Theorem ressxms 20231

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 2454	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
2		eqid 2454	. . . . . 6 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
3	1, 2	xmsxmet 20162	. . . . 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 20070	. . . 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 5230	. . . . 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 5079	. . . . . 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 5214	. . . . 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 2483	. . . 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 2454	. . . . . . 7 |- ( K ↾s A ) = ( K ↾s A )
12		eqid 2454	. . . . . . 7 |- ( dist ` K ) = ( dist ` K )
13	11, 12	ressds 14470	. . . . . 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 3650	. . . . . . 7 |- ( ( Base ` K ) i^i A ) = ( A i^i ( Base ` K ) )
16	11, 1	ressbas 14346	. . . . . . . 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 2507	. . . . . 6 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( Base ` ( K ↾s A ) ) )
19	18, 18	xpeq12d 4972	. . . . 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 5218	. . . 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 2507	. . 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 5802	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( *Met ` ( ( Base ` K ) i^i A ) ) = ( *Met ` ( Base ` ( K ↾s A ) ) ) )
23	6, 21, 22	3eltr3d 2556	. 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 2454	. . . . . . 7 |- ( TopOpen ` K ) = ( TopOpen ` K )
25	24, 1, 2	xmstopn 20157	. . . . . 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 6214	. . . 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 3677	. . . . 5 |- ( ( Base ` K ) i^i A ) C_ ( Base ` K )
29		xpss12 5052	. . . . . . . . 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 5246	. . . . . . . 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 2486	. . . . . 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 2454	. . . . . 6 |- ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
35		eqid 2454	. . . . . 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 20230	. . . . 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 2495	. . 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 20159	. . . . . . . . 9 |- ( K e. *MetSp -> K e. TopSp )
40	1, 24	tpsuni 18674	. . . . . . . . 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 3659	. . . . . 6 |- ( ( K e. *MetSp /\ A e. V ) -> ( A i^i ( Base ` K ) ) = ( A i^i U. ( TopOpen ` K ) ) )
44	15, 43	syl5eq 2507	. . . . 5 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( A i^i U. ( TopOpen ` K ) ) )
45	44	oveq2d 6215	. . . 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 18672	. . . . . 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 2454	. . . . . 6 |- U. ( TopOpen ` K ) = U. ( TopOpen ` K )
49	48	restin 18901	. . . . 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 2498	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t ( ( Base ` K ) i^i A ) ) = ( ( TopOpen ` K ) ↾t A ) )
52	21	fveq2d 5802	. . 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 2503	. 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 18921	. . 3 |- ( ( TopOpen ` K ) ↾t A ) = ( TopOpen ` ( K ↾s A ) )
55		eqid 2454	. . 3 |- ( Base ` ( K ↾s A ) ) = ( Base ` ( K ↾s A ) )
56		eqid 2454	. . 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 20154	. 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 1370   e. wcel 1758   i^i cin 3434   C_ wss 3435   U.cuni 4198   X. cxp 4945   |` cres 4949   ` cfv 5525  (class class class)co 6199   Basecbs 14291   ↾_s cress 14292   distcds 14365   ↾_t crest 14477   TopOpenctopn 14478   *Metcxmt 17925   MetOpencmopn 17930  TopOnctopon 18630   TopSpctps 18632   *MetSpcxme 20023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-tset 14375  df-ds 14378  df-rest 14479  df-topn 14480  df-topgen 14500  df-psmet 17933  df-xmet 17934  df-bl 17936  df-mopn 17937  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-xms 20026

This theorem is referenced by:  ressms  20232  qqhcn  26564  qqhucn  26565