Theorem cfilucfil3OLD 21625

Description: Given a metric D and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.)

Assertion

Ref	Expression
cfilucfil3OLD	|- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( ( C e. ( Fil ` X ) /\ C e. (CauFilu ` (metUnifOLD ` D ) ) ) <-> C e. (CauFil ` D ) ) )

Proof of Theorem cfilucfil3OLD

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

Step	Hyp	Ref	Expression
1		cfilucfil2OLD 20944	. . . 4 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( C e. (CauFilu ` (metUnifOLD ` D ) ) <-> ( C e. ( fBas ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
2	1	anbi2d 703	. . 3 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( ( C e. ( Fil ` X ) /\ C e. (CauFilu ` (metUnifOLD ` D ) ) ) <-> ( C e. ( Fil ` X ) /\ ( C e. ( fBas ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) ) )
3		filfbas 20217	. . . . . 6 |- ( C e. ( Fil ` X ) -> C e. ( fBas ` X ) )
4	3	pm4.71i 632	. . . . 5 |- ( C e. ( Fil ` X ) <-> ( C e. ( Fil ` X ) /\ C e. ( fBas ` X ) ) )
5	4	anbi1i 695	. . . 4 |- ( ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) <-> ( ( C e. ( Fil ` X ) /\ C e. ( fBas ` X ) ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) )
6		anass 649	. . . 4 |- ( ( ( C e. ( Fil ` X ) /\ C e. ( fBas ` X ) ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) <-> ( C e. ( Fil ` X ) /\ ( C e. ( fBas ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
7	5, 6	bitr2i 250	. . 3 |- ( ( C e. ( Fil ` X ) /\ ( C e. ( fBas ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) <-> ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) )
8	2, 7	syl6bb 261	. 2 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( ( C e. ( Fil ` X ) /\ C e. (CauFilu ` (metUnifOLD ` D ) ) ) <-> ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
9		iscfil 21572	. . 3 |- ( D e. ( *Met ` X ) -> ( C e. (CauFil ` D ) <-> ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
10	9	adantl 466	. 2 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( C e. (CauFil ` D ) <-> ( C e. ( Fil ` X ) /\ A. x e. RR+ E. y e. C ( D " ( y X. y ) ) C_ ( 0 [,) x ) ) ) )
11	8, 10	bitr4d 256	1 |- ( ( X =/= (/) /\ D e. ( *Met ` X ) ) -> ( ( C e. ( Fil ` X ) /\ C e. (CauFilu ` (metUnifOLD ` D ) ) ) <-> C e. (CauFil ` D ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1767   =/= wne 2662   A.wral 2817   E.wrex 2818   C_ wss 3481   (/)c0 3790   X. cxp 5003   "cima 5008   ` cfv 5594  (class class class)co 6295   0cc0 9504   RR+crp 11232   [,)cico 11543   *Metcxmt 18273   fBascfbas 18276  metUnif_OLDcmetuOLD 18279   Filcfil 20214  CauFil_uccfilu 20657  CauFilccfil 21559

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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ico 11547  df-xmet 18282  df-fbas 18286  df-fg 18287  df-metuOLD 18288  df-fil 20215  df-ust 20571  df-cfilu 20658  df-cfil 21562

This theorem is referenced by:  cfilucfil4OLD  21627