Theorem blpnfctr 15540

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 2238	. . . . 5 |- ( `' D " RR ) = ( `' D " RR )
2	1	xmeter 15537	. . . 4 |- ( D e. ( *Met ` X ) -> ( `' D " RR ) Er X )
3	2	3ad2ant1 1049	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( `' D " RR ) Er X )
4		simp3 1030	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. ( P ( ball ` D ) +oo ) )
5	1	xmetec 15538	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> [ P ] ( `' D " RR ) = ( P ( ball ` D ) +oo ) )
6	5	3adant3 1048	. . . . 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 2318	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. [ P ] ( `' D " RR ) )
8		elecg 6847	. . . . . 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 1046	. . . 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 6855	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> [ P ] ( `' D " RR ) = [ A ] ( `' D " RR ) )
13		pnfxr 8378	. . . . . 6 |- +oo e. RR*
14		blssm 15522	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ +oo e. RR* ) -> ( P ( ball ` D ) +oo ) C_ X )
15	13, 14	mp3an3 1367	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( P ( ball ` D ) +oo ) C_ X )
16	15	sselda 3248	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. X )
17	1	xmetec 15538	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
18	17	adantlr 481	. . . 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 1225	. 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 2279	1 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( P ( ball ` D ) +oo ) = ( A ( ball ` D ) +oo ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   class class class wbr 4130   `'ccnv 4773   "cima 4777   ` cfv 5377  (class class class)co 6085   Er wer 6804   [cec 6805   RRcr 8178   +oocpnf 8357   RR*cxr 8359   *Metcxmet 14873   ballcbl 14875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-er 6807  df-ec 6809  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-2 9363  df-xadd 10175  df-psmet 14880  df-xmet 14881  df-bl 14883

This theorem is used by: (None)