Theorem xmetresbl 14760

Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 14757, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance +oo from each other. (Contributed by Mario Carneiro, 23-Aug-2015.)

Hypothesis

Ref	Expression
xmetresbl.1	|- B = ( P ( ball ` D ) R )

Assertion

Ref	Expression
xmetresbl	|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) e. ( Met ` B ) )

Proof of Theorem xmetresbl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 999	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> D e. ( *Met ` X ) )
2		xmetresbl.1	. . . 4 |- B = ( P ( ball ` D ) R )
3		blssm 14741	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X )
4	2, 3	eqsstrid 3230	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ X )
5		xmetres2 14699	. . 3 |- ( ( D e. ( *Met ` X ) /\ B C_ X ) -> ( D |` ( B X. B ) ) e. ( *Met ` B ) )
6	1, 4, 5	syl2anc 411	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) e. ( *Met ` B ) )
7		xmetf 14670	. . . . . 6 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
8	1, 7	syl 14	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> D : ( X X. X ) --> RR* )
9		xpss12 4771	. . . . . 6 |- ( ( B C_ X /\ B C_ X ) -> ( B X. B ) C_ ( X X. X ) )
10	4, 4, 9	syl2anc 411	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( B X. B ) C_ ( X X. X ) )
11	8, 10	fssresd 5437	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) : ( B X. B ) --> RR* )
12	11	ffnd 5411	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) Fn ( B X. B ) )
13		ovres 6067	. . . . . 6 |- ( ( x e. B /\ y e. B ) -> ( x ( D |` ( B X. B ) ) y ) = ( x D y ) )
14	13	adantl 277	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> ( x ( D |` ( B X. B ) ) y ) = ( x D y ) )
15		simpl1 1002	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> D e. ( *Met ` X ) )
16		eqid 2196	. . . . . . . . . 10 |- ( `' D " RR ) = ( `' D " RR )
17	16	xmeter 14756	. . . . . . . . 9 |- ( D e. ( *Met ` X ) -> ( `' D " RR ) Er X )
18	15, 17	syl 14	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> ( `' D " RR ) Er X )
19	16	blssec 14758	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ [ P ] ( `' D " RR ) )
20	2, 19	eqsstrid 3230	. . . . . . . . . . 11 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ [ P ] ( `' D " RR ) )
21	20	sselda 3184	. . . . . . . . . 10 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. B ) -> x e. [ P ] ( `' D " RR ) )
22	21	adantrr 479	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> x e. [ P ] ( `' D " RR ) )
23		simpl2 1003	. . . . . . . . . 10 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> P e. X )
24		elecg 6641	. . . . . . . . . 10 |- ( ( x e. [ P ] ( `' D " RR ) /\ P e. X ) -> ( x e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) x ) )
25	22, 23, 24	syl2anc 411	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> ( x e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) x ) )
26	22, 25	mpbid 147	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> P ( `' D " RR ) x )
27	20	sselda 3184	. . . . . . . . . 10 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ y e. B ) -> y e. [ P ] ( `' D " RR ) )
28	27	adantrl 478	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> y e. [ P ] ( `' D " RR ) )
29		elecg 6641	. . . . . . . . . 10 |- ( ( y e. [ P ] ( `' D " RR ) /\ P e. X ) -> ( y e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) y ) )
30	28, 23, 29	syl2anc 411	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> ( y e. [ P ] ( `' D " RR ) <-> P ( `' D " RR ) y ) )
31	28, 30	mpbid 147	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> P ( `' D " RR ) y )
32	18, 26, 31	ertr3d 6619	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> x ( `' D " RR ) y )
33	16	xmeterval 14755	. . . . . . . 8 |- ( D e. ( *Met ` X ) -> ( x ( `' D " RR ) y <-> ( x e. X /\ y e. X /\ ( x D y ) e. RR ) ) )
34	15, 33	syl 14	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> ( x ( `' D " RR ) y <-> ( x e. X /\ y e. X /\ ( x D y ) e. RR ) ) )
35	32, 34	mpbid 147	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> ( x e. X /\ y e. X /\ ( x D y ) e. RR ) )
36	35	simp3d 1013	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> ( x D y ) e. RR )
37	14, 36	eqeltrd 2273	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> ( x ( D |` ( B X. B ) ) y ) e. RR )
38	37	ralrimivva 2579	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> A. x e. B A. y e. B ( x ( D |` ( B X. B ) ) y ) e. RR )
39		ffnov 6030	. . 3 |- ( ( D |` ( B X. B ) ) : ( B X. B ) --> RR <-> ( ( D |` ( B X. B ) ) Fn ( B X. B ) /\ A. x e. B A. y e. B ( x ( D |` ( B X. B ) ) y ) e. RR ) )
40	12, 38, 39	sylanbrc 417	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) : ( B X. B ) --> RR )
41		ismet2 14674	. 2 |- ( ( D |` ( B X. B ) ) e. ( Met ` B ) <-> ( ( D |` ( B X. B ) ) e. ( *Met ` B ) /\ ( D |` ( B X. B ) ) : ( B X. B ) --> RR ) )
42	6, 40, 41	sylanbrc 417	1 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) e. ( Met ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   class class class wbr 4034   X. cxp 4662   `'ccnv 4663   |` cres 4666   "cima 4667   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   Er wer 6598   [cec 6599   RRcr 7895   RR*cxr 8077   *Metcxmet 14168   Metcmet 14169   ballcbl 14170

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-er 6601  df-ec 6603  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-2 9066  df-xneg 9864  df-xadd 9865  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178

This theorem is referenced by: (None)