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Theorem isxms2 18480
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 18479 . 2  |-  ( K  e.  * MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
52, 1istps 17003 . . . 4  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
6 df-mopn 16700 . . . . . . . . . 10  |-  MetOpen  =  ( x  e.  U. ran  * Met  |->  ( topGen `  ran  ( ball `  x )
) )
76dmmptss 5368 . . . . . . . . 9  |-  dom  MetOpen  C_  U. ran  * Met
8 toponmax 16995 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
98adantl 454 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  e.  J )
10 simpl 445 . . . . . . . . . . 11  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  J  =  ( MetOpen `  D )
)
119, 10eleqtrd 2514 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  e.  ( MetOpen `  D )
)
12 elfvdm 5759 . . . . . . . . . 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 3348 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  U.
ran  * Met )
15 xmetunirn 18369 . . . . . . . 8  |-  ( D  e.  U. ran  * Met 
<->  D  e.  ( * Met `  dom  dom  D ) )
1614, 15sylib 190 . . . . . . 7  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  ( * Met `  dom  dom 
D ) )
17 eqid 2438 . . . . . . . . . . . . 13  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1817mopntopon 18471 . . . . . . . . . . . 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 2512 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  J  e.  (TopOn `  dom  dom  D
) )
21 toponuni 16994 . . . . . . . . . 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 16994 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2423adantl 454 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  =  U. J )
2522, 24eqtr4d 2473 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  dom  dom  D  =  X )
2625fveq2d 5734 . . . . . . 7  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  ( * Met `  dom  dom  D
)  =  ( * Met `  X ) )
2716, 26eleqtrd 2514 . . . . . 6  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  ( * Met `  X
) )
2827ex 425 . . . . 5  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  ->  D  e.  ( * Met `  X ) ) )
2917mopntopon 18471 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  ( MetOpen
`  D )  e.  (TopOn `  X )
)
30 eleq1 2498 . . . . . 6  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  <->  (
MetOpen `  D )  e.  (TopOn `  X )
) )
3129, 30syl5ibr 214 . . . . 5  |-  ( J  =  ( MetOpen `  D
)  ->  ( D  e.  ( * Met `  X
)  ->  J  e.  (TopOn `  X ) ) )
3228, 31impbid 185 . . . 4  |-  ( J  =  ( MetOpen `  D
)  ->  ( J  e.  (TopOn `  X )  <->  D  e.  ( * Met `  X ) ) )
335, 32syl5bb 250 . . 3  |-  ( J  =  ( MetOpen `  D
)  ->  ( K  e.  TopSp 
<->  D  e.  ( * Met `  X ) ) )
3433pm5.32ri 621 . 2  |-  ( ( K  e.  TopSp  /\  J  =  ( MetOpen `  D
) )  <->  ( D  e.  ( * Met `  X
)  /\  J  =  ( MetOpen `  D )
) )
354, 34bitri 242 1  |-  ( K  e.  * MetSp  <->  ( D  e.  ( * Met `  X
)  /\  J  =  ( MetOpen `  D )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   U.cuni 4017    X. cxp 4878   dom cdm 4880   ran crn 4881    |` cres 4882   ` cfv 5456   Basecbs 13471   distcds 13540   TopOpenctopn 13651   topGenctg 13667   * Metcxmt 16688   ballcbl 16690   MetOpencmopn 16693  TopOnctopon 16961   TopSpctps 16963   *
MetSpcxme 18349
This theorem is referenced by:  isms2  18482  xmsxmet  18488  setsxms  18511  tmsxms  18518  imasf1oxms  18521  ressxms  18557  prdsxms  18562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-topgen 13669  df-psmet 16696  df-xmet 16697  df-bl 16699  df-mopn 16700  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-xms 18352
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