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Theorem xmsuspOLD 21257
Description: If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
xmsusp.x  |-  X  =  ( Base `  F
)
xmsusp.d  |-  D  =  ( ( dist `  F
)  |`  ( X  X.  X ) )
xmsusp.u  |-  U  =  (UnifSt `  F )
Assertion
Ref Expression
xmsuspOLD  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  F  e. UnifSp )

Proof of Theorem xmsuspOLD
StepHypRef Expression
1 simp3 996 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  U  =  (metUnifOLD `  D ) )
2 simp1 994 . . . 4  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  X  =/=  (/) )
3 xmsusp.x . . . . . 6  |-  X  =  ( Base `  F
)
4 xmsusp.d . . . . . 6  |-  D  =  ( ( dist `  F
)  |`  ( X  X.  X ) )
53, 4xmsxmet 21128 . . . . 5  |-  ( F  e.  *MetSp  ->  D  e.  ( *Met `  X ) )
653ad2ant2 1016 . . . 4  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  D  e.  ( *Met `  X
) )
7 metuustOLD 21243 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(metUnifOLD `  D )  e.  (UnifOn `  X ) )
82, 6, 7syl2anc 659 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  (metUnifOLD
`  D )  e.  (UnifOn `  X )
)
91, 8eqeltrd 2542 . 2  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  U  e.  (UnifOn `  X ) )
10 metutopOLD 21254 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(unifTop `  (metUnifOLD
`  D ) )  =  ( MetOpen `  D
) )
112, 6, 10syl2anc 659 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  (unifTop `  (metUnifOLD
`  D ) )  =  ( MetOpen `  D
) )
121fveq2d 5852 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  (unifTop `  U )  =  (unifTop `  (metUnifOLD
`  D ) ) )
13 eqid 2454 . . . . 5  |-  ( TopOpen `  F )  =  (
TopOpen `  F )
1413, 3, 4xmstopn 21123 . . . 4  |-  ( F  e.  *MetSp  ->  ( TopOpen
`  F )  =  ( MetOpen `  D )
)
15143ad2ant2 1016 . . 3  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  ( TopOpen `  F
)  =  ( MetOpen `  D ) )
1611, 12, 153eqtr4rd 2506 . 2  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  ( TopOpen `  F
)  =  (unifTop `  U
) )
17 xmsusp.u . . 3  |-  U  =  (UnifSt `  F )
183, 17, 13isusp 20933 . 2  |-  ( F  e. UnifSp 
<->  ( U  e.  (UnifOn `  X )  /\  ( TopOpen
`  F )  =  (unifTop `  U )
) )
199, 16, 18sylanbrc 662 1  |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD
`  D ) )  ->  F  e. UnifSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   (/)c0 3783    X. cxp 4986    |` cres 4990   ` cfv 5570   Basecbs 14719   distcds 14796   TopOpenctopn 14914   *Metcxmt 18601   MetOpencmopn 18606  metUnifOLDcmetuOLD 18607  UnifOncust 20871  unifTopcutop 20902  UnifStcuss 20925  UnifSpcusp 20926   *MetSpcxme 20989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ico 11538  df-topgen 14936  df-psmet 18609  df-xmet 18610  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-metuOLD 18616  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-fil 20516  df-ust 20872  df-utop 20903  df-usp 20929  df-xms 20992
This theorem is referenced by:  cmetcusp1OLD  21960
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