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Theorem isxms2 17994
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 17993 . 2  |-  ( K  e.  * MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
52, 1istps 16674 . . . 4  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
6 df-mopn 16376 . . . . . . . . . 10  |-  MetOpen  =  ( x  e.  U. ran  * Met  |->  ( topGen `  ran  ( ball `  x )
) )
76dmmptss 5169 . . . . . . . . 9  |-  dom  MetOpen  C_  U. ran  * Met
8 toponmax 16666 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
98adantl 452 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  e.  J )
10 simpl 443 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  J  =  ( MetOpen `  D )
)
119, 10eleqtrd 2359 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  e.  ( MetOpen `  D )
)
12 elfvdm 5554 . . . . . . . . . 10  |-  ( X  e.  ( MetOpen `  D
)  ->  D  e.  dom 
MetOpen )
1311, 12syl 15 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  dom 
MetOpen )
147, 13sseldi 3178 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  U.
ran  * Met )
15 xmetunirn 17902 . . . . . . . 8  |-  ( D  e.  U. ran  * Met 
<->  D  e.  ( * Met `  dom  dom  D ) )
1614, 15sylib 188 . . . . . . 7  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  ( * Met `  dom  dom 
D ) )
17 eqid 2283 . . . . . . . . . . . . 13  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1817mopntopon 17985 . . . . . . . . . . . 12  |-  ( D  e.  ( * Met ` 
dom  dom  D )  -> 
( MetOpen `  D )  e.  (TopOn `  dom  dom  D
) )
1916, 18syl 15 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  ( MetOpen `  D )  e.  (TopOn `  dom  dom  D )
)
2010, 19eqeltrd 2357 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  J  e.  (TopOn `  dom  dom  D
) )
21 toponuni 16665 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  dom  dom 
D )  ->  dom  dom 
D  =  U. J
)
2220, 21syl 15 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  dom  dom  D  =  U. J )
23 toponuni 16665 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2423adantl 452 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  =  U. J )
2522, 24eqtr4d 2318 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  dom  dom  D  =  X )
2625fveq2d 5529 . . . . . . 7  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  ( * Met `  dom  dom  D
)  =  ( * Met `  X ) )
2716, 26eleqtrd 2359 . . . . . 6  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  ( * Met `  X
) )
2827ex 423 . . . . 5  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  ->  D  e.  ( * Met `  X ) ) )
2917mopntopon 17985 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  ( MetOpen
`  D )  e.  (TopOn `  X )
)
30 eleq1 2343 . . . . . 6  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  <->  (
MetOpen `  D )  e.  (TopOn `  X )
) )
3129, 30syl5ibr 212 . . . . 5  |-  ( J  =  ( MetOpen `  D
)  ->  ( D  e.  ( * Met `  X
)  ->  J  e.  (TopOn `  X ) ) )
3228, 31impbid 183 . . . 4  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  <->  D  e.  ( * Met `  X ) ) )
335, 32syl5bb 248 . . 3  |-  ( J  =  ( MetOpen `  D
)  ->  ( K  e.  TopSp 
<->  D  e.  ( * Met `  X ) ) )
3433pm5.32ri 619 . 2  |-  ( ( K  e.  TopSp  /\  J  =  ( MetOpen `  D
) )  <->  ( D  e.  ( * Met `  X
)  /\  J  =  ( MetOpen `  D )
) )
354, 34bitri 240 1  |-  ( K  e.  * MetSp  <->  ( D  e.  ( * Met `  X
)  /\  J  =  ( MetOpen `  D )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   U.cuni 3827    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   ` cfv 5255   Basecbs 13148   distcds 13217   TopOpenctopn 13326   topGenctg 13342   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372  TopOnctopon 16632   TopSpctps 16634   *
MetSpcxme 17882
This theorem is referenced by:  isms2  17996  xmsxmet  18002  setsxms  18025  tmsxms  18032  imasf1oxms  18035  ressxms  18071  prdsxms  18076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885
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