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Theorem setsxms 21418
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 21417 . . . 4  |-  ( ph  ->  ( MetOpen `  D )  =  ( TopOpen `  K
) )
61, 2, 3setsmsds 21415 . . . . . . 7  |-  ( ph  ->  ( dist `  M
)  =  ( dist `  K ) )
71, 2, 3setsmsbas 21414 . . . . . . . 8  |-  ( ph  ->  X  =  ( Base `  K ) )
87sqxpeqd 4871 . . . . . . 7  |-  ( ph  ->  ( X  X.  X
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
96, 8reseq12d 5117 . . . . . 6  |-  ( ph  ->  ( ( dist `  M
)  |`  ( X  X.  X ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
102, 9eqtrd 2461 . . . . 5  |-  ( ph  ->  D  =  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )
1110fveq2d 5876 . . . 4  |-  ( ph  ->  ( MetOpen `  D )  =  ( MetOpen `  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
125, 11eqtr3d 2463 . . 3  |-  ( ph  ->  ( TopOpen `  K )  =  ( MetOpen `  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
13 eqid 2420 . . . . 5  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
14 eqid 2420 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
15 eqid 2420 . . . . 5  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1613, 14, 15isxms2 21387 . . . 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 914 . . 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 17 . 2  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( *Met `  ( Base `  K ) ) ) )
197fveq2d 5876 . . 3  |-  ( ph  ->  ( *Met `  X )  =  ( *Met `  ( Base `  K ) ) )
2010, 19eleq12d 2502 . 2  |-  ( ph  ->  ( D  e.  ( *Met `  X
)  <->  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( *Met `  ( Base `  K )
) ) )
2118, 20bitr4d 259 1  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
D  e.  ( *Met `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1867   <.cop 3999    X. cxp 4843    |` cres 4847   ` cfv 5592  (class class class)co 6296   ndxcnx 15070   sSet csts 15071   Basecbs 15073  TopSetcts 15148   distcds 15151   TopOpenctopn 15272   *Metcxmt 18883   MetOpencmopn 18888   *MetSpcxme 21256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-tset 15161  df-ds 15164  df-rest 15273  df-topn 15274  df-topgen 15294  df-psmet 18890  df-xmet 18891  df-bl 18893  df-mopn 18894  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-xms 21259
This theorem is referenced by:  setsms  21419
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