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Theorem isxms2 21077
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 21076 . 2  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
52, 1istps 19564 . . . 4  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
6 df-mopn 18542 . . . . . . . . . 10  |-  MetOpen  =  ( x  e.  U. ran  *Met  |->  ( topGen `  ran  ( ball `  x )
) )
76dmmptss 5509 . . . . . . . . 9  |-  dom  MetOpen  C_  U. ran  *Met
8 toponmax 19556 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
98adantl 466 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  e.  J )
10 simpl 457 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  J  =  ( MetOpen `  D )
)
119, 10eleqtrd 2547 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  e.  ( MetOpen `  D )
)
12 elfvdm 5898 . . . . . . . . . 10  |-  ( X  e.  ( MetOpen `  D
)  ->  D  e.  dom 
MetOpen )
1311, 12syl 16 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  dom 
MetOpen )
147, 13sseldi 3497 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  U.
ran  *Met )
15 xmetunirn 20966 . . . . . . . 8  |-  ( D  e.  U. ran  *Met 
<->  D  e.  ( *Met `  dom  dom  D ) )
1614, 15sylib 196 . . . . . . 7  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  ( *Met `  dom  dom 
D ) )
17 eqid 2457 . . . . . . . . . . . . 13  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1817mopntopon 21068 . . . . . . . . . . . 12  |-  ( D  e.  ( *Met ` 
dom  dom  D )  -> 
( MetOpen `  D )  e.  (TopOn `  dom  dom  D
) )
1916, 18syl 16 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  ( MetOpen `  D )  e.  (TopOn `  dom  dom  D )
)
2010, 19eqeltrd 2545 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  J  e.  (TopOn `  dom  dom  D
) )
21 toponuni 19555 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  dom  dom 
D )  ->  dom  dom 
D  =  U. J
)
2220, 21syl 16 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  dom  dom  D  =  U. J )
23 toponuni 19555 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2423adantl 466 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  =  U. J )
2522, 24eqtr4d 2501 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  dom  dom  D  =  X )
2625fveq2d 5876 . . . . . . 7  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  ( *Met `  dom  dom  D
)  =  ( *Met `  X ) )
2716, 26eleqtrd 2547 . . . . . 6  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  ( *Met `  X
) )
2827ex 434 . . . . 5  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  ->  D  e.  ( *Met `  X ) ) )
2917mopntopon 21068 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  D )  e.  (TopOn `  X )
)
30 eleq1 2529 . . . . . 6  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  <->  (
MetOpen `  D )  e.  (TopOn `  X )
) )
3129, 30syl5ibr 221 . . . . 5  |-  ( J  =  ( MetOpen `  D
)  ->  ( D  e.  ( *Met `  X )  ->  J  e.  (TopOn `  X )
) )
3228, 31impbid 191 . . . 4  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  <->  D  e.  ( *Met `  X ) ) )
335, 32syl5bb 257 . . 3  |-  ( J  =  ( MetOpen `  D
)  ->  ( K  e.  TopSp 
<->  D  e.  ( *Met `  X ) ) )
3433pm5.32ri 638 . 2  |-  ( ( K  e.  TopSp  /\  J  =  ( MetOpen `  D
) )  <->  ( D  e.  ( *Met `  X )  /\  J  =  ( MetOpen `  D
) ) )
354, 34bitri 249 1  |-  ( K  e.  *MetSp  <->  ( D  e.  ( *Met `  X )  /\  J  =  ( MetOpen `  D
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   U.cuni 4251    X. cxp 5006   dom cdm 5008   ran crn 5009    |` cres 5010   ` cfv 5594   Basecbs 14644   distcds 14721   TopOpenctopn 14839   topGenctg 14855   *Metcxmt 18530   ballcbl 18532   MetOpencmopn 18535  TopOnctopon 19522   TopSpctps 19524   *MetSpcxme 20946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-xms 20949
This theorem is referenced by:  isms2  21079  xmsxmet  21085  setsxms  21108  tmsxms  21115  imasf1oxms  21118  ressxms  21154  prdsxms  21159
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