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Theorem isxms2 18010
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 18009 . 2  |-  ( K  e.  * MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
52, 1istps 16690 . . . 4  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
6 df-mopn 16392 . . . . . . . . . 10  |-  MetOpen  =  ( x  e.  U. ran  * Met  |->  ( topGen `  ran  ( ball `  x )
) )
76dmmptss 5185 . . . . . . . . 9  |-  dom  MetOpen  C_  U. ran  * Met
8 toponmax 16682 . . . . . . . . . . . 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 2372 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  e.  ( MetOpen `  D )
)
12 elfvdm 5570 . . . . . . . . . 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 3191 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  D  e.  U.
ran  * Met )
15 xmetunirn 17918 . . . . . . . 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 2296 . . . . . . . . . . . . 13  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1817mopntopon 18001 . . . . . . . . . . . 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 2370 . . . . . . . . . 10  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  J  e.  (TopOn `  dom  dom  D
) )
21 toponuni 16681 . . . . . . . . . 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 16681 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2423adantl 452 . . . . . . . . 9  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  X  =  U. J )
2522, 24eqtr4d 2331 . . . . . . . 8  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  dom  dom  D  =  X )
2625fveq2d 5545 . . . . . . 7  |-  ( ( J  =  ( MetOpen `  D )  /\  J  e.  (TopOn `  X )
)  ->  ( * Met `  dom  dom  D
)  =  ( * Met `  X ) )
2716, 26eleqtrd 2372 . . . . . 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 18001 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  ( MetOpen
`  D )  e.  (TopOn `  X )
)
30 eleq1 2356 . . . . . 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 1632    e. wcel 1696   U.cuni 3843    X. cxp 4703   dom cdm 4705   ran crn 4706    |` cres 4707   ` cfv 5271   Basecbs 13164   distcds 13233   TopOpenctopn 13342   topGenctg 13358   * Metcxmt 16385   ballcbl 16387   MetOpencmopn 16388  TopOnctopon 16648   TopSpctps 16650   *
MetSpcxme 17898
This theorem is referenced by:  isms2  18012  xmsxmet  18018  setsxms  18041  tmsxms  18048  imasf1oxms  18051  ressxms  18087  prdsxms  18092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-topgen 13360  df-xmet 16389  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901
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