Theorem equivcmet 23114

Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23098, metss2 22317, this theorem does not have a one-directional form - it is possible for a metric C that is strongly finer than the complete metric D to be incomplete and vice versa. Consider D = the metric on RR induced by the usual homeomorphism from ( 0 , 1 ) against the usual metric C on RR and against the discrete metric E on RR. Then both C and E are complete but D is not, and C is strongly finer than D, which is strongly finer than E. (Contributed by Mario Carneiro, 15-Sep-2015.)

Hypotheses

Ref	Expression
equivcmet.1	|- ( ph -> C e. ( Met ` X ) )
equivcmet.2	|- ( ph -> D e. ( Met ` X ) )
equivcmet.3	|- ( ph -> R e. RR+ )
equivcmet.4	|- ( ph -> S e. RR+ )
equivcmet.5	|- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x C y ) <_ ( R x. ( x D y ) ) )
equivcmet.6	|- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x D y ) <_ ( S x. ( x C y ) ) )

Assertion

Ref	Expression
equivcmet	|- ( ph -> ( C e. ( CMet ` X ) <-> D e. ( CMet ` X ) ) )

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

Proof of Theorem equivcmet

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equivcmet.1	. . . 4 |- ( ph -> C e. ( Met ` X ) )
2		equivcmet.2	. . . 4 |- ( ph -> D e. ( Met ` X ) )
3	1, 2	2thd 255	. . 3 |- ( ph -> ( C e. ( Met ` X ) <-> D e. ( Met ` X ) ) )
4		equivcmet.4	. . . . . 6 |- ( ph -> S e. RR+ )
5		equivcmet.6	. . . . . 6 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x D y ) <_ ( S x. ( x C y ) ) )
6	2, 1, 4, 5	equivcfil 23097	. . . . 5 |- ( ph -> (CauFil ` C ) C_ (CauFil ` D ) )
7		equivcmet.3	. . . . . 6 |- ( ph -> R e. RR+ )
8		equivcmet.5	. . . . . 6 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x C y ) <_ ( R x. ( x D y ) ) )
9	1, 2, 7, 8	equivcfil 23097	. . . . 5 |- ( ph -> (CauFil ` D ) C_ (CauFil ` C ) )
10	6, 9	eqssd 3620	. . . 4 |- ( ph -> (CauFil ` C ) = (CauFil ` D ) )
11		eqid 2622	. . . . . . . 8 |- ( MetOpen ` C ) = ( MetOpen ` C )
12		eqid 2622	. . . . . . . 8 |- ( MetOpen ` D ) = ( MetOpen ` D )
13	11, 12, 1, 2, 7, 8	metss2 22317	. . . . . . 7 |- ( ph -> ( MetOpen ` C ) C_ ( MetOpen ` D ) )
14	12, 11, 2, 1, 4, 5	metss2 22317	. . . . . . 7 |- ( ph -> ( MetOpen ` D ) C_ ( MetOpen ` C ) )
15	13, 14	eqssd 3620	. . . . . 6 |- ( ph -> ( MetOpen ` C ) = ( MetOpen ` D ) )
16	15	oveq1d 6665	. . . . 5 |- ( ph -> ( ( MetOpen ` C ) fLim f ) = ( ( MetOpen ` D ) fLim f ) )
17	16	neeq1d 2853	. . . 4 |- ( ph -> ( ( ( MetOpen ` C ) fLim f ) =/= (/) <-> ( ( MetOpen ` D ) fLim f ) =/= (/) ) )
18	10, 17	raleqbidv 3152	. . 3 |- ( ph -> ( A. f e. (CauFil ` C ) ( ( MetOpen ` C ) fLim f ) =/= (/) <-> A. f e. (CauFil ` D ) ( ( MetOpen ` D ) fLim f ) =/= (/) ) )
19	3, 18	anbi12d 747	. 2 |- ( ph -> ( ( C e. ( Met ` X ) /\ A. f e. (CauFil ` C ) ( ( MetOpen ` C ) fLim f ) =/= (/) ) <-> ( D e. ( Met ` X ) /\ A. f e. (CauFil ` D ) ( ( MetOpen ` D ) fLim f ) =/= (/) ) ) )
20	11	iscmet 23082	. 2 |- ( C e. ( CMet ` X ) <-> ( C e. ( Met ` X ) /\ A. f e. (CauFil ` C ) ( ( MetOpen ` C ) fLim f ) =/= (/) ) )
21	12	iscmet 23082	. 2 |- ( D e. ( CMet ` X ) <-> ( D e. ( Met ` X ) /\ A. f e. (CauFil ` D ) ( ( MetOpen ` D ) fLim f ) =/= (/) ) )
22	19, 20, 21	3bitr4g 303	1 |- ( ph -> ( C e. ( CMet ` X ) <-> D e. ( CMet ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   x. cmul 9941   <_ cle 10075   RR+crp 11832   Metcme 19732   MetOpencmopn 19736   fLim cflim 21738  CauFilccfil 23050   CMetcms 23052

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  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-iun 4522  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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-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-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-bases 20750  df-fil 21650  df-cfil 23053  df-cmet 23055

This theorem is referenced by: (None)