Theorem xblpnfps 15209

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 8291	. . 3 |- +oo e. RR*
2		elblps 15201	. . 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 15137	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> ( P D A ) e. RR* )
5		psmetge0 15142	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ A e. X ) -> 0 <_ ( P D A ) )
6		ge0nemnf 10120	. . . . . . . 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 10114	. . . . . . . 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 10115	. . . . . 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 2202   =/= wne 2403   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8091   0cc0 8092   +oocpnf 8270   -oocmnf 8271   RR*cxr 8272   < clt 8273   <_ cle 8274  PsMetcpsmet 14631   ballcbl 14634

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208  ax-pre-mulgt0 8209

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-2 9261  df-xadd 10069  df-psmet 14639  df-bl 14642

This theorem is referenced by:  xblss2ps  15215