Theorem ressxms 20080

Description: The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
ressxms	|- ( ( K e. *MetSp /\ A e. V ) -> ( K ↾s A ) e. *MetSp )

Proof of Theorem ressxms

Step	Hyp	Ref	Expression
1		eqid 2438	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
2		eqid 2438	. . . . . 6 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
3	1, 2	xmsxmet 20011	. . . . 5 |- ( K e. *MetSp -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( *Met ` ( Base ` K ) ) )
4	3	adantr 465	. . . 4 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( *Met ` ( Base ` K ) ) )
5		xmetres 19919	. . . 4 |- ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( *Met ` ( Base ` K ) ) -> ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) e. ( *Met ` ( ( Base ` K ) i^i A ) ) )
6	4, 5	syl 16	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) e. ( *Met ` ( ( Base ` K ) i^i A ) ) )
7		resres 5118	. . . . 5 |- ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) = ( ( dist ` K ) |` ( ( ( Base ` K ) X. ( Base ` K ) ) i^i ( A X. A ) ) )
8		inxp 4967	. . . . . 6 |- ( ( ( Base ` K ) X. ( Base ` K ) ) i^i ( A X. A ) ) = ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) )
9	8	reseq2i 5102	. . . . 5 |- ( ( dist ` K ) |` ( ( ( Base ` K ) X. ( Base ` K ) ) i^i ( A X. A ) ) ) = ( ( dist ` K ) |` ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) )
10	7, 9	eqtri 2458	. . . 4 |- ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) = ( ( dist ` K ) |` ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) )
11		eqid 2438	. . . . . . 7 |- ( K ↾s A ) = ( K ↾s A )
12		eqid 2438	. . . . . . 7 |- ( dist ` K ) = ( dist ` K )
13	11, 12	ressds 14344	. . . . . 6 |- ( A e. V -> ( dist ` K ) = ( dist ` ( K ↾s A ) ) )
14	13	adantl 466	. . . . 5 |- ( ( K e. *MetSp /\ A e. V ) -> ( dist ` K ) = ( dist ` ( K ↾s A ) ) )
15		incom 3538	. . . . . . 7 |- ( ( Base ` K ) i^i A ) = ( A i^i ( Base ` K ) )
16	11, 1	ressbas 14220	. . . . . . . 8 |- ( A e. V -> ( A i^i ( Base ` K ) ) = ( Base ` ( K ↾s A ) ) )
17	16	adantl 466	. . . . . . 7 |- ( ( K e. *MetSp /\ A e. V ) -> ( A i^i ( Base ` K ) ) = ( Base ` ( K ↾s A ) ) )
18	15, 17	syl5eq 2482	. . . . . 6 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( Base ` ( K ↾s A ) ) )
19	18, 18	xpeq12d 4860	. . . . 5 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) = ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) )
20	14, 19	reseq12d 5106	. . . 4 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( dist ` K ) |` ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) ) = ( ( dist ` ( K ↾s A ) ) |` ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) ) )
21	10, 20	syl5eq 2482	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) = ( ( dist ` ( K ↾s A ) ) |` ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) ) )
22	18	fveq2d 5690	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( *Met ` ( ( Base ` K ) i^i A ) ) = ( *Met ` ( Base ` ( K ↾s A ) ) ) )
23	6, 21, 22	3eltr3d 2518	. 2 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( dist ` ( K ↾s A ) ) |` ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) ) e. ( *Met ` ( Base ` ( K ↾s A ) ) ) )
24		eqid 2438	. . . . . . 7 |- ( TopOpen ` K ) = ( TopOpen ` K )
25	24, 1, 2	xmstopn 20006	. . . . . 6 |- ( K e. *MetSp -> ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) )
26	25	adantr 465	. . . . 5 |- ( ( K e. *MetSp /\ A e. V ) -> ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) )
27	26	oveq1d 6101	. . . 4 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t ( ( Base ` K ) i^i A ) ) = ( ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) ↾t ( ( Base ` K ) i^i A ) ) )
28		inss1 3565	. . . . 5 |- ( ( Base ` K ) i^i A ) C_ ( Base ` K )
29		xpss12 4940	. . . . . . . . 9 |- ( ( ( ( Base ` K ) i^i A ) C_ ( Base ` K ) /\ ( ( Base ` K ) i^i A ) C_ ( Base ` K ) ) -> ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) C_ ( ( Base ` K ) X. ( Base ` K ) ) )
30	28, 28, 29	mp2an 672	. . . . . . . 8 |- ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) C_ ( ( Base ` K ) X. ( Base ` K ) )
31		resabs1 5134	. . . . . . . 8 |- ( ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) C_ ( ( Base ` K ) X. ( Base ` K ) ) -> ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) ) = ( ( dist ` K ) |` ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) ) )
32	30, 31	ax-mp 5	. . . . . . 7 |- ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) ) = ( ( dist ` K ) |` ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) )
33	10, 32	eqtr4i 2461	. . . . . 6 |- ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) = ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( ( ( Base ` K ) i^i A ) X. ( ( Base ` K ) i^i A ) ) )
34		eqid 2438	. . . . . 6 |- ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
35		eqid 2438	. . . . . 6 |- ( MetOpen ` ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) ) = ( MetOpen ` ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) )
36	33, 34, 35	metrest 20079	. . . . 5 |- ( ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( *Met ` ( Base ` K ) ) /\ ( ( Base ` K ) i^i A ) C_ ( Base ` K ) ) -> ( ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) ↾t ( ( Base ` K ) i^i A ) ) = ( MetOpen ` ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) ) )
37	4, 28, 36	sylancl 662	. . . 4 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) ↾t ( ( Base ` K ) i^i A ) ) = ( MetOpen ` ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) ) )
38	27, 37	eqtrd 2470	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t ( ( Base ` K ) i^i A ) ) = ( MetOpen ` ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) ) )
39		xmstps 20008	. . . . . . . . 9 |- ( K e. *MetSp -> K e. TopSp )
40	1, 24	tpsuni 18523	. . . . . . . . 9 |- ( K e. TopSp -> ( Base ` K ) = U. ( TopOpen ` K ) )
41	39, 40	syl 16	. . . . . . . 8 |- ( K e. *MetSp -> ( Base ` K ) = U. ( TopOpen ` K ) )
42	41	adantr 465	. . . . . . 7 |- ( ( K e. *MetSp /\ A e. V ) -> ( Base ` K ) = U. ( TopOpen ` K ) )
43	42	ineq2d 3547	. . . . . 6 |- ( ( K e. *MetSp /\ A e. V ) -> ( A i^i ( Base ` K ) ) = ( A i^i U. ( TopOpen ` K ) ) )
44	15, 43	syl5eq 2482	. . . . 5 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( A i^i U. ( TopOpen ` K ) ) )
45	44	oveq2d 6102	. . . 4 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t ( ( Base ` K ) i^i A ) ) = ( ( TopOpen ` K ) ↾t ( A i^i U. ( TopOpen ` K ) ) ) )
46	1, 24	istps 18521	. . . . . 6 |- ( K e. TopSp <-> ( TopOpen ` K ) e. (TopOn ` ( Base ` K ) ) )
47	39, 46	sylib 196	. . . . 5 |- ( K e. *MetSp -> ( TopOpen ` K ) e. (TopOn ` ( Base ` K ) ) )
48		eqid 2438	. . . . . 6 |- U. ( TopOpen ` K ) = U. ( TopOpen ` K )
49	48	restin 18750	. . . . 5 |- ( ( ( TopOpen ` K ) e. (TopOn ` ( Base ` K ) ) /\ A e. V ) -> ( ( TopOpen ` K ) ↾t A ) = ( ( TopOpen ` K ) ↾t ( A i^i U. ( TopOpen ` K ) ) ) )
50	47, 49	sylan 471	. . . 4 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t A ) = ( ( TopOpen ` K ) ↾t ( A i^i U. ( TopOpen ` K ) ) ) )
51	45, 50	eqtr4d 2473	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t ( ( Base ` K ) i^i A ) ) = ( ( TopOpen ` K ) ↾t A ) )
52	21	fveq2d 5690	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( MetOpen ` ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |` ( A X. A ) ) ) = ( MetOpen ` ( ( dist ` ( K ↾s A ) ) |` ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) ) ) )
53	38, 51, 52	3eqtr3d 2478	. 2 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t A ) = ( MetOpen ` ( ( dist ` ( K ↾s A ) ) |` ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) ) ) )
54	11, 24	resstopn 18770	. . 3 |- ( ( TopOpen ` K ) ↾t A ) = ( TopOpen ` ( K ↾s A ) )
55		eqid 2438	. . 3 |- ( Base ` ( K ↾s A ) ) = ( Base ` ( K ↾s A ) )
56		eqid 2438	. . 3 |- ( ( dist ` ( K ↾s A ) ) |` ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) ) = ( ( dist ` ( K ↾s A ) ) |` ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) )
57	54, 55, 56	isxms2 20003	. 2 |- ( ( K ↾s A ) e. *MetSp <-> ( ( ( dist ` ( K ↾s A ) ) |` ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) ) e. ( *Met ` ( Base ` ( K ↾s A ) ) ) /\ ( ( TopOpen ` K ) ↾t A ) = ( MetOpen ` ( ( dist ` ( K ↾s A ) ) |` ( ( Base ` ( K ↾s A ) ) X. ( Base ` ( K ↾s A ) ) ) ) ) ) )
58	23, 53, 57	sylanbrc 664	1 |- ( ( K e. *MetSp /\ A e. V ) -> ( K ↾s A ) e. *MetSp )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   i^i cin 3322   C_ wss 3323   U.cuni 4086   X. cxp 4833   |` cres 4837   ` cfv 5413  (class class class)co 6086   Basecbs 14166   ↾_s cress 14167   distcds 14239   ↾_t crest 14351   TopOpenctopn 14352   *Metcxmt 17781   MetOpencmopn 17786  TopOnctopon 18479   TopSpctps 18481   *MetSpcxme 19872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-tset 14249  df-ds 14252  df-rest 14353  df-topn 14354  df-topgen 14374  df-psmet 17789  df-xmet 17790  df-bl 17792  df-mopn 17793  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-xms 19875

This theorem is referenced by:  ressms  20081  qqhcn  26393  qqhucn  26394