Theorem blpnfctr 15304

Description: The infinity ball in an extended metric acts like an ultrametric ball in that every point in the ball is also its center. (Contributed by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
blpnfctr	|- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( P ( ball ` D ) +oo ) = ( A ( ball ` D ) +oo ) )

Proof of Theorem blpnfctr

Step	Hyp	Ref	Expression
1		eqid 2232	. . . . 5 |- ( `' D " RR ) = ( `' D " RR )
2	1	xmeter 15301	. . . 4 |- ( D e. ( *Met ` X ) -> ( `' D " RR ) Er X )
3	2	3ad2ant1 1045	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( `' D " RR ) Er X )
4		simp3 1026	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. ( P ( ball ` D ) +oo ) )
5	1	xmetec 15302	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> [ P ] ( `' D " RR ) = ( P ( ball ` D ) +oo ) )
6	5	3adant3 1044	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> [ P ] ( `' D " RR ) = ( P ( ball ` D ) +oo ) )
7	4, 6	eleqtrrd 2312	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. [ P ] ( `' D " RR ) )
8		elecg 6807	. . . . . 6 |- ( ( A e. ( P ( ball ` D ) +oo ) /\ P e. X ) -> ( A e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) A ) )
9	8	ancoms 268	. . . . 5 |- ( ( P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( A e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) A ) )
10	9	3adant1 1042	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( A e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) A ) )
11	7, 10	mpbid 147	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> P ( `' D " RR ) A )
12	3, 11	erthi 6815	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> [ P ] ( `' D " RR ) = [ A ] ( `' D " RR ) )
13		pnfxr 8326	. . . . . 6 |- +oo e. RR*
14		blssm 15286	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ +oo e. RR* ) -> ( P ( ball ` D ) +oo ) C_ X )
15	13, 14	mp3an3 1363	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( P ( ball ` D ) +oo ) C_ X )
16	15	sselda 3238	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. X )
17	1	xmetec 15302	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
18	17	adantlr 477	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. X ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
19	16, 18	syldan 282	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. ( P ( ball ` D ) +oo ) ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
20	19	3impa 1221	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
21	12, 6, 20	3eqtr3d 2273	1 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( P ( ball ` D ) +oo ) = ( A ( ball ` D ) +oo ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   C_ wss 3211   class class class wbr 4109   `'ccnv 4748   "cima 4752   ` cfv 5352  (class class class)co 6050   Er wer 6764   [cec 6765   RRcr 8126   +oocpnf 8305   RR*cxr 8307   *Metcxmet 14684   ballcbl 14686

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-er 6767  df-ec 6769  df-map 6884  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-2 9296  df-xadd 10106  df-psmet 14691  df-xmet 14692  df-bl 14694

This theorem is referenced by: (None)