Theorem blvalps 15056

Description: The ball around a point P is the set of all points whose distance from P is less than the ball's radius R. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
blvalps	|- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) = { x e. X | ( P D x ) < R } )

Distinct variable groups:   x, P   x, D   x, R   x, X

Proof of Theorem blvalps

Dummy variables r y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		blfvalps 15053	. . 3 |- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( y e. X , r e. RR* |-> { x e. X | ( y D x ) < r } ) )
2	1	3ad2ant1 1042	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( ball ` D ) = ( y e. X , r e. RR* |-> { x e. X | ( y D x ) < r } ) )
3		simprl 529	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> y = P )
4	3	oveq1d 6015	. . . 4 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> ( y D x ) = ( P D x ) )
5		simprr 531	. . . 4 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> r = R )
6	4, 5	breq12d 4095	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> ( ( y D x ) < r <-> ( P D x ) < R ) )
7	6	rabbidv 2788	. 2 |- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> { x e. X | ( y D x ) < r } = { x e. X | ( P D x ) < R } )
8		simp2 1022	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> P e. X )
9		simp3 1023	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> R e. RR* )
10		psmetrel 14990	. . . . 5 |- Rel PsMet
11		relelfvdm 5658	. . . . 5 |- ( ( Rel PsMet /\ D e. (PsMet ` X ) ) -> X e. dom PsMet )
12	10, 11	mpan 424	. . . 4 |- ( D e. (PsMet ` X ) -> X e. dom PsMet )
13	12	3ad2ant1 1042	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> X e. dom PsMet )
14		rabexg 4226	. . 3 |- ( X e. dom PsMet -> { x e. X | ( P D x ) < R } e. _V )
15	13, 14	syl 14	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> { x e. X | ( P D x ) < R } e. _V )
16	2, 7, 8, 9, 15	ovmpod 6131	1 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) = { x e. X | ( P D x ) < R } )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   class class class wbr 4082   dom cdm 4718   Rel wrel 4723   ` cfv 5317  (class class class)co 6000   e. cmpo 6002   RR*cxr 8176   < clt 8177  PsMetcpsmet 14493   ballcbl 14496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-map 6795  df-pnf 8179  df-mnf 8180  df-xr 8181  df-psmet 14501  df-bl 14504

This theorem is referenced by:  elblps  15058