Theorem hhsscms 28136

Description: The induced metric of a closed subspace is complete. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hhssims2.1	|- W = <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >.
hhssims2.3	|- D = ( IndMet ` W )
hhsscms.3	|- H e. CH

Assertion

Ref	Expression
hhsscms	|- D e. ( CMet ` H )

Proof of Theorem hhsscms

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( MetOpen ` D ) = ( MetOpen ` D )
2		hhssims2.1	. . 3 |- W = <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >.
3		hhssims2.3	. . 3 |- D = ( IndMet ` W )
4		hhsscms.3	. . . 4 |- H e. CH
5	4	chshii 28084	. . 3 |- H e. SH
6	2, 3, 5	hhssmet 28134	. 2 |- D e. ( Met ` H )
7		simpl 473	. . . . . . . . . 10 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f e. ( Cau ` D ) )
8	2, 3, 5	hhssims2 28133	. . . . . . . . . . 11 |- D = ( ( normh o. -h ) |` ( H X. H ) )
9	8	fveq2i 6194	. . . . . . . . . 10 |- ( Cau ` D ) = ( Cau ` ( ( normh o. -h ) |` ( H X. H ) ) )
10	7, 9	syl6eleq 2711	. . . . . . . . 9 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f e. ( Cau ` ( ( normh o. -h ) |` ( H X. H ) ) ) )
11		eqid 2622	. . . . . . . . . . 11 |- ( normh o. -h ) = ( normh o. -h )
12	11	hilxmet 28052	. . . . . . . . . 10 |- ( normh o. -h ) e. ( *Met ` ~H )
13		simpr 477	. . . . . . . . . 10 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f : NN --> H )
14		causs 23096	. . . . . . . . . 10 |- ( ( ( normh o. -h ) e. ( *Met ` ~H ) /\ f : NN --> H ) -> ( f e. ( Cau ` ( normh o. -h ) ) <-> f e. ( Cau ` ( ( normh o. -h ) |` ( H X. H ) ) ) ) )
15	12, 13, 14	sylancr 695	. . . . . . . . 9 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> ( f e. ( Cau ` ( normh o. -h ) ) <-> f e. ( Cau ` ( ( normh o. -h ) |` ( H X. H ) ) ) ) )
16	10, 15	mpbird 247	. . . . . . . 8 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f e. ( Cau ` ( normh o. -h ) ) )
17	4	chssii 28088	. . . . . . . . . 10 |- H C_ ~H
18		fss 6056	. . . . . . . . . 10 |- ( ( f : NN --> H /\ H C_ ~H ) -> f : NN --> ~H )
19	13, 17, 18	sylancl 694	. . . . . . . . 9 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f : NN --> ~H )
20		ax-hilex 27856	. . . . . . . . . 10 |- ~H e. _V
21		nnex 11026	. . . . . . . . . 10 |- NN e. _V
22	20, 21	elmap 7886	. . . . . . . . 9 |- ( f e. ( ~H ^m NN ) <-> f : NN --> ~H )
23	19, 22	sylibr 224	. . . . . . . 8 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f e. ( ~H ^m NN ) )
24		eqid 2622	. . . . . . . . . 10 |- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >.
25	24, 11	hhims 28029	. . . . . . . . . 10 |- ( normh o. -h ) = ( IndMet ` <. <. +h , .h >. , normh >. )
26	24, 25	hhcau 28055	. . . . . . . . 9 |- Cauchy = ( ( Cau ` ( normh o. -h ) ) i^i ( ~H ^m NN ) )
27	26	elin2 3801	. . . . . . . 8 |- ( f e. Cauchy <-> ( f e. ( Cau ` ( normh o. -h ) ) /\ f e. ( ~H ^m NN ) ) )
28	16, 23, 27	sylanbrc 698	. . . . . . 7 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f e. Cauchy )
29		ax-hcompl 28059	. . . . . . 7 |- ( f e. Cauchy -> E. x e. ~H f ~~>v x )
30		vex 3203	. . . . . . . . 9 |- f e. _V
31		vex 3203	. . . . . . . . 9 |- x e. _V
32	30, 31	breldm 5329	. . . . . . . 8 |- ( f ~~>v x -> f e. dom ~~>v )
33	32	rexlimivw 3029	. . . . . . 7 |- ( E. x e. ~H f ~~>v x -> f e. dom ~~>v )
34	28, 29, 33	3syl 18	. . . . . 6 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f e. dom ~~>v )
35		hlimf 28094	. . . . . . 7 |- ~~>v : dom ~~>v --> ~H
36		ffun 6048	. . . . . . 7 |- ( ~~>v : dom ~~>v --> ~H -> Fun ~~>v )
37		funfvbrb 6330	. . . . . . 7 |- ( Fun ~~>v -> ( f e. dom ~~>v <-> f ~~>v ( ~~>v ` f ) ) )
38	35, 36, 37	mp2b 10	. . . . . 6 |- ( f e. dom ~~>v <-> f ~~>v ( ~~>v ` f ) )
39	34, 38	sylib 208	. . . . 5 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f ~~>v ( ~~>v ` f ) )
40		eqid 2622	. . . . . . . 8 |- ( MetOpen ` ( normh o. -h ) ) = ( MetOpen ` ( normh o. -h ) )
41	24, 25, 40	hhlm 28056	. . . . . . 7 |- ~~>v = ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) )
42		resss 5422	. . . . . . 7 |- ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) ) C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
43	41, 42	eqsstri 3635	. . . . . 6 |- ~~>v C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
44	43	ssbri 4697	. . . . 5 |- ( f ~~>v ( ~~>v ` f ) -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) ( ~~>v ` f ) )
45	39, 44	syl 17	. . . 4 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) ( ~~>v ` f ) )
46	8, 40, 1	metrest 22329	. . . . . . 7 |- ( ( ( normh o. -h ) e. ( *Met ` ~H ) /\ H C_ ~H ) -> ( ( MetOpen ` ( normh o. -h ) ) ↾t H ) = ( MetOpen ` D ) )
47	12, 17, 46	mp2an 708	. . . . . 6 |- ( ( MetOpen ` ( normh o. -h ) ) ↾t H ) = ( MetOpen ` D )
48	47	eqcomi 2631	. . . . 5 |- ( MetOpen ` D ) = ( ( MetOpen ` ( normh o. -h ) ) ↾t H )
49		nnuz 11723	. . . . 5 |- NN = ( ZZ>= ` 1 )
50	4	a1i 11	. . . . 5 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> H e. CH )
51	40	mopntop 22245	. . . . . 6 |- ( ( normh o. -h ) e. ( *Met ` ~H ) -> ( MetOpen ` ( normh o. -h ) ) e. Top )
52	12, 51	mp1i 13	. . . . 5 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> ( MetOpen ` ( normh o. -h ) ) e. Top )
53		fvex 6201	. . . . . . 7 |- ( ~~>v ` f ) e. _V
54	53	chlimi 28091	. . . . . 6 |- ( ( H e. CH /\ f : NN --> H /\ f ~~>v ( ~~>v ` f ) ) -> ( ~~>v ` f ) e. H )
55	50, 13, 39, 54	syl3anc 1326	. . . . 5 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> ( ~~>v ` f ) e. H )
56		1zzd 11408	. . . . 5 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> 1 e. ZZ )
57	48, 49, 50, 52, 55, 56, 13	lmss 21102	. . . 4 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> ( f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) ( ~~>v ` f ) <-> f ( ~~> t ` ( MetOpen ` D ) ) ( ~~>v ` f ) ) )
58	45, 57	mpbid 222	. . 3 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f ( ~~> t ` ( MetOpen ` D ) ) ( ~~>v ` f ) )
59	30, 53	breldm 5329	. . 3 |- ( f ( ~~> t ` ( MetOpen ` D ) ) ( ~~>v ` f ) -> f e. dom ( ~~> t ` ( MetOpen ` D ) ) )
60	58, 59	syl 17	. 2 |- ( ( f e. ( Cau ` D ) /\ f : NN --> H ) -> f e. dom ( ~~> t ` ( MetOpen ` D ) ) )
61	1, 6, 60	iscmet3i 23110	1 |- D e. ( CMet ` H )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114   |` cres 5116   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   1c1 9937   NNcn 11020   ↾_t crest 16081   *Metcxmt 19731   MetOpencmopn 19736   Topctop 20698   ~~> tclm 21030   Caucca 23051   CMetcms 23052   IndMetcims 27446   ~Hchil 27776   +h cva 27777   .h csm 27778   normhcno 27780   -h cmv 27782   Cauchyccau 27783   ~~>v chli 27784   CHcch 27786

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-inf2 8538  ax-cc 9257  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  ax-addf 10015  ax-mulf 10016  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942  ax-hcompl 28059

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-se 5074  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-isom 5897  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-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  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-n0 11293  df-z 11378  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-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-rest 16083  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-bases 20750  df-ntr 20824  df-nei 20902  df-lm 21033  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-cfil 23053  df-cau 23054  df-cmet 23055  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-ssp 27577  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-hlim 27829  df-hcau 27830  df-sh 28064  df-ch 28078  df-ch0 28110

This theorem is referenced by:  hhssbn  28137