Theorem isxms2 13246

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 13245	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )
5	2, 1	istps 12824	. . . 4 |- ( K e. TopSp <-> J e. (TopOn ` X ) )
6		df-mopn 12785	. . . . . . . . . 10 |- MetOpen = ( x e. U. ran *Met |-> ( topGen ` ran ( ball ` x ) ) )
7	6	dmmptss 5107	. . . . . . . . 9 |- dom MetOpen C_ U. ran *Met
8		mopnrel 13235	. . . . . . . . . 10 |- Rel MetOpen
9		toponmax 12817	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X e. J )
10	9	adantl 275	. . . . . . . . . . 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 2249	. . . . . . . . . 10 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X e. ( MetOpen ` D ) )
13		relelfvdm 5528	. . . . . . . . . 10 |- ( ( Rel MetOpen /\ X e. ( MetOpen ` D ) ) -> D e. dom MetOpen )
14	8, 12, 13	sylancr 412	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. dom MetOpen )
15	7, 14	sselid 3145	. . . . . . . 8 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. U. ran *Met )
16		xmetunirn 13152	. . . . . . . 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 2170	. . . . . . . . . . . . 13 |- ( MetOpen ` D ) = ( MetOpen ` D )
19	18	mopntopon 13237	. . . . . . . . . . . 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 2247	. . . . . . . . . 10 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> J e. (TopOn ` dom dom D ) )
22		toponuni 12807	. . . . . . . . . 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 12807	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> X = U. J )
25	24	adantl 275	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X = U. J )
26	23, 25	eqtr4d 2206	. . . . . . . 8 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> dom dom D = X )
27	26	fveq2d 5500	. . . . . . 7 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> ( *Met ` dom dom D ) = ( *Met ` X ) )
28	17, 27	eleqtrd 2249	. . . . . 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 13237	. . . . . 6 |- ( D e. ( *Met ` X ) -> ( MetOpen ` D ) e. (TopOn ` X ) )
31		eleq1 2233	. . . . . 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 452	. 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 1348   e. wcel 2141   U.cuni 3796   X. cxp 4609   dom cdm 4611   ran crn 4612   |` cres 4613   Rel wrel 4616   ` cfv 5198   Basecbs 12416   distcds 12489   TopOpenctopn 12580   topGenctg 12594   *Metcxmet 12774   ballcbl 12776   MetOpencmopn 12779  TopOnctopon 12802   TopSpctps 12822   *MetSpcxms 13130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-9 8944  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-ndx 12419  df-slot 12420  df-base 12422  df-tset 12499  df-rest 12581  df-topn 12582  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-topsp 12823  df-bases 12835  df-xms 13133

This theorem is referenced by:  isms2  13248  xmsxmet  13254