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Theorem blvalps 21331
 Description: The ball around a point is the set of all points whose distance from is less than the ball's radius . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blvalps PsMet
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem blvalps
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blfvalps 21329 . . 3 PsMet
3 simprl 762 . . . . 5 PsMet
43oveq1d 6320 . . . 4 PsMet
5 simprr 764 . . . 4 PsMet
64, 5breq12d 4439 . . 3 PsMet
76rabbidv 3079 . 2 PsMet
8 simp2 1006 . 2 PsMet
9 simp3 1007 . 2 PsMet
10 elfvdm 5907 . . . 4 PsMet PsMet
11103ad2ant1 1026 . . 3 PsMet PsMet
12 rabexg 4575 . . 3 PsMet
1311, 12syl 17 . 2 PsMet
142, 7, 8, 9, 13ovmpt2d 6438 1 PsMet
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wceq 1437   wcel 1870  crab 2786  cvv 3087   class class class wbr 4426   cdm 4854  cfv 5601  (class class class)co 6305   cmpt2 6307  cxr 9673   clt 9674  PsMetcpsmet 18889  cbl 18892 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-xr 9678  df-psmet 18897  df-bl 18900 This theorem is referenced by:  elblps  21333  blval2  21508
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