Theorem blvalps 14945

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 14942	. . 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 5977	. . . 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 4067	. . 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 2762	. 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 14879	. . . . 5 |- Rel PsMet
11		relelfvdm 5626	. . . . 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 4198	. . 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 6091	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 2177   {crab 2489   _Vcvv 2773   class class class wbr 4054   dom cdm 4688   Rel wrel 4693   ` cfv 5285  (class class class)co 5962   e. cmpo 5964   RR*cxr 8136   < clt 8137  PsMetcpsmet 14382   ballcbl 14385

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-map 6755  df-pnf 8139  df-mnf 8140  df-xr 8141  df-psmet 14390  df-bl 14393

This theorem is referenced by:  elblps  14947