Theorem ressxms 20791

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 2467	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
2		eqid 2467	. . . . . 6 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
3	1, 2	xmsxmet 20722	. . . . 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 20630	. . . 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 5286	. . . . 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 5135	. . . . . 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 5270	. . . . 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 2496	. . . 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 2467	. . . . . . 7 |- ( K ↾s A ) = ( K ↾s A )
12		eqid 2467	. . . . . . 7 |- ( dist ` K ) = ( dist ` K )
13	11, 12	ressds 14669	. . . . . 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 3691	. . . . . . 7 |- ( ( Base ` K ) i^i A ) = ( A i^i ( Base ` K ) )
16	11, 1	ressbas 14545	. . . . . . . 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 2520	. . . . . 6 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( Base ` ( K ↾s A ) ) )
19	18, 18	xpeq12d 5024	. . . . 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 5274	. . . 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 2520	. . 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 5870	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( *Met ` ( ( Base ` K ) i^i A ) ) = ( *Met ` ( Base ` ( K ↾s A ) ) ) )
23	6, 21, 22	3eltr3d 2569	. 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 2467	. . . . . . 7 |- ( TopOpen ` K ) = ( TopOpen ` K )
25	24, 1, 2	xmstopn 20717	. . . . . 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 6299	. . . 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 3718	. . . . 5 |- ( ( Base ` K ) i^i A ) C_ ( Base ` K )
29		xpss12 5108	. . . . . . . . 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 5302	. . . . . . . 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 2499	. . . . . 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 2467	. . . . . 6 |- ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
35		eqid 2467	. . . . . 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 20790	. . . . 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 2508	. . 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 20719	. . . . . . . . 9 |- ( K e. *MetSp -> K e. TopSp )
40	1, 24	tpsuni 19234	. . . . . . . . 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 3700	. . . . . 6 |- ( ( K e. *MetSp /\ A e. V ) -> ( A i^i ( Base ` K ) ) = ( A i^i U. ( TopOpen ` K ) ) )
44	15, 43	syl5eq 2520	. . . . 5 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( A i^i U. ( TopOpen ` K ) ) )
45	44	oveq2d 6300	. . . 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 19232	. . . . . 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 2467	. . . . . 6 |- U. ( TopOpen ` K ) = U. ( TopOpen ` K )
49	48	restin 19461	. . . . 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 2511	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t ( ( Base ` K ) i^i A ) ) = ( ( TopOpen ` K ) ↾t A ) )
52	21	fveq2d 5870	. . 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 2516	. 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 19481	. . 3 |- ( ( TopOpen ` K ) ↾t A ) = ( TopOpen ` ( K ↾s A ) )
55		eqid 2467	. . 3 |- ( Base ` ( K ↾s A ) ) = ( Base ` ( K ↾s A ) )
56		eqid 2467	. . 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 20714	. 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 1379   e. wcel 1767   i^i cin 3475   C_ wss 3476   U.cuni 4245   X. cxp 4997   |` cres 5001   ` cfv 5588  (class class class)co 6284   Basecbs 14490   ↾_s cress 14491   distcds 14564   ↾_t crest 14676   TopOpenctopn 14677   *Metcxmt 18202   MetOpencmopn 18207  TopOnctopon 19190   TopSpctps 19192   *MetSpcxme 20583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-tset 14574  df-ds 14577  df-rest 14678  df-topn 14679  df-topgen 14699  df-psmet 18210  df-xmet 18211  df-bl 18213  df-mopn 18214  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-xms 20586

This theorem is referenced by:  ressms  20792  qqhcn  27636  qqhucn  27637