Theorem xblpnfps 15389

Description: The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
xblpnfps	|- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( A e. ( P ( ball ` D ) +oo ) <-> ( A e. X /\ ( P D A ) e. RR ) ) )

Proof of Theorem xblpnfps

Step	Hyp	Ref	Expression
1		pnfxr 8342	. . 3 |- +oo e. RR*
2		elblps 15381	. . 3 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ +oo e. RR* ) -> ( A e. ( P ( ball ` D ) +oo ) <-> ( A e. X /\ ( P D A ) < +oo ) ) )
3	1, 2	mp3an3 1363	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( A e. ( P ( ball ` D ) +oo ) <-> ( A e. X /\ ( P D A ) < +oo ) ) )
4		psmetcl 15317	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> ( P D A ) e. RR* )
5		psmetge0 15322	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> 0 <_ ( P D A ) )
6		ge0nemnf 10176	. . . . . . . 8 |- ( ( ( P D A ) e. RR* /\ 0 <_ ( P D A ) ) -> ( P D A ) =/= -oo )
7	4, 5, 6	syl2anc 411	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> ( P D A ) =/= -oo )
8		nmnfgt 10170	. . . . . . . 8 |- ( ( P D A ) e. RR* -> ( -oo < ( P D A ) <-> ( P D A ) =/= -oo ) )
9	4, 8	syl 14	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> ( -oo < ( P D A ) <-> ( P D A ) =/= -oo ) )
10	7, 9	mpbird 167	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> -oo < ( P D A ) )
11	10	biantrurd 305	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> ( ( P D A ) < +oo <-> ( -oo < ( P D A ) /\ ( P D A ) < +oo ) ) )
12		xrrebnd 10171	. . . . . 6 |- ( ( P D A ) e. RR* -> ( ( P D A ) e. RR <-> ( -oo < ( P D A ) /\ ( P D A ) < +oo ) ) )
13	4, 12	syl 14	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> ( ( P D A ) e. RR <-> ( -oo < ( P D A ) /\ ( P D A ) < +oo ) ) )
14	11, 13	bitr4d 191	. . . 4 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> ( ( P D A ) < +oo <-> ( P D A ) e. RR ) )
15	14	3expa 1230	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ A e. X ) -> ( ( P D A ) < +oo <-> ( P D A ) e. RR ) )
16	15	pm5.32da 452	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( ( A e. X /\ ( P D A ) < +oo ) <-> ( A e. X /\ ( P D A ) e. RR ) ) )
17	3, 16	bitrd 188	1 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( A e. ( P ( ball ` D ) +oo ) <-> ( A e. X /\ ( P D A ) e. RR ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   e. wcel 2205   =/= wne 2414   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   < clt 8324   <_ cle 8325  PsMetcpsmet 14809   ballcbl 14812

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-2 9313  df-xadd 10125  df-psmet 14817  df-bl 14820

This theorem is referenced by:  xblss2ps  15395