Theorem equivcfil 23097

Description: If the metric D is "strongly finer" than C (meaning that there is a positive real constant R such that C ( x , y ) <_ R x. D ( x , y )), all the D-Cauchy filters are also C-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)

Hypotheses

Ref	Expression
equivcau.1	|- ( ph -> C e. ( Met ` X ) )
equivcau.2	|- ( ph -> D e. ( Met ` X ) )
equivcau.3	|- ( ph -> R e. RR+ )
equivcau.4	|- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x C y ) <_ ( R x. ( x D y ) ) )

Assertion

Ref	Expression
equivcfil	|- ( ph -> (CauFil ` D ) C_ (CauFil ` C ) )

Distinct variable groups:   x, y, C   x, D, y   ph, x, y   x, R, y   x, X, y

Proof of Theorem equivcfil

Dummy variables f r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) -> r e. RR+ )
2		equivcau.3	. . . . . . . . 9 |- ( ph -> R e. RR+ )
3	2	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) -> R e. RR+ )
4	1, 3	rpdivcld 11889	. . . . . . 7 |- ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) -> ( r / R ) e. RR+ )
5		oveq2 6658	. . . . . . . . . 10 |- ( s = ( r / R ) -> ( x ( ball ` D ) s ) = ( x ( ball ` D ) ( r / R ) ) )
6	5	eleq1d 2686	. . . . . . . . 9 |- ( s = ( r / R ) -> ( ( x ( ball ` D ) s ) e. f <-> ( x ( ball ` D ) ( r / R ) ) e. f ) )
7	6	rexbidv 3052	. . . . . . . 8 |- ( s = ( r / R ) -> ( E. x e. X ( x ( ball ` D ) s ) e. f <-> E. x e. X ( x ( ball ` D ) ( r / R ) ) e. f ) )
8	7	rspcv 3305	. . . . . . 7 |- ( ( r / R ) e. RR+ -> ( A. s e. RR+ E. x e. X ( x ( ball ` D ) s ) e. f -> E. x e. X ( x ( ball ` D ) ( r / R ) ) e. f ) )
9	4, 8	syl 17	. . . . . 6 |- ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) -> ( A. s e. RR+ E. x e. X ( x ( ball ` D ) s ) e. f -> E. x e. X ( x ( ball ` D ) ( r / R ) ) e. f ) )
10		simpllr 799	. . . . . . . 8 |- ( ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) /\ x e. X ) -> f e. ( Fil ` X ) )
11		eqid 2622	. . . . . . . . . . . 12 |- ( MetOpen ` C ) = ( MetOpen ` C )
12		eqid 2622	. . . . . . . . . . . 12 |- ( MetOpen ` D ) = ( MetOpen ` D )
13		equivcau.1	. . . . . . . . . . . 12 |- ( ph -> C e. ( Met ` X ) )
14		equivcau.2	. . . . . . . . . . . 12 |- ( ph -> D e. ( Met ` X ) )
15		equivcau.4	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x C y ) <_ ( R x. ( x D y ) ) )
16	11, 12, 13, 14, 2, 15	metss2lem 22316	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. X /\ r e. RR+ ) ) -> ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) )
17	16	ancom2s 844	. . . . . . . . . 10 |- ( ( ph /\ ( r e. RR+ /\ x e. X ) ) -> ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) )
18	17	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ f e. ( Fil ` X ) ) /\ ( r e. RR+ /\ x e. X ) ) -> ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) )
19	18	anassrs 680	. . . . . . . 8 |- ( ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) /\ x e. X ) -> ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) )
20	13	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) /\ x e. X ) -> C e. ( Met ` X ) )
21		metxmet 22139	. . . . . . . . . 10 |- ( C e. ( Met ` X ) -> C e. ( *Met ` X ) )
22	20, 21	syl 17	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) /\ x e. X ) -> C e. ( *Met ` X ) )
23		simpr 477	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) /\ x e. X ) -> x e. X )
24		rpxr 11840	. . . . . . . . . 10 |- ( r e. RR+ -> r e. RR* )
25	24	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) /\ x e. X ) -> r e. RR* )
26		blssm 22223	. . . . . . . . 9 |- ( ( C e. ( *Met ` X ) /\ x e. X /\ r e. RR* ) -> ( x ( ball ` C ) r ) C_ X )
27	22, 23, 25, 26	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) /\ x e. X ) -> ( x ( ball ` C ) r ) C_ X )
28		filss 21657	. . . . . . . . . 10 |- ( ( f e. ( Fil ` X ) /\ ( ( x ( ball ` D ) ( r / R ) ) e. f /\ ( x ( ball ` C ) r ) C_ X /\ ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) ) ) -> ( x ( ball ` C ) r ) e. f )
29	28	3exp2 1285	. . . . . . . . 9 |- ( f e. ( Fil ` X ) -> ( ( x ( ball ` D ) ( r / R ) ) e. f -> ( ( x ( ball ` C ) r ) C_ X -> ( ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) -> ( x ( ball ` C ) r ) e. f ) ) ) )
30	29	com24 95	. . . . . . . 8 |- ( f e. ( Fil ` X ) -> ( ( x ( ball ` D ) ( r / R ) ) C_ ( x ( ball ` C ) r ) -> ( ( x ( ball ` C ) r ) C_ X -> ( ( x ( ball ` D ) ( r / R ) ) e. f -> ( x ( ball ` C ) r ) e. f ) ) ) )
31	10, 19, 27, 30	syl3c 66	. . . . . . 7 |- ( ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) /\ x e. X ) -> ( ( x ( ball ` D ) ( r / R ) ) e. f -> ( x ( ball ` C ) r ) e. f ) )
32	31	reximdva 3017	. . . . . 6 |- ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) -> ( E. x e. X ( x ( ball ` D ) ( r / R ) ) e. f -> E. x e. X ( x ( ball ` C ) r ) e. f ) )
33	9, 32	syld 47	. . . . 5 |- ( ( ( ph /\ f e. ( Fil ` X ) ) /\ r e. RR+ ) -> ( A. s e. RR+ E. x e. X ( x ( ball ` D ) s ) e. f -> E. x e. X ( x ( ball ` C ) r ) e. f ) )
34	33	ralrimdva 2969	. . . 4 |- ( ( ph /\ f e. ( Fil ` X ) ) -> ( A. s e. RR+ E. x e. X ( x ( ball ` D ) s ) e. f -> A. r e. RR+ E. x e. X ( x ( ball ` C ) r ) e. f ) )
35	34	imdistanda 729	. . 3 |- ( ph -> ( ( f e. ( Fil ` X ) /\ A. s e. RR+ E. x e. X ( x ( ball ` D ) s ) e. f ) -> ( f e. ( Fil ` X ) /\ A. r e. RR+ E. x e. X ( x ( ball ` C ) r ) e. f ) ) )
36		metxmet 22139	. . . 4 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
37		iscfil3 23071	. . . 4 |- ( D e. ( *Met ` X ) -> ( f e. (CauFil ` D ) <-> ( f e. ( Fil ` X ) /\ A. s e. RR+ E. x e. X ( x ( ball ` D ) s ) e. f ) ) )
38	14, 36, 37	3syl 18	. . 3 |- ( ph -> ( f e. (CauFil ` D ) <-> ( f e. ( Fil ` X ) /\ A. s e. RR+ E. x e. X ( x ( ball ` D ) s ) e. f ) ) )
39		iscfil3 23071	. . . 4 |- ( C e. ( *Met ` X ) -> ( f e. (CauFil ` C ) <-> ( f e. ( Fil ` X ) /\ A. r e. RR+ E. x e. X ( x ( ball ` C ) r ) e. f ) ) )
40	13, 21, 39	3syl 18	. . 3 |- ( ph -> ( f e. (CauFil ` C ) <-> ( f e. ( Fil ` X ) /\ A. r e. RR+ E. x e. X ( x ( ball ` C ) r ) e. f ) ) )
41	35, 38, 40	3imtr4d 283	. 2 |- ( ph -> ( f e. (CauFil ` D ) -> f e. (CauFil ` C ) ) )
42	41	ssrdv 3609	1 |- ( ph -> (CauFil ` D ) C_ (CauFil ` C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   x. cmul 9941   RR*cxr 10073   <_ cle 10075   / cdiv 10684   RR+crp 11832   *Metcxmt 19731   Metcme 19732   ballcbl 19733   MetOpencmopn 19736   Filcfil 21649  CauFilccfil 23050

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  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

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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-fbas 19743  df-fil 21650  df-cfil 23053

This theorem is referenced by:  equivcmet  23114