Theorem blvalps 14860

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 14857	. . 3 |- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( y e. X , r e. RR* |-> { x e. X | ( y D x ) < r } ) )
2	1	3ad2ant1 1021	. 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 5959	. . . 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 4057	. . 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 2761	. 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 1001	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> P e. X )
9		simp3 1002	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> R e. RR* )
10		psmetrel 14794	. . . . 5 |- Rel PsMet
11		relelfvdm 5608	. . . . 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 1021	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> X e. dom PsMet )
14		rabexg 4187	. . 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 6073	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 981   = wceq 1373   e. wcel 2176   {crab 2488   _Vcvv 2772   class class class wbr 4044   dom cdm 4675   Rel wrel 4680   ` cfv 5271  (class class class)co 5944   e. cmpo 5946   RR*cxr 8106   < clt 8107  PsMetcpsmet 14297   ballcbl 14300

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-pnf 8109  df-mnf 8110  df-xr 8111  df-psmet 14305  df-bl 14308

This theorem is referenced by:  elblps  14862