Theorem isxms2 22253

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 22252	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )
5	2, 1	istps 20738	. . . 4 |- ( K e. TopSp <-> J e. (TopOn ` X ) )
6		df-mopn 19742	. . . . . . . . . 10 |- MetOpen = ( x e. U. ran *Met |-> ( topGen ` ran ( ball ` x ) ) )
7	6	dmmptss 5631	. . . . . . . . 9 |- dom MetOpen C_ U. ran *Met
8		toponmax 20730	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X e. J )
9	8	adantl 482	. . . . . . . . . . 11 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X e. J )
10		simpl 473	. . . . . . . . . . 11 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> J = ( MetOpen ` D ) )
11	9, 10	eleqtrd 2703	. . . . . . . . . 10 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X e. ( MetOpen ` D ) )
12		elfvdm 6220	. . . . . . . . . 10 |- ( X e. ( MetOpen ` D ) -> D e. dom MetOpen )
13	11, 12	syl 17	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. dom MetOpen )
14	7, 13	sseldi 3601	. . . . . . . 8 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. U. ran *Met )
15		xmetunirn 22142	. . . . . . . 8 |- ( D e. U. ran *Met <-> D e. ( *Met ` dom dom D ) )
16	14, 15	sylib 208	. . . . . . 7 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. ( *Met ` dom dom D ) )
17		eqid 2622	. . . . . . . . . . . . 13 |- ( MetOpen ` D ) = ( MetOpen ` D )
18	17	mopntopon 22244	. . . . . . . . . . . 12 |- ( D e. ( *Met ` dom dom D ) -> ( MetOpen ` D ) e. (TopOn ` dom dom D ) )
19	16, 18	syl 17	. . . . . . . . . . 11 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> ( MetOpen ` D ) e. (TopOn ` dom dom D ) )
20	10, 19	eqeltrd 2701	. . . . . . . . . 10 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> J e. (TopOn ` dom dom D ) )
21		toponuni 20719	. . . . . . . . . 10 |- ( J e. (TopOn ` dom dom D ) -> dom dom D = U. J )
22	20, 21	syl 17	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> dom dom D = U. J )
23		toponuni 20719	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> X = U. J )
24	23	adantl 482	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X = U. J )
25	22, 24	eqtr4d 2659	. . . . . . . 8 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> dom dom D = X )
26	25	fveq2d 6195	. . . . . . 7 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> ( *Met ` dom dom D ) = ( *Met ` X ) )
27	16, 26	eleqtrd 2703	. . . . . 6 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. ( *Met ` X ) )
28	27	ex 450	. . . . 5 |- ( J = ( MetOpen ` D ) -> ( J e. (TopOn ` X ) -> D e. ( *Met ` X ) ) )
29	17	mopntopon 22244	. . . . . 6 |- ( D e. ( *Met ` X ) -> ( MetOpen ` D ) e. (TopOn ` X ) )
30		eleq1 2689	. . . . . 6 |- ( J = ( MetOpen ` D ) -> ( J e. (TopOn ` X ) <-> ( MetOpen ` D ) e. (TopOn ` X ) ) )
31	29, 30	syl5ibr 236	. . . . 5 |- ( J = ( MetOpen ` D ) -> ( D e. ( *Met ` X ) -> J e. (TopOn ` X ) ) )
32	28, 31	impbid 202	. . . 4 |- ( J = ( MetOpen ` D ) -> ( J e. (TopOn ` X ) <-> D e. ( *Met ` X ) ) )
33	5, 32	syl5bb 272	. . 3 |- ( J = ( MetOpen ` D ) -> ( K e. TopSp <-> D e. ( *Met ` X ) ) )
34	33	pm5.32ri 670	. 2 |- ( ( K e. TopSp /\ J = ( MetOpen ` D ) ) <-> ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) )
35	4, 34	bitri 264	1 |- ( K e. *MetSp <-> ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   U.cuni 4436   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   ` cfv 5888   Basecbs 15857   distcds 15950   TopOpenctopn 16082   topGenctg 16098   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736  TopOnctopon 20715   TopSpctps 20736   *MetSpcxme 22122

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-topgen 16104  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125

This theorem is referenced by:  isms2  22255  xmsxmet  22261  setsxms  22284  tmsxms  22291  imasf1oxms  22294  ressxms  22330  prdsxms  22335