Theorem ressxms 21154

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 2457	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
2		eqid 2457	. . . . . 6 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
3	1, 2	xmsxmet 21085	. . . . 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 20993	. . . 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 5296	. . . . 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 5145	. . . . . 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 5280	. . . . 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 2486	. . . 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 2457	. . . . . . 7 |- ( K ↾s A ) = ( K ↾s A )
12		eqid 2457	. . . . . . 7 |- ( dist ` K ) = ( dist ` K )
13	11, 12	ressds 14830	. . . . . 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 3687	. . . . . . 7 |- ( ( Base ` K ) i^i A ) = ( A i^i ( Base ` K ) )
16	11, 1	ressbas 14701	. . . . . . . 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 2510	. . . . . 6 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( Base ` ( K ↾s A ) ) )
19	18	sqxpeqd 5034	. . . . 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 5284	. . . 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 2510	. . 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 5876	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( *Met ` ( ( Base ` K ) i^i A ) ) = ( *Met ` ( Base ` ( K ↾s A ) ) ) )
23	6, 21, 22	3eltr3d 2559	. 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 2457	. . . . . . 7 |- ( TopOpen ` K ) = ( TopOpen ` K )
25	24, 1, 2	xmstopn 21080	. . . . . 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 6311	. . . 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 3714	. . . . 5 |- ( ( Base ` K ) i^i A ) C_ ( Base ` K )
29		xpss12 5117	. . . . . . . . 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 5312	. . . . . . . 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 2489	. . . . . 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 2457	. . . . . 6 |- ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
35		eqid 2457	. . . . . 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 21153	. . . . 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 2498	. . 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 21082	. . . . . . . . 9 |- ( K e. *MetSp -> K e. TopSp )
40	1, 24	tpsuni 19566	. . . . . . . . 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 3696	. . . . . 6 |- ( ( K e. *MetSp /\ A e. V ) -> ( A i^i ( Base ` K ) ) = ( A i^i U. ( TopOpen ` K ) ) )
44	15, 43	syl5eq 2510	. . . . 5 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( Base ` K ) i^i A ) = ( A i^i U. ( TopOpen ` K ) ) )
45	44	oveq2d 6312	. . . 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 19564	. . . . . 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 2457	. . . . . 6 |- U. ( TopOpen ` K ) = U. ( TopOpen ` K )
49	48	restin 19794	. . . . 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 2501	. . 3 |- ( ( K e. *MetSp /\ A e. V ) -> ( ( TopOpen ` K ) ↾t ( ( Base ` K ) i^i A ) ) = ( ( TopOpen ` K ) ↾t A ) )
52	21	fveq2d 5876	. . 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 2506	. 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 19814	. . 3 |- ( ( TopOpen ` K ) ↾t A ) = ( TopOpen ` ( K ↾s A ) )
55		eqid 2457	. . 3 |- ( Base ` ( K ↾s A ) ) = ( Base ` ( K ↾s A ) )
56		eqid 2457	. . 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 21077	. 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 1395   e. wcel 1819   i^i cin 3470   C_ wss 3471   U.cuni 4251   X. cxp 5006   |` cres 5010   ` cfv 5594  (class class class)co 6296   Basecbs 14644   ↾_s cress 14645   distcds 14721   ↾_t crest 14838   TopOpenctopn 14839   *Metcxmt 18530   MetOpencmopn 18535  TopOnctopon 19522   TopSpctps 19524   *MetSpcxme 20946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-tset 14731  df-ds 14734  df-rest 14840  df-topn 14841  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-xms 20949

This theorem is referenced by:  ressms  21155  qqhcn  28133  qqhucn  28134