Theorem elbl4 22368

Description: Membership in a ball, alternative definition. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
elbl4	|- ( ( ( D e. (PsMet ` X ) /\ R e. RR+ ) /\ ( A e. X /\ B e. X ) ) -> ( B e. ( A ( ball ` D ) R ) <-> B ( `' D " ( 0 [,) R ) ) A ) )

Proof of Theorem elbl4

Step	Hyp	Ref	Expression
1		rpxr 11840	. . 3 |- ( R e. RR+ -> R e. RR* )
2		blcomps 22198	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ R e. RR* ) /\ ( A e. X /\ B e. X ) ) -> ( B e. ( A ( ball ` D ) R ) <-> A e. ( B ( ball ` D ) R ) ) )
3	1, 2	sylanl2 683	. 2 |- ( ( ( D e. (PsMet ` X ) /\ R e. RR+ ) /\ ( A e. X /\ B e. X ) ) -> ( B e. ( A ( ball ` D ) R ) <-> A e. ( B ( ball ` D ) R ) ) )
4		simpll 790	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ R e. RR+ ) /\ ( A e. X /\ B e. X ) ) -> D e. (PsMet ` X ) )
5		simprr 796	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ R e. RR+ ) /\ ( A e. X /\ B e. X ) ) -> B e. X )
6		simplr 792	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ R e. RR+ ) /\ ( A e. X /\ B e. X ) ) -> R e. RR+ )
7		blval2 22367	. . . 4 |- ( ( D e. (PsMet ` X ) /\ B e. X /\ R e. RR+ ) -> ( B ( ball ` D ) R ) = ( ( `' D " ( 0 [,) R ) ) " { B } ) )
8	7	eleq2d 2687	. . 3 |- ( ( D e. (PsMet ` X ) /\ B e. X /\ R e. RR+ ) -> ( A e. ( B ( ball ` D ) R ) <-> A e. ( ( `' D " ( 0 [,) R ) ) " { B } ) ) )
9	4, 5, 6, 8	syl3anc 1326	. 2 |- ( ( ( D e. (PsMet ` X ) /\ R e. RR+ ) /\ ( A e. X /\ B e. X ) ) -> ( A e. ( B ( ball ` D ) R ) <-> A e. ( ( `' D " ( 0 [,) R ) ) " { B } ) ) )
10		elimasng 5491	. . . . 5 |- ( ( B e. X /\ A e. X ) -> ( A e. ( ( `' D " ( 0 [,) R ) ) " { B } ) <-> <. B , A >. e. ( `' D " ( 0 [,) R ) ) ) )
11		df-br 4654	. . . . 5 |- ( B ( `' D " ( 0 [,) R ) ) A <-> <. B , A >. e. ( `' D " ( 0 [,) R ) ) )
12	10, 11	syl6bbr 278	. . . 4 |- ( ( B e. X /\ A e. X ) -> ( A e. ( ( `' D " ( 0 [,) R ) ) " { B } ) <-> B ( `' D " ( 0 [,) R ) ) A ) )
13	12	ancoms 469	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A e. ( ( `' D " ( 0 [,) R ) ) " { B } ) <-> B ( `' D " ( 0 [,) R ) ) A ) )
14	13	adantl 482	. 2 |- ( ( ( D e. (PsMet ` X ) /\ R e. RR+ ) /\ ( A e. X /\ B e. X ) ) -> ( A e. ( ( `' D " ( 0 [,) R ) ) " { B } ) <-> B ( `' D " ( 0 [,) R ) ) A ) )
15	3, 9, 14	3bitrd 294	1 |- ( ( ( D e. (PsMet ` X ) /\ R e. RR+ ) /\ ( A e. X /\ B e. X ) ) -> ( B e. ( A ( ball ` D ) R ) <-> B ( `' D " ( 0 [,) R ) ) A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   {csn 4177   <.cop 4183   class class class wbr 4653   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   RR+crp 11832   [,)cico 12177  PsMetcpsmet 19730   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-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-ico 12181  df-psmet 19738  df-bl 19741

This theorem is referenced by:  metucn  22376