Theorem xmetresbl 14987

Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 14984, 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 1000	. . 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 14968	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X )
4	2, 3	eqsstrid 3243	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ X )
5		xmetres2 14926	. . 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 14897	. . . . . 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 4790	. . . . . 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 5464	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) : ( B X. B ) --> RR* )
12	11	ffnd 5436	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) Fn ( B X. B ) )
13		ovres 6099	. . . . . 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 1003	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> D e. ( *Met ` X ) )
16		eqid 2206	. . . . . . . . . 10 |- ( `' D " RR ) = ( `' D " RR )
17	16	xmeter 14983	. . . . . . . . 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 14985	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ [ P ] ( `' D " RR ) )
20	2, 19	eqsstrid 3243	. . . . . . . . . . 11 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ [ P ] ( `' D " RR ) )
21	20	sselda 3197	. . . . . . . . . 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 1004	. . . . . . . . . 10 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> P e. X )
24		elecg 6673	. . . . . . . . . 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 3197	. . . . . . . . . 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 6673	. . . . . . . . . 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 6651	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> x ( `' D " RR ) y )
33	16	xmeterval 14982	. . . . . . . 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 1014	. . . . 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 2283	. . . 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 2589	. . 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 6062	. . 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 14901	. 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 981   = wceq 1373   e. wcel 2177   A.wral 2485   C_ wss 3170   class class class wbr 4051   X. cxp 4681   `'ccnv 4682   |` cres 4685   "cima 4686   Fn wfn 5275   -->wf 5276   ` cfv 5280  (class class class)co 5957   Er wer 6630   [cec 6631   RRcr 7944   RR*cxr 8126   *Metcxmet 14373   Metcmet 14374   ballcbl 14375

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-po 4351  df-iso 4352  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-er 6633  df-ec 6635  df-map 6750  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-2 9115  df-xneg 9914  df-xadd 9915  df-psmet 14380  df-xmet 14381  df-met 14382  df-bl 14383

This theorem is referenced by: (None)