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Theorem isxms2 21405
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.
StepHypRef 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 ) )
41, 2, 3isxms 21404 . 2  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
52, 1istps 19893 . . . 4  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
6 df-mopn 18909 . . . . . . . . . 10  |-  MetOpen  =  ( x  e.  U. ran  *Met  |->  ( topGen `  ran  ( ball `  x )
) )
76dmmptss 5293 . . . . . . . . 9  |-  dom  MetOpen  C_  U. ran  *Met
8 toponmax 19885 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
98adantl 467 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  e.  J )
10 simpl 458 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  J  =  ( MetOpen `  D )
)
119, 10eleqtrd 2508 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  e.  ( MetOpen `  D )
)
12 elfvdm 5851 . . . . . . . . . 10  |-  ( X  e.  ( MetOpen `  D
)  ->  D  e.  dom 
MetOpen )
1311, 12syl 17 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  dom 
MetOpen )
147, 13sseldi 3405 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  U.
ran  *Met )
15 xmetunirn 21294 . . . . . . . 8  |-  ( D  e.  U. ran  *Met 
<->  D  e.  ( *Met `  dom  dom  D ) )
1614, 15sylib 199 . . . . . . 7  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  ( *Met `  dom  dom 
D ) )
17 eqid 2428 . . . . . . . . . . . . 13  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1817mopntopon 21396 . . . . . . . . . . . 12  |-  ( D  e.  ( *Met ` 
dom  dom  D )  -> 
( MetOpen `  D )  e.  (TopOn `  dom  dom  D
) )
1916, 18syl 17 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  ( MetOpen `  D )  e.  (TopOn `  dom  dom  D )
)
2010, 19eqeltrd 2506 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  J  e.  (TopOn `  dom  dom  D
) )
21 toponuni 19884 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  dom  dom 
D )  ->  dom  dom 
D  =  U. J
)
2220, 21syl 17 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  dom  dom  D  =  U. J )
23 toponuni 19884 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2423adantl 467 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  =  U. J )
2522, 24eqtr4d 2465 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  dom  dom  D  =  X )
2625fveq2d 5829 . . . . . . 7  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  ( *Met `  dom  dom  D
)  =  ( *Met `  X ) )
2716, 26eleqtrd 2508 . . . . . 6  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  ( *Met `  X
) )
2827ex 435 . . . . 5  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  ->  D  e.  ( *Met `  X ) ) )
2917mopntopon 21396 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  D )  e.  (TopOn `  X )
)
30 eleq1 2494 . . . . . 6  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  <->  (
MetOpen `  D )  e.  (TopOn `  X )
) )
3129, 30syl5ibr 224 . . . . 5  |-  ( J  =  ( MetOpen `  D
)  ->  ( D  e.  ( *Met `  X )  ->  J  e.  (TopOn `  X )
) )
3228, 31impbid 193 . . . 4  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  <->  D  e.  ( *Met `  X ) ) )
335, 32syl5bb 260 . . 3  |-  ( J  =  ( MetOpen `  D
)  ->  ( K  e.  TopSp 
<->  D  e.  ( *Met `  X ) ) )
3433pm5.32ri 642 . 2  |-  ( ( K  e.  TopSp  /\  J  =  ( MetOpen `  D
) )  <->  ( D  e.  ( *Met `  X )  /\  J  =  ( MetOpen `  D
) ) )
354, 34bitri 252 1  |-  ( K  e.  *MetSp  <->  ( D  e.  ( *Met `  X )  /\  J  =  ( MetOpen `  D
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   U.cuni 4162    X. cxp 4794   dom cdm 4796   ran crn 4797    |` cres 4798   ` cfv 5544   Basecbs 15064   distcds 15142   TopOpenctopn 15263   topGenctg 15279   *Metcxmt 18898   ballcbl 18900   MetOpencmopn 18903  TopOnctopon 19860   TopSpctps 19861   *MetSpcxme 21274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-n0 10821  df-z 10889  df-uz 11111  df-q 11216  df-rp 11254  df-xneg 11360  df-xadd 11361  df-xmul 11362  df-topgen 15285  df-psmet 18905  df-xmet 18906  df-bl 18908  df-mopn 18909  df-top 19863  df-bases 19864  df-topon 19865  df-topsp 19866  df-xms 21277
This theorem is referenced by:  isms2  21407  xmsxmet  21413  setsxms  21436  tmsxms  21443  imasf1oxms  21446  ressxms  21482  prdsxms  21487
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