Theorem xmetresbl 13943

Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 13940, 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 997	. . 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 13924	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X )
4	2, 3	eqsstrid 3202	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ X )
5		xmetres2 13882	. . 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 13853	. . . . . 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 4734	. . . . . 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 5393	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) : ( B X. B ) --> RR* )
12	11	ffnd 5367	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) Fn ( B X. B ) )
13		ovres 6014	. . . . . 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 1000	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> D e. ( *Met ` X ) )
16		eqid 2177	. . . . . . . . . 10 |- ( `' D " RR ) = ( `' D " RR )
17	16	xmeter 13939	. . . . . . . . 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 13941	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ [ P ] ( `' D " RR ) )
20	2, 19	eqsstrid 3202	. . . . . . . . . . 11 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ [ P ] ( `' D " RR ) )
21	20	sselda 3156	. . . . . . . . . 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 1001	. . . . . . . . . 10 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> P e. X )
24		elecg 6573	. . . . . . . . . 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 3156	. . . . . . . . . 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 6573	. . . . . . . . . 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 6553	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> x ( `' D " RR ) y )
33	16	xmeterval 13938	. . . . . . . 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 1011	. . . . 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 2254	. . . 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 2559	. . 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 5979	. . 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 13857	. 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 978   = wceq 1353   e. wcel 2148   A.wral 2455   C_ wss 3130   class class class wbr 4004   X. cxp 4625   `'ccnv 4626   |` cres 4629   "cima 4630   Fn wfn 5212   -->wf 5213   ` cfv 5217  (class class class)co 5875   Er wer 6532   [cec 6533   RRcr 7810   RR*cxr 7991   *Metcxmet 13443   Metcmet 13444   ballcbl 13445

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-po 4297  df-iso 4298  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-er 6535  df-ec 6537  df-map 6650  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-2 8978  df-xneg 9772  df-xadd 9773  df-psmet 13450  df-xmet 13451  df-met 13452  df-bl 13453

This theorem is referenced by: (None)