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Theorem blssex 17989
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 17988 . . . . . . 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 3200 . . . . . . . . 9  |-  ( ( ( P ( ball `  D ) r ) 
C_  x  /\  x  C_  A )  ->  ( P ( ball `  D
) r )  C_  A )
32expcom 424 . . . . . . . 8  |-  ( x 
C_  A  ->  (
( P ( ball `  D ) r ) 
C_  x  ->  ( P ( ball `  D
) r )  C_  A ) )
43reximdv 2667 . . . . . . 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 26 . . . . . 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 1151 . . . . 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 586 . . . 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 695 . . 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 2680 . 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 730 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  D  e.  ( * Met `  X ) )
11 simplr 731 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  P  e.  X )
12 rpxr 10377 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e. 
RR* )
1312ad2antrl 708 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR* )
14 blelrn 17983 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  r  e.  RR* )  ->  ( P ( ball `  D ) r )  e.  ran  ( ball `  D ) )
1510, 11, 13, 14syl3anc 1182 . . . . 5  |-  ( ( ( 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 732 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
r  e.  RR+ )
17 blcntr 17980 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  r  e.  RR+ )  ->  P  e.  ( P ( ball `  D
) r ) )
1810, 11, 16, 17syl3anc 1182 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  ->  P  e.  ( P
( ball `  D )
r ) )
19 simprr 733 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
r  e.  RR+  /\  ( P ( ball `  D
) r )  C_  A ) )  -> 
( P ( ball `  D ) r ) 
C_  A )
20 eleq2 2357 . . . . . . 7  |-  ( x  =  ( P (
ball `  D )
r )  ->  ( P  e.  x  <->  P  e.  ( P ( ball `  D
) r ) ) )
21 sseq1 3212 . . . . . . 7  |-  ( x  =  ( P (
ball `  D )
r )  ->  (
x  C_  A  <->  ( P
( ball `  D )
r )  C_  A
) )
2220, 21anbi12d 691 . . . . . 6  |-  ( x  =  ( P (
ball `  D )
r )  ->  (
( P  e.  x  /\  x  C_  A )  <-> 
( P  e.  ( P ( ball `  D
) r )  /\  ( P ( ball `  D
) r )  C_  A ) ) )
2322rspcev 2897 . . . . 5  |-  ( ( ( 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 1180 . . . 4  |-  ( ( ( 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 ) )
2524expr 598 . . 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 ) ) )
2625rexlimdva 2680 . 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 ) ) )
279, 26impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ran crn 4706   ` cfv 5271  (class class class)co 5874   RR*cxr 8882   RR+crp 10370   * Metcxmt 16385   ballcbl 16387
This theorem is referenced by:  blbas  17992  elmopn2  18007  mopni2  18055  metss  18070  tgioo  18318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-xmet 16389  df-bl 16391
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