MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  setsxms Unicode version

Theorem setsxms 18077
Description: The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
Hypotheses
Ref Expression
setsms.x  |-  ( ph  ->  X  =  ( Base `  M ) )
setsms.d  |-  ( ph  ->  D  =  ( (
dist `  M )  |`  ( X  X.  X
) ) )
setsms.k  |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
setsms.m  |-  ( ph  ->  M  e.  V )
Assertion
Ref Expression
setsxms  |-  ( ph  ->  ( K  e.  * MetSp  <-> 
D  e.  ( * Met `  X ) ) )

Proof of Theorem setsxms
StepHypRef Expression
1 setsms.x . . . . 5  |-  ( ph  ->  X  =  ( Base `  M ) )
2 setsms.d . . . . 5  |-  ( ph  ->  D  =  ( (
dist `  M )  |`  ( X  X.  X
) ) )
3 setsms.k . . . . 5  |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
4 setsms.m . . . . 5  |-  ( ph  ->  M  e.  V )
51, 2, 3, 4setsmstopn 18076 . . . 4  |-  ( ph  ->  ( MetOpen `  D )  =  ( TopOpen `  K
) )
61, 2, 3setsmsds 18074 . . . . . . 7  |-  ( ph  ->  ( dist `  M
)  =  ( dist `  K ) )
71, 2, 3setsmsbas 18073 . . . . . . . 8  |-  ( ph  ->  X  =  ( Base `  K ) )
87, 7xpeq12d 4751 . . . . . . 7  |-  ( ph  ->  ( X  X.  X
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
96, 8reseq12d 4993 . . . . . 6  |-  ( ph  ->  ( ( dist `  M
)  |`  ( X  X.  X ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
102, 9eqtrd 2348 . . . . 5  |-  ( ph  ->  D  =  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )
1110fveq2d 5567 . . . 4  |-  ( ph  ->  ( MetOpen `  D )  =  ( MetOpen `  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
125, 11eqtr3d 2350 . . 3  |-  ( ph  ->  ( TopOpen `  K )  =  ( MetOpen `  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
13 eqid 2316 . . . . 5  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
14 eqid 2316 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
15 eqid 2316 . . . . 5  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1613, 14, 15isxms2 18046 . . . 4  |-  ( K  e.  * MetSp  <->  ( (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( * Met `  ( Base `  K
) )  /\  ( TopOpen
`  K )  =  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) ) )
1716rbaib 873 . . 3  |-  ( (
TopOpen `  K )  =  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )  ->  ( K  e.  * MetSp  <->  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( * Met `  ( Base `  K ) ) ) )
1812, 17syl 15 . 2  |-  ( ph  ->  ( K  e.  * MetSp  <-> 
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( * Met `  ( Base `  K ) ) ) )
197fveq2d 5567 . . 3  |-  ( ph  ->  ( * Met `  X
)  =  ( * Met `  ( Base `  K ) ) )
2010, 19eleq12d 2384 . 2  |-  ( ph  ->  ( D  e.  ( * Met `  X
)  <->  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( * Met `  ( Base `  K ) ) ) )
2118, 20bitr4d 247 1  |-  ( ph  ->  ( K  e.  * MetSp  <-> 
D  e.  ( * Met `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701   <.cop 3677    X. cxp 4724    |` cres 4728   ` cfv 5292  (class class class)co 5900   ndxcnx 13192   sSet csts 13193   Basecbs 13195  TopSetcts 13261   distcds 13264   TopOpenctopn 13375   * Metcxmt 16418   MetOpencmopn 16423   *
MetSpcxme 17934
This theorem is referenced by:  setsms  18078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-tset 13274  df-ds 13277  df-rest 13376  df-topn 13377  df-topgen 13393  df-xmet 16425  df-bl 16427  df-mopn 16428  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-xms 17937
  Copyright terms: Public domain W3C validator