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Theorem resubmet 21597
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 xpss12 4928 . . . . . 6  |-  ( ( A  C_  RR  /\  A  C_  RR )  ->  ( A  X.  A )  C_  ( RR  X.  RR ) )
21anidms 643 . . . . 5  |-  ( A 
C_  RR  ->  ( A  X.  A )  C_  ( RR  X.  RR ) )
32resabs1d 5122 . . . 4  |-  ( A 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A ) )  =  ( ( abs 
o.  -  )  |`  ( A  X.  A ) ) )
43fveq2d 5852 . . 3  |-  ( A 
C_  RR  ->  ( MetOpen `  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A
) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( A  X.  A
) ) ) )
5 resubmet.2 . . 3  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( A  X.  A
) ) )
64, 5syl6reqr 2462 . 2  |-  ( A 
C_  RR  ->  J  =  ( MetOpen `  ( (
( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A ) ) ) )
7 eqid 2402 . . . 4  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
87rexmet 21586 . . 3  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
9 eqid 2402 . . . 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 2402 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
127, 11tgioo 21591 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
1310, 12eqtri 2431 . . . 4  |-  R  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
14 eqid 2402 . . . 4  |-  ( MetOpen `  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A
) ) )  =  ( MetOpen `  ( (
( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A ) ) )
159, 13, 14metrest 21317 . . 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 668 . 2  |-  ( A 
C_  RR  ->  ( Rt  A )  =  ( MetOpen `  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  |`  ( A  X.  A
) ) ) )
176, 16eqtr4d 2446 1  |-  ( A 
C_  RR  ->  J  =  ( Rt  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    C_ wss 3413    X. cxp 4820   ran crn 4823    |` cres 4824    o. ccom 4826   ` cfv 5568  (class class class)co 6277   RRcr 9520    - cmin 9840   (,)cioo 11581   abscabs 13214   ↾t crest 15033   topGenctg 15050   *Metcxmt 18721   MetOpencmopn 18726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ioo 11585  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-rest 15035  df-topgen 15056  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-top 19689  df-bases 19691  df-topon 19692
This theorem is referenced by:  dfii2  21676  icoopnst  21729  iocopnst  21730  evthicc  22161
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