Theorem ressxms 18508

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 2404	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
2		eqid 2404	. . . . . 6 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
3	1, 2	xmsxmet 18439	. . . . 5 |- ( K e. * MetSp -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( * Met ` ( Base ` K ) ) )
4	3	adantr 452	. . . 4 |- ( ( K e. * MetSp /\ A e. V ) -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( * Met ` ( Base ` K ) ) )
5		xmetres 18347	. . . 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 4966	. . . . . 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 2424	. . . 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 2404	. . . . . . 7 |- ( K ↾s A ) = ( K ↾s A )
12		eqid 2404	. . . . . . 7 |- ( dist ` K ) = ( dist ` K )
13	11, 12	ressds 13596	. . . . . 6 |- ( A e. V -> ( dist ` K ) = ( dist ` ( K ↾s A ) ) )
14	13	adantl 453	. . . . 5 |- ( ( K e. * MetSp /\ A e. V ) -> ( dist ` K ) = ( dist ` ( K ↾s A ) ) )
15		incom 3493	. . . . . . 7 |- ( ( Base ` K ) i^i A ) = ( A i^i ( Base ` K ) )
16	11, 1	ressbas 13474	. . . . . . . 8 |- ( A e. V -> ( A i^i ( Base ` K ) ) = ( Base ` ( K ↾s A ) ) )
17	16	adantl 453	. . . . . . 7 |- ( ( K e. * MetSp /\ A e. V ) -> ( A i^i ( Base ` K ) ) = ( Base ` ( K ↾s A ) ) )
18	15, 17	syl5eq 2448	. . . . . 6 |- ( ( K e. * MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( Base ` ( K ↾s A ) ) )
19	18, 18	xpeq12d 4862	. . . . 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 2448	. . 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 5691	. . 3 |- ( ( K e. * MetSp /\ A e. V ) -> ( * Met ` ( ( Base ` K ) i^i A ) ) = ( * Met ` ( Base ` ( K ↾s A ) ) ) )
23	6, 21, 22	3eltr3d 2484	. 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 2404	. . . . . . 7 |- ( TopOpen ` K ) = ( TopOpen ` K )
25	24, 1, 2	xmstopn 18434	. . . . . 6 |- ( K e. * MetSp -> ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) )
26	25	adantr 452	. . . . 5 |- ( ( K e. * MetSp /\ A e. V ) -> ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) )
27	26	oveq1d 6055	. . . 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 3521	. . . . 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 654	. . . . . . . 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 8	. . . . . . 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 2427	. . . . . 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 2404	. . . . . 6 |- ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
35		eqid 2404	. . . . . 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 18507	. . . . 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 644	. . . 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 2436	. . 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 18436	. . . . . . . . 9 |- ( K e. * MetSp -> K e. TopSp )
40	1, 24	tpsuni 16958	. . . . . . . . 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 452	. . . . . . 7 |- ( ( K e. * MetSp /\ A e. V ) -> ( Base ` K ) = U. ( TopOpen ` K ) )
43	42	ineq2d 3502	. . . . . 6 |- ( ( K e. * MetSp /\ A e. V ) -> ( A i^i ( Base ` K ) ) = ( A i^i U. ( TopOpen ` K ) ) )
44	15, 43	syl5eq 2448	. . . . 5 |- ( ( K e. * MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( A i^i U. ( TopOpen ` K ) ) )
45	44	oveq2d 6056	. . . 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 16956	. . . . . 6 |- ( K e. TopSp <-> ( TopOpen ` K ) e. (TopOn ` ( Base ` K ) ) )
47	39, 46	sylib 189	. . . . 5 |- ( K e. * MetSp -> ( TopOpen ` K ) e. (TopOn ` ( Base ` K ) ) )
48		eqid 2404	. . . . . 6 |- U. ( TopOpen ` K ) = U. ( TopOpen ` K )
49	48	restin 17184	. . . . 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 458	. . . 4 |- ( ( K e. * MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t A ) = ( ( TopOpen ` K ) ↾t ( A i^i U. ( TopOpen ` K ) ) ) )
51	45, 50	eqtr4d 2439	. . 3 |- ( ( K e. * MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t ( ( Base ` K ) i^i A ) ) = ( ( TopOpen ` K ) ↾t A ) )
52	21	fveq2d 5691	. . 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 2444	. 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 17204	. . 3 |- ( ( TopOpen ` K ) ↾t A ) = ( TopOpen ` ( K ↾s A ) )
55		eqid 2404	. . 3 |- ( Base ` ( K ↾s A ) ) = ( Base ` ( K ↾s A ) )
56		eqid 2404	. . 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 18431	. 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 646	1 |- ( ( K e. * MetSp /\ A e. V ) -> ( K ↾s A ) e. * MetSp )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   i^i cin 3279   C_ wss 3280   U.cuni 3975   X. cxp 4835   |` cres 4839   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾_s cress 13425   distcds 13493   ↾_t crest 13603   TopOpenctopn 13604   * Metcxmt 16641   MetOpencmopn 16646  TopOnctopon 16914   TopSpctps 16916   * MetSpcxme 18300

This theorem is referenced by:  ressms  18509  qqhcn  24328  qqhucn  24329  dya2icoseg2  24581

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-tset 13503  df-ds 13506  df-rest 13605  df-topn 13606  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-xms 18303