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Theorem blssexps 21095
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 21093 . . . . . . 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 3497 . . . . . . . . 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 2928 . . . . . . 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 30 . . . . . 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 1194 . . . . 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 601 . . . 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 712 . . 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 2946 . 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 751 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  D  e.  (PsMet `  X
) )
11 simplr 753 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  P  e.  X )
12 rpxr 11228 . . . . . 6  |-  ( r  e.  RR+  ->  r  e. 
RR* )
1312ad2antrl 725 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR* )
14 blelrnps 21085 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR* )  ->  ( P ( ball `  D
) r )  e. 
ran  ( ball `  D
) )
1510, 11, 13, 14syl3anc 1226 . . . 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 754 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR+ )
17 blcntrps 21081 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  P  e.  ( P ( ball `  D ) r ) )
1810, 11, 16, 17syl3anc 1226 . . . 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 755 . . . 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 2527 . . . . . 6  |-  ( x  =  ( P (
ball `  D )
r )  ->  ( P  e.  x  <->  P  e.  ( P ( ball `  D
) r ) ) )
21 sseq1 3510 . . . . . 6  |-  ( x  =  ( P (
ball `  D )
r )  ->  (
x  C_  A  <->  ( P
( ball `  D )
r )  C_  A
) )
2220, 21anbi12d 708 . . . . 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 3207 . . . 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 1224 . . 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 2947 . 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 191 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805    C_ wss 3461   ran crn 4989   ` cfv 5570  (class class class)co 6270   RR*cxr 9616   RR+crp 11221  PsMetcpsmet 18597   ballcbl 18600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-psmet 18606  df-bl 18609
This theorem is referenced by:  metustbl  21250  psmetutop  21252
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