Theorem blpnfctr 22241

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 2622	. . . . 5 |- ( `' D " RR ) = ( `' D " RR )
2	1	xmeter 22238	. . . 4 |- ( D e. ( *Met ` X ) -> ( `' D " RR ) Er X )
3	2	3ad2ant1 1082	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( `' D " RR ) Er X )
4		simp3 1063	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. ( P ( ball ` D ) +oo ) )
5	1	xmetec 22239	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> [ P ] ( `' D " RR ) = ( P ( ball ` D ) +oo ) )
6	5	3adant3 1081	. . . . 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 2704	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. [ P ] ( `' D " RR ) )
8		elecg 7785	. . . . . 6 |- ( ( A e. ( P ( ball ` D ) +oo ) /\ P e. X ) -> ( A e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) A ) )
9	8	ancoms 469	. . . . 5 |- ( ( P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> ( A e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) A ) )
10	9	3adant1 1079	. . . 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 222	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> P ( `' D " RR ) A )
12	3, 11	erthi 7793	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ A e. ( P ( ball ` D ) +oo ) ) -> [ P ] ( `' D " RR ) = [ A ] ( `' D " RR ) )
13		pnfxr 10092	. . . . . 6 |- +oo e. RR*
14		blssm 22223	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ +oo e. RR* ) -> ( P ( ball ` D ) +oo ) C_ X )
15	13, 14	mp3an3 1413	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( P ( ball ` D ) +oo ) C_ X )
16	15	sselda 3603	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. ( P ( ball ` D ) +oo ) ) -> A e. X )
17	1	xmetec 22239	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
18	17	adantlr 751	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. X ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
19	16, 18	syldan 487	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ A e. ( P ( ball ` D ) +oo ) ) -> [ A ] ( `' D " RR ) = ( A ( ball ` D ) +oo ) )
20	19	3impa 1259	. 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 2664	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   Er wer 7739   [cec 7740   RRcr 9935   +oocpnf 10071   RR*cxr 10073   *Metcxmt 19731   ballcbl 19733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-ec 7744  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-psmet 19738  df-xmet 19739  df-bl 19741

This theorem is referenced by:  metdstri  22654