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Theorem resubmet 14401
Description: The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
Hypotheses
Ref Expression
resubmet.1  |-  R  =  ( topGen `  ran  (,) )
resubmet.2  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( A  X.  A
) ) )
Assertion
Ref Expression
resubmet  |-  ( A 
C_  RR  ->  J  =  ( Rt  A ) )

Proof of Theorem resubmet
StepHypRef Expression
1 resubmet.2 . . 3  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( A  X.  A
) ) )
2 xpss12 4745 . . . . . 6  |-  ( ( A  C_  RR  /\  A  C_  RR )  ->  ( A  X.  A )  C_  ( RR  X.  RR ) )
32anidms 397 . . . . 5  |-  ( A 
C_  RR  ->  ( A  X.  A )  C_  ( RR  X.  RR ) )
43resabs1d 4949 . . . 4  |-  ( A 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A ) )  =  ( ( abs 
o.  -  )  |`  ( A  X.  A ) ) )
54fveq2d 5531 . . 3  |-  ( A 
C_  RR  ->  ( MetOpen `  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A
) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( A  X.  A
) ) ) )
61, 5eqtr4id 2239 . 2  |-  ( A 
C_  RR  ->  J  =  ( MetOpen `  ( (
( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A ) ) ) )
7 eqid 2187 . . . 4  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
87rexmet 14394 . . 3  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
9 eqid 2187 . . . 4  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A ) )
10 resubmet.1 . . . . 5  |-  R  =  ( topGen `  ran  (,) )
11 eqid 2187 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
127, 11tgioo 14399 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
1310, 12eqtri 2208 . . . 4  |-  R  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
14 eqid 2187 . . . 4  |-  ( MetOpen `  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A
) ) )  =  ( MetOpen `  ( (
( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A ) ) )
159, 13, 14metrest 14359 . . 3  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( *Met `  RR )  /\  A  C_  RR )  ->  ( Rt  A )  =  (
MetOpen `  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A ) ) ) )
168, 15mpan 424 . 2  |-  ( A 
C_  RR  ->  ( Rt  A )  =  ( MetOpen `  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A
) ) ) )
176, 16eqtr4d 2223 1  |-  ( A 
C_  RR  ->  J  =  ( Rt  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158    C_ wss 3141    X. cxp 4636   ran crn 4639    |` cres 4640    o. ccom 4642   ` cfv 5228  (class class class)co 5888   RRcr 7824    - cmin 8142   (,)cioo 9902   abscabs 11020   ↾t crest 12706   topGenctg 12721   *Metcxmet 13779   MetOpencmopn 13784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-map 6664  df-sup 6997  df-inf 6998  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-xneg 9786  df-xadd 9787  df-ioo 9906  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-rest 12708  df-topgen 12727  df-psmet 13786  df-xmet 13787  df-met 13788  df-bl 13789  df-mopn 13790  df-top 13851  df-topon 13864  df-bases 13896
This theorem is referenced by: (None)
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