Theorem xmetresbl 14676

Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 14673, 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 14657	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X )
4	2, 3	eqsstrid 3229	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ X )
5		xmetres2 14615	. . 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 14586	. . . . . 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 4770	. . . . . 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 5434	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) : ( B X. B ) --> RR* )
12	11	ffnd 5408	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) Fn ( B X. B ) )
13		ovres 6063	. . . . . 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 14672	. . . . . . . . 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 14674	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ [ P ] ( `' D " RR ) )
20	2, 19	eqsstrid 3229	. . . . . . . . . . 11 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ [ P ] ( `' D " RR ) )
21	20	sselda 3183	. . . . . . . . . 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 6632	. . . . . . . . . 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 3183	. . . . . . . . . 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 6632	. . . . . . . . . 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 6610	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> x ( `' D " RR ) y )
33	16	xmeterval 14671	. . . . . . . 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 6026	. . 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 14590	. 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 4033   X. cxp 4661   `'ccnv 4662   |` cres 4665   "cima 4666   Fn wfn 5253   -->wf 5254   ` cfv 5258  (class class class)co 5922   Er wer 6589   [cec 6590   RRcr 7878   RR*cxr 8060   *Metcxmet 14092   Metcmet 14093   ballcbl 14094

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996

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 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-er 6592  df-ec 6594  df-map 6709  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-2 9049  df-xneg 9847  df-xadd 9848  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102

This theorem is referenced by: (None)