Theorem iscau 23074

Description: Express the property " F is a Cauchy sequence of metric D." Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition F C_ ( CC X. X ) allows us to use objects more general than sequences when convenient; see the comment in df-lm 21033. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
iscau	|- ( D e. ( *Met ` X ) -> ( F e. ( Cau ` D ) <-> ( F e. ( X ^pm CC ) /\ A. x e. RR+ E. k e. ZZ ( F |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( F ` k ) ( ball ` D ) x ) ) ) )

Distinct variable groups:   x, k, D   k, F, x   k, X, x

Proof of Theorem iscau

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caufval 23073	. . 3 |- ( D e. ( *Met ` X ) -> ( Cau ` D ) = { f e. ( X ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) } )
2	1	eleq2d 2687	. 2 |- ( D e. ( *Met ` X ) -> ( F e. ( Cau ` D ) <-> F e. { f e. ( X ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) } ) )
3		reseq1 5390	. . . . . 6 |- ( f = F -> ( f |` ( ZZ>= ` k ) ) = ( F |` ( ZZ>= ` k ) ) )
4		eqidd 2623	. . . . . 6 |- ( f = F -> ( ZZ>= ` k ) = ( ZZ>= ` k ) )
5		fveq1 6190	. . . . . . 7 |- ( f = F -> ( f ` k ) = ( F ` k ) )
6	5	oveq1d 6665	. . . . . 6 |- ( f = F -> ( ( f ` k ) ( ball ` D ) x ) = ( ( F ` k ) ( ball ` D ) x ) )
7	3, 4, 6	feq123d 6034	. . . . 5 |- ( f = F -> ( ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) <-> ( F |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( F ` k ) ( ball ` D ) x ) ) )
8	7	rexbidv 3052	. . . 4 |- ( f = F -> ( E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) <-> E. k e. ZZ ( F |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( F ` k ) ( ball ` D ) x ) ) )
9	8	ralbidv 2986	. . 3 |- ( f = F -> ( A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) <-> A. x e. RR+ E. k e. ZZ ( F |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( F ` k ) ( ball ` D ) x ) ) )
10	9	elrab 3363	. 2 |- ( F e. { f e. ( X ^pm CC ) | A. x e. RR+ E. k e. ZZ ( f |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( f ` k ) ( ball ` D ) x ) } <-> ( F e. ( X ^pm CC ) /\ A. x e. RR+ E. k e. ZZ ( F |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( F ` k ) ( ball ` D ) x ) ) )
11	2, 10	syl6bb 276	1 |- ( D e. ( *Met ` X ) -> ( F e. ( Cau ` D ) <-> ( F e. ( X ^pm CC ) /\ A. x e. RR+ E. k e. ZZ ( F |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( F ` k ) ( ball ` D ) x ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   *Metcxmt 19731   ballcbl 19733   Caucca 23051

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-xmet 19739  df-cau 23054

This theorem is referenced by:  iscau2  23075  caufpm  23080  lmcau  23111