Theorem blpnfctr 12611

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 2139	. . . . 5 |- ( `' D " RR ) = ( `' D " RR )
2	1	xmeter 12608	. . . 4 |- ( D e. ( *Met ` X ) -> ( `' D " RR ) Er X )
3	2	3ad2ant1 1002	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( `' D " RR ) Er X )
4		simp3 983	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. ( P ( ball ` D ) +oo ) )
5	1	xmetec 12609	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> [ P ] ( `' D " RR ) = ( P ( ball ` D ) +oo ) )
6	5	3adant3 1001	. . . . 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 2219	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. [ P ] ( `' D " RR ) )
8		elecg 6467	. . . . . 6 |- ( ( A e. ( P ( ball ` D ) +oo ) /\ P e. X ) -> ( A e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) A ) )
9	8	ancoms 266	. . . . 5 |- ( ( P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( A e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) A ) )
10	9	3adant1 999	. . . 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 146	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> P ( `' D " RR ) A )
12	3, 11	erthi 6475	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> [ P ] ( `' D " RR ) = [ A ] ( `' D " RR ) )
13		pnfxr 7821	. . . . . 6 |- +oo e. RR*
14		blssm 12593	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ +oo e. RR* ) -> ( P ( ball ` D ) +oo ) C_ X )
15	13, 14	mp3an3 1304	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( P ( ball ` D ) +oo ) C_ X )
16	15	sselda 3097	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. X )
17	1	xmetec 12609	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
18	17	adantlr 468	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. X ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
19	16, 18	syldan 280	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. ( P ( ball ` D ) +oo ) ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
20	19	3impa 1176	. 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 2180	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 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   C_ wss 3071   class class class wbr 3929   `'ccnv 4538   "cima 4542   ` cfv 5123  (class class class)co 5774   Er wer 6426   [cec 6427   RRcr 7622   +oocpnf 7800   RR*cxr 7802   *Metcxmet 12152   ballcbl 12154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-mulrcl 7722  ax-addcom 7723  ax-mulcom 7724  ax-addass 7725  ax-mulass 7726  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-1rid 7730  ax-0id 7731  ax-rnegex 7732  ax-precex 7733  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-ltwlin 7736  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739  ax-pre-mulgt0 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-er 6429  df-ec 6431  df-map 6544  df-pnf 7805  df-mnf 7806  df-xr 7807  df-ltxr 7808  df-le 7809  df-sub 7938  df-neg 7939  df-2 8782  df-xadd 9563  df-psmet 12159  df-xmet 12160  df-bl 12162

This theorem is referenced by: (None)