Theorem xmetresbl 15431

Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 15428, 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 1024	. . 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 15412	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X )
4	2, 3	eqsstrid 3288	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ X )
5		xmetres2 15370	. . 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 15341	. . . . . 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 4862	. . . . . 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 5546	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) : ( B X. B ) --> RR* )
12	11	ffnd 5514	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( D |` ( B X. B ) ) Fn ( B X. B ) )
13		ovres 6202	. . . . . 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 1027	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> D e. ( *Met ` X ) )
16		eqid 2234	. . . . . . . . . 10 |- ( `' D " RR ) = ( `' D " RR )
17	16	xmeter 15427	. . . . . . . . 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 15429	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ [ P ] ( `' D " RR ) )
20	2, 19	eqsstrid 3288	. . . . . . . . . . 11 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> B C_ [ P ] ( `' D " RR ) )
21	20	sselda 3242	. . . . . . . . . 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 1028	. . . . . . . . . 10 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> P e. X )
24		elecg 6820	. . . . . . . . . 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 3242	. . . . . . . . . 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 6820	. . . . . . . . . 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 6798	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( x e. B /\ y e. B ) ) -> x ( `' D " RR ) y )
33	16	xmeterval 15426	. . . . . . . 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 1038	. . . . 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 2311	. . . 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 2626	. . 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 6165	. . 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 15345	. 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 1005   = wceq 1398   e. wcel 2205   A.wral 2522   C_ wss 3214   class class class wbr 4114   X. cxp 4752   `'ccnv 4753   |` cres 4756   "cima 4757   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   Er wer 6777   [cec 6778   RRcr 8142   RR*cxr 8323   *Metcxmet 14810   Metcmet 14811   ballcbl 14812

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-er 6780  df-ec 6782  df-map 6897  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-2 9313  df-xneg 10124  df-xadd 10125  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820

This theorem is referenced by: (None)