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Theorem blininf 12623
Description: The intersection of two balls with the same center is the smaller of them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
blininf  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( ( P (
ball `  D ) R )  i^i  ( P ( ball `  D
) S ) )  =  ( P (
ball `  D )inf ( { R ,  S } ,  RR* ,  <  ) ) )

Proof of Theorem blininf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xmetcl 12551 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
213expa 1182 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  ( P D x )  e. 
RR* )
32adantlr 469 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  ( P D x )  e.  RR* )
4 simplrl 525 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  R  e.  RR* )
5 simplrr 526 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  S  e.  RR* )
6 xrltmininf 11067 . . . . 5  |-  ( ( ( P D x )  e.  RR*  /\  R  e.  RR*  /\  S  e. 
RR* )  ->  (
( P D x )  < inf ( { R ,  S } ,  RR* ,  <  )  <->  ( ( P D x )  <  R  /\  ( P D x )  <  S ) ) )
73, 4, 5, 6syl3anc 1217 . . . 4  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  ( ( P D x )  < inf ( { R ,  S } ,  RR* ,  <  )  <->  ( ( P D x )  <  R  /\  ( P D x )  <  S ) ) )
87pm5.32da 448 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( ( x  e.  X  /\  ( P D x )  < inf ( { R ,  S } ,  RR* ,  <  ) )  <->  ( x  e.  X  /\  ( ( P D x )  <  R  /\  ( P D x )  < 
S ) ) ) )
9 xrmincl 11063 . . . 4  |-  ( ( R  e.  RR*  /\  S  e.  RR* )  -> inf ( { R ,  S } ,  RR* ,  <  )  e.  RR* )
10 elbl 12590 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\ inf ( { R ,  S } ,  RR* ,  <  )  e.  RR* )  ->  (
x  e.  ( P ( ball `  D
)inf ( { R ,  S } ,  RR* ,  <  ) )  <->  ( x  e.  X  /\  ( P D x )  < inf ( { R ,  S } ,  RR* ,  <  ) ) ) )
11103expa 1182 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\ inf ( { R ,  S } ,  RR* ,  <  )  e.  RR* )  ->  (
x  e.  ( P ( ball `  D
)inf ( { R ,  S } ,  RR* ,  <  ) )  <->  ( x  e.  X  /\  ( P D x )  < inf ( { R ,  S } ,  RR* ,  <  ) ) ) )
129, 11sylan2 284 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( P ( ball `  D
)inf ( { R ,  S } ,  RR* ,  <  ) )  <->  ( x  e.  X  /\  ( P D x )  < inf ( { R ,  S } ,  RR* ,  <  ) ) ) )
13 elbl 12590 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
14133expa 1182 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  R  e.  RR* )  ->  (
x  e.  ( P ( ball `  D
) R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
1514adantrr 471 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( P ( ball `  D
) R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
16 elbl 12590 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) S )  <->  ( x  e.  X  /\  ( P D x )  < 
S ) ) )
17163expa 1182 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  S  e.  RR* )  ->  (
x  e.  ( P ( ball `  D
) S )  <->  ( x  e.  X  /\  ( P D x )  < 
S ) ) )
1817adantrl 470 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( P ( ball `  D
) S )  <->  ( x  e.  X  /\  ( P D x )  < 
S ) ) )
1915, 18anbi12d 465 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  ( P ( ball `  D
) S ) )  <-> 
( ( x  e.  X  /\  ( P D x )  < 
R )  /\  (
x  e.  X  /\  ( P D x )  <  S ) ) ) )
20 elin 3260 . . . 4  |-  ( x  e.  ( ( P ( ball `  D
) R )  i^i  ( P ( ball `  D ) S ) )  <->  ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  ( P ( ball `  D
) S ) ) )
21 anandi 580 . . . 4  |-  ( ( x  e.  X  /\  ( ( P D x )  <  R  /\  ( P D x )  <  S ) )  <->  ( ( x  e.  X  /\  ( P D x )  < 
R )  /\  (
x  e.  X  /\  ( P D x )  <  S ) ) )
2219, 20, 213bitr4g 222 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( ( P ( ball `  D ) R )  i^i  ( P (
ball `  D ) S ) )  <->  ( x  e.  X  /\  (
( P D x )  <  R  /\  ( P D x )  <  S ) ) ) )
238, 12, 223bitr4rd 220 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( ( P ( ball `  D ) R )  i^i  ( P (
ball `  D ) S ) )  <->  x  e.  ( P ( ball `  D
)inf ( { R ,  S } ,  RR* ,  <  ) ) ) )
2423eqrdv 2138 1  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( ( P (
ball `  D ) R )  i^i  ( P ( ball `  D
) S ) )  =  ( P (
ball `  D )inf ( { R ,  S } ,  RR* ,  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481    i^i cin 3071   {cpr 3529   class class class wbr 3933   ` cfv 5127  (class class class)co 5778  infcinf 6874   RR*cxr 7819    < clt 7820   *Metcxmet 12179   ballcbl 12181
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506  ax-cnex 7731  ax-resscn 7732  ax-1cn 7733  ax-1re 7734  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-mulrcl 7739  ax-addcom 7740  ax-mulcom 7741  ax-addass 7742  ax-mulass 7743  ax-distr 7744  ax-i2m1 7745  ax-0lt1 7746  ax-1rid 7747  ax-0id 7748  ax-rnegex 7749  ax-precex 7750  ax-cnre 7751  ax-pre-ltirr 7752  ax-pre-ltwlin 7753  ax-pre-lttrn 7754  ax-pre-apti 7755  ax-pre-ltadd 7756  ax-pre-mulgt0 7757  ax-pre-mulext 7758  ax-arch 7759  ax-caucvg 7760
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-ilim 4295  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-isom 5136  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-frec 6292  df-map 6548  df-sup 6875  df-inf 6876  df-pnf 7822  df-mnf 7823  df-xr 7824  df-ltxr 7825  df-le 7826  df-sub 7955  df-neg 7956  df-reap 8357  df-ap 8364  df-div 8453  df-inn 8741  df-2 8799  df-3 8800  df-4 8801  df-n0 8998  df-z 9075  df-uz 9347  df-rp 9467  df-xneg 9585  df-seqfrec 10246  df-exp 10320  df-cj 10642  df-re 10643  df-im 10644  df-rsqrt 10798  df-abs 10799  df-psmet 12186  df-xmet 12187  df-bl 12189
This theorem is referenced by:  blin2  12631
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