Theorem cmsss 23147

Description: The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

Ref	Expression
cmsss.h	|- K = ( M ↾s A )
cmsss.x	|- X = ( Base ` M )
cmsss.j	|- J = ( TopOpen ` M )

Assertion

Ref	Expression
cmsss	|- ( ( M e. CMetSp /\ A C_ X ) -> ( K e. CMetSp <-> A e. ( Clsd ` J ) ) )

Proof of Theorem cmsss

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . 7 |- ( ( M e. CMetSp /\ A C_ X ) -> A C_ X )
2		xpss12 5225	. . . . . . 7 |- ( ( A C_ X /\ A C_ X ) -> ( A X. A ) C_ ( X X. X ) )
3	1, 2	sylancom 701	. . . . . 6 |- ( ( M e. CMetSp /\ A C_ X ) -> ( A X. A ) C_ ( X X. X ) )
4	3	resabs1d 5428	. . . . 5 |- ( ( M e. CMetSp /\ A C_ X ) -> ( ( ( dist ` M ) |` ( X X. X ) ) |` ( A X. A ) ) = ( ( dist ` M ) |` ( A X. A ) ) )
5		cmsss.x	. . . . . . . . . 10 |- X = ( Base ` M )
6		fvex 6201	. . . . . . . . . 10 |- ( Base ` M ) e. _V
7	5, 6	eqeltri 2697	. . . . . . . . 9 |- X e. _V
8	7	ssex 4802	. . . . . . . 8 |- ( A C_ X -> A e. _V )
9	8	adantl 482	. . . . . . 7 |- ( ( M e. CMetSp /\ A C_ X ) -> A e. _V )
10		cmsss.h	. . . . . . . 8 |- K = ( M ↾s A )
11		eqid 2622	. . . . . . . 8 |- ( dist ` M ) = ( dist ` M )
12	10, 11	ressds 16073	. . . . . . 7 |- ( A e. _V -> ( dist ` M ) = ( dist ` K ) )
13	9, 12	syl 17	. . . . . 6 |- ( ( M e. CMetSp /\ A C_ X ) -> ( dist ` M ) = ( dist ` K ) )
14	10, 5	ressbas2 15931	. . . . . . . 8 |- ( A C_ X -> A = ( Base ` K ) )
15	14	adantl 482	. . . . . . 7 |- ( ( M e. CMetSp /\ A C_ X ) -> A = ( Base ` K ) )
16	15	sqxpeqd 5141	. . . . . 6 |- ( ( M e. CMetSp /\ A C_ X ) -> ( A X. A ) = ( ( Base ` K ) X. ( Base ` K ) ) )
17	13, 16	reseq12d 5397	. . . . 5 |- ( ( M e. CMetSp /\ A C_ X ) -> ( ( dist ` M ) |` ( A X. A ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
18	4, 17	eqtrd 2656	. . . 4 |- ( ( M e. CMetSp /\ A C_ X ) -> ( ( ( dist ` M ) |` ( X X. X ) ) |` ( A X. A ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
19	15	fveq2d 6195	. . . 4 |- ( ( M e. CMetSp /\ A C_ X ) -> ( CMet ` A ) = ( CMet ` ( Base ` K ) ) )
20	18, 19	eleq12d 2695	. . 3 |- ( ( M e. CMetSp /\ A C_ X ) -> ( ( ( ( dist ` M ) |` ( X X. X ) ) |` ( A X. A ) ) e. ( CMet ` A ) <-> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( CMet ` ( Base ` K ) ) ) )
21		eqid 2622	. . . . . 6 |- ( ( dist ` M ) |` ( X X. X ) ) = ( ( dist ` M ) |` ( X X. X ) )
22	5, 21	cmscmet 23143	. . . . 5 |- ( M e. CMetSp -> ( ( dist ` M ) |` ( X X. X ) ) e. ( CMet ` X ) )
23	22	adantr 481	. . . 4 |- ( ( M e. CMetSp /\ A C_ X ) -> ( ( dist ` M ) |` ( X X. X ) ) e. ( CMet ` X ) )
24		eqid 2622	. . . . 5 |- ( MetOpen ` ( ( dist ` M ) |` ( X X. X ) ) ) = ( MetOpen ` ( ( dist ` M ) |` ( X X. X ) ) )
25	24	cmetss 23113	. . . 4 |- ( ( ( dist ` M ) |` ( X X. X ) ) e. ( CMet ` X ) -> ( ( ( ( dist ` M ) |` ( X X. X ) ) |` ( A X. A ) ) e. ( CMet ` A ) <-> A e. ( Clsd ` ( MetOpen ` ( ( dist ` M ) |` ( X X. X ) ) ) ) ) )
26	23, 25	syl 17	. . 3 |- ( ( M e. CMetSp /\ A C_ X ) -> ( ( ( ( dist ` M ) |` ( X X. X ) ) |` ( A X. A ) ) e. ( CMet ` A ) <-> A e. ( Clsd ` ( MetOpen ` ( ( dist ` M ) |` ( X X. X ) ) ) ) ) )
27	20, 26	bitr3d 270	. 2 |- ( ( M e. CMetSp /\ A C_ X ) -> ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( CMet ` ( Base ` K ) ) <-> A e. ( Clsd ` ( MetOpen ` ( ( dist ` M ) |` ( X X. X ) ) ) ) ) )
28		cmsms 23145	. . . 4 |- ( M e. CMetSp -> M e. MetSp )
29		ressms 22331	. . . . 5 |- ( ( M e. MetSp /\ A e. _V ) -> ( M ↾s A ) e. MetSp )
30	10, 29	syl5eqel 2705	. . . 4 |- ( ( M e. MetSp /\ A e. _V ) -> K e. MetSp )
31	28, 8, 30	syl2an 494	. . 3 |- ( ( M e. CMetSp /\ A C_ X ) -> K e. MetSp )
32		eqid 2622	. . . . 5 |- ( Base ` K ) = ( Base ` K )
33		eqid 2622	. . . . 5 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
34	32, 33	iscms 23142	. . . 4 |- ( K e. CMetSp <-> ( K e. MetSp /\ ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( CMet ` ( Base ` K ) ) ) )
35	34	baib 944	. . 3 |- ( K e. MetSp -> ( K e. CMetSp <-> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( CMet ` ( Base ` K ) ) ) )
36	31, 35	syl 17	. 2 |- ( ( M e. CMetSp /\ A C_ X ) -> ( K e. CMetSp <-> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( CMet ` ( Base ` K ) ) ) )
37	28	adantr 481	. . . . 5 |- ( ( M e. CMetSp /\ A C_ X ) -> M e. MetSp )
38		cmsss.j	. . . . . 6 |- J = ( TopOpen ` M )
39	38, 5, 21	mstopn 22257	. . . . 5 |- ( M e. MetSp -> J = ( MetOpen ` ( ( dist ` M ) |` ( X X. X ) ) ) )
40	37, 39	syl 17	. . . 4 |- ( ( M e. CMetSp /\ A C_ X ) -> J = ( MetOpen ` ( ( dist ` M ) |` ( X X. X ) ) ) )
41	40	fveq2d 6195	. . 3 |- ( ( M e. CMetSp /\ A C_ X ) -> ( Clsd ` J ) = ( Clsd ` ( MetOpen ` ( ( dist ` M ) |` ( X X. X ) ) ) ) )
42	41	eleq2d 2687	. 2 |- ( ( M e. CMetSp /\ A C_ X ) -> ( A e. ( Clsd ` J ) <-> A e. ( Clsd ` ( MetOpen ` ( ( dist ` M ) |` ( X X. X ) ) ) ) ) )
43	27, 36, 42	3bitr4d 300	1 |- ( ( M e. CMetSp /\ A C_ X ) -> ( K e. CMetSp <-> A e. ( Clsd ` J ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   distcds 15950   TopOpenctopn 16082   MetOpencmopn 19736   Clsdccld 20820   MetSpcmt 22123   CMetcms 23052  CMetSpccms 23129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-tset 15960  df-ds 15964  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-haus 21119  df-fil 21650  df-flim 21743  df-xms 22125  df-ms 22126  df-cfil 23053  df-cmet 23055  df-cms 23132

This theorem is referenced by:  lssbn  23148  resscdrg  23154  srabn  23156  ishl2  23166  recms  23168  pjthlem2  23209