Theorem isxms2 12380

Description: Express the predicate " <. X , D >. is an extended metric space" with underlying set X and distance function D. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
isms.j	|- J = ( TopOpen ` K )
isms.x	|- X = ( Base ` K )
isms.d	|- D = ( ( dist ` K ) |` ( X X. X ) )

Assertion

Ref	Expression
isxms2	|- ( K e. *MetSp <-> ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) )

Proof of Theorem isxms2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isms.j	. . 3 |- J = ( TopOpen ` K )
2		isms.x	. . 3 |- X = ( Base ` K )
3		isms.d	. . 3 |- D = ( ( dist ` K ) |` ( X X. X ) )
4	1, 2, 3	isxms 12379	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )
5	2, 1	istps 11981	. . . 4 |- ( K e. TopSp <-> J e. (TopOn ` X ) )
6		df-mopn 11942	. . . . . . . . . 10 |- MetOpen = ( x e. U. ran *Met |-> ( topGen ` ran ( ball ` x ) ) )
7	6	dmmptss 4971	. . . . . . . . 9 |- dom MetOpen C_ U. ran *Met
8		mopnrel 12369	. . . . . . . . . 10 |- Rel MetOpen
9		toponmax 11974	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X e. J )
10	9	adantl 273	. . . . . . . . . . 11 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X e. J )
11		simpl 108	. . . . . . . . . . 11 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> J = ( MetOpen ` D ) )
12	10, 11	eleqtrd 2178	. . . . . . . . . 10 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X e. ( MetOpen ` D ) )
13		relelfvdm 5385	. . . . . . . . . 10 |- ( ( Rel MetOpen /\ X e. ( MetOpen ` D ) ) -> D e. dom MetOpen )
14	8, 12, 13	sylancr 408	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. dom MetOpen )
15	7, 14	sseldi 3045	. . . . . . . 8 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. U. ran *Met )
16		xmetunirn 12286	. . . . . . . 8 |- ( D e. U. ran *Met <-> D e. ( *Met ` dom dom D ) )
17	15, 16	sylib 121	. . . . . . 7 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. ( *Met ` dom dom D ) )
18		eqid 2100	. . . . . . . . . . . . 13 |- ( MetOpen ` D ) = ( MetOpen ` D )
19	18	mopntopon 12371	. . . . . . . . . . . 12 |- ( D e. ( *Met ` dom dom D ) -> ( MetOpen ` D ) e. (TopOn ` dom dom D ) )
20	17, 19	syl 14	. . . . . . . . . . 11 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> ( MetOpen ` D ) e. (TopOn ` dom dom D ) )
21	11, 20	eqeltrd 2176	. . . . . . . . . 10 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> J e. (TopOn ` dom dom D ) )
22		toponuni 11964	. . . . . . . . . 10 |- ( J e. (TopOn ` dom dom D ) -> dom dom D = U. J )
23	21, 22	syl 14	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> dom dom D = U. J )
24		toponuni 11964	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> X = U. J )
25	24	adantl 273	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X = U. J )
26	23, 25	eqtr4d 2135	. . . . . . . 8 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> dom dom D = X )
27	26	fveq2d 5357	. . . . . . 7 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> ( *Met ` dom dom D ) = ( *Met ` X ) )
28	17, 27	eleqtrd 2178	. . . . . 6 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. ( *Met ` X ) )
29	28	ex 114	. . . . 5 |- ( J = ( MetOpen ` D ) -> ( J e. (TopOn ` X ) -> D e. ( *Met ` X ) ) )
30	18	mopntopon 12371	. . . . . 6 |- ( D e. ( *Met ` X ) -> ( MetOpen ` D ) e. (TopOn ` X ) )
31		eleq1 2162	. . . . . 6 |- ( J = ( MetOpen ` D ) -> ( J e. (TopOn ` X ) <-> ( MetOpen ` D ) e. (TopOn ` X ) ) )
32	30, 31	syl5ibr 155	. . . . 5 |- ( J = ( MetOpen ` D ) -> ( D e. ( *Met ` X ) -> J e. (TopOn ` X ) ) )
33	29, 32	impbid 128	. . . 4 |- ( J = ( MetOpen ` D ) -> ( J e. (TopOn ` X ) <-> D e. ( *Met ` X ) ) )
34	5, 33	syl5bb 191	. . 3 |- ( J = ( MetOpen ` D ) -> ( K e. TopSp <-> D e. ( *Met ` X ) ) )
35	34	pm5.32ri 446	. 2 |- ( ( K e. TopSp /\ J = ( MetOpen ` D ) ) <-> ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) )
36	4, 35	bitri 183	1 |- ( K e. *MetSp <-> ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   U.cuni 3683   X. cxp 4475   dom cdm 4477   ran crn 4478   |` cres 4479   Rel wrel 4482   ` cfv 5059   Basecbs 11741   distcds 11812   TopOpenctopn 11903   topGenctg 11917   *Metcxmet 11931   ballcbl 11933   MetOpencmopn 11936  TopOnctopon 11959   TopSpctps 11979   *MetSpcxms 12264

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-map 6474  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-5 8640  df-6 8641  df-7 8642  df-8 8643  df-9 8644  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-xneg 9400  df-xadd 9401  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-ndx 11744  df-slot 11745  df-base 11747  df-tset 11822  df-rest 11904  df-topn 11905  df-topgen 11923  df-psmet 11938  df-xmet 11939  df-bl 11941  df-mopn 11942  df-top 11947  df-topon 11960  df-topsp 11980  df-bases 11992  df-xms 12267

This theorem is referenced by:  isms2  12382  xmsxmet  12388