Theorem isxms2 15443

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 15442	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )
5	2, 1	istps 15023	. . . 4 |- ( K e. TopSp <-> J e. (TopOn ` X ) )
6		df-mopn 14821	. . . . . . . . . 10 |- MetOpen = ( x e. U. ran *Met |-> ( topGen ` ran ( ball ` x ) ) )
7	6	dmmptss 5264	. . . . . . . . 9 |- dom MetOpen C_ U. ran *Met
8		mopnrel 15432	. . . . . . . . . 10 |- Rel MetOpen
9		toponmax 15016	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X e. J )
10	9	adantl 277	. . . . . . . . . . 11 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X e. J )
11		simpl 109	. . . . . . . . . . 11 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> J = ( MetOpen ` D ) )
12	10, 11	eleqtrd 2313	. . . . . . . . . 10 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X e. ( MetOpen ` D ) )
13		relelfvdm 5707	. . . . . . . . . 10 |- ( ( Rel MetOpen /\ X e. ( MetOpen ` D ) ) -> D e. dom MetOpen )
14	8, 12, 13	sylancr 414	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. dom MetOpen )
15	7, 14	sselid 3240	. . . . . . . 8 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. U. ran *Met )
16		xmetunirn 15349	. . . . . . . 8 |- ( D e. U. ran *Met <-> D e. ( *Met ` dom dom D ) )
17	15, 16	sylib 122	. . . . . . 7 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. ( *Met ` dom dom D ) )
18		eqid 2234	. . . . . . . . . . . . 13 |- ( MetOpen ` D ) = ( MetOpen ` D )
19	18	mopntopon 15434	. . . . . . . . . . . 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 2311	. . . . . . . . . 10 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> J e. (TopOn ` dom dom D ) )
22		toponuni 15006	. . . . . . . . . 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 15006	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> X = U. J )
25	24	adantl 277	. . . . . . . . 9 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> X = U. J )
26	23, 25	eqtr4d 2270	. . . . . . . 8 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> dom dom D = X )
27	26	fveq2d 5679	. . . . . . 7 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> ( *Met ` dom dom D ) = ( *Met ` X ) )
28	17, 27	eleqtrd 2313	. . . . . 6 |- ( ( J = ( MetOpen ` D ) /\ J e. (TopOn ` X ) ) -> D e. ( *Met ` X ) )
29	28	ex 115	. . . . 5 |- ( J = ( MetOpen ` D ) -> ( J e. (TopOn ` X ) -> D e. ( *Met ` X ) ) )
30	18	mopntopon 15434	. . . . . 6 |- ( D e. ( *Met ` X ) -> ( MetOpen ` D ) e. (TopOn ` X ) )
31		eleq1 2297	. . . . . 6 |- ( J = ( MetOpen ` D ) -> ( J e. (TopOn ` X ) <-> ( MetOpen ` D ) e. (TopOn ` X ) ) )
32	30, 31	imbitrrid 156	. . . . 5 |- ( J = ( MetOpen ` D ) -> ( D e. ( *Met ` X ) -> J e. (TopOn ` X ) ) )
33	29, 32	impbid 129	. . . 4 |- ( J = ( MetOpen ` D ) -> ( J e. (TopOn ` X ) <-> D e. ( *Met ` X ) ) )
34	5, 33	bitrid 192	. . 3 |- ( J = ( MetOpen ` D ) -> ( K e. TopSp <-> D e. ( *Met ` X ) ) )
35	34	pm5.32ri 455	. 2 |- ( ( K e. TopSp /\ J = ( MetOpen ` D ) ) <-> ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) )
36	4, 35	bitri 184	1 |- ( K e. *MetSp <-> ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   U.cuni 3919   X. cxp 4752   dom cdm 4754   ran crn 4755   |` cres 4756   Rel wrel 4759   ` cfv 5357   Basecbs 13296   distcds 13383   TopOpenctopn 13537   topGenctg 13551   *Metcxmet 14810   ballcbl 14812   MetOpencmopn 14815  TopOnctopon 15001   TopSpctps 15021   *MetSpcxms 15327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-map 6897  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-ndx 13299  df-slot 13300  df-base 13302  df-tset 13393  df-rest 13538  df-topn 13539  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-bl 14820  df-mopn 14821  df-top 14989  df-topon 15002  df-topsp 15022  df-bases 15034  df-xms 15330

This theorem is referenced by:  isms2  15445  xmsxmet  15451