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR+ )
17 blcntrps 13209 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  P  e.  ( P ( ball `  D ) r ) )
1810, 11, 16, 17syl3anc 1233 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  P  e.  ( P
( ball `  D )
r ) )
19 simprr 527 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
( P ( ball `  D ) r ) 
C_  A )
20 eleq2 2234 . . . . . 6  |-  ( x  =  ( P (
ball `  D )
r )  ->  ( P  e.  x  <->  P  e.  ( P ( ball `  D
) r ) ) )
21 sseq1 3170 . . . . . 6  |-  ( x  =  ( P (
ball `  D )
r )  ->  (
x  C_  A  <->  ( P
( ball `  D )
r )  C_  A
) )
2220, 21anbi12d 470 . . . . 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 2834 . . . 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 1231 . . 3  |-  ( ( ( D  e.  (PsMet `  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 2588 . 2  |-  ( ( D  e.  (PsMet `  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 128 1  |-  ( ( D  e.  (PsMet `  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    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   E.wrex 2449    C_ wss 3121   ran crn 4612   ` cfv 5198  (class class class)co 5853   RR*cxr 7953   RR+crp 9610  PsMetcpsmet 12773   ballcbl 12776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-psmet 12781  df-bl 12784
This theorem is referenced by: (None)
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