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Theorem xmeter 12364
Description: The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
Hypothesis
Ref Expression
xmeter.1  |-  .~  =  ( `' D " RR )
Assertion
Ref Expression
xmeter  |-  ( D  e.  ( *Met `  X )  ->  .~  Er  X )

Proof of Theorem xmeter
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmeter.1 . . . . 5  |-  .~  =  ( `' D " RR )
2 cnvimass 4838 . . . . 5  |-  ( `' D " RR ) 
C_  dom  D
31, 2eqsstri 3079 . . . 4  |-  .~  C_  dom  D
4 xmetf 12278 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
53, 4fssdm 5223 . . 3  |-  ( D  e.  ( *Met `  X )  ->  .~  C_  ( X  X.  X ) )
6 relxp 4586 . . 3  |-  Rel  ( X  X.  X )
7 relss 4564 . . 3  |-  (  .~  C_  ( X  X.  X
)  ->  ( Rel  ( X  X.  X
)  ->  Rel  .~  )
)
85, 6, 7mpisyl 1390 . 2  |-  ( D  e.  ( *Met `  X )  ->  Rel  .~  )
91xmeterval 12363 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
x  .~  y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x D y )  e.  RR ) ) )
109biimpa 292 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x  e.  X  /\  y  e.  X  /\  (
x D y )  e.  RR ) )
1110simp2d 962 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  y  e.  X )
1210simp1d 961 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  x  e.  X )
13 simpl 108 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  D  e.  ( *Met `  X
) )
14 xmetsym 12296 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x D y )  =  ( y D x ) )
1513, 12, 11, 14syl3anc 1184 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x D y )  =  ( y D x ) )
1610simp3d 963 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x D y )  e.  RR )
1715, 16eqeltrrd 2177 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( y D x )  e.  RR )
181xmeterval 12363 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
y  .~  x  <->  ( y  e.  X  /\  x  e.  X  /\  (
y D x )  e.  RR ) ) )
1918adantr 272 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( y  .~  x  <->  ( y  e.  X  /\  x  e.  X  /\  ( y D x )  e.  RR ) ) )
2011, 12, 17, 19mpbir3and 1132 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  y  .~  x )
2112adantrr 466 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  x  e.  X )
221xmeterval 12363 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  (
y  .~  z  <->  ( y  e.  X  /\  z  e.  X  /\  (
y D z )  e.  RR ) ) )
2322biimpa 292 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  y  .~  z
)  ->  ( y  e.  X  /\  z  e.  X  /\  (
y D z )  e.  RR ) )
2423adantrl 465 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
y  e.  X  /\  z  e.  X  /\  ( y D z )  e.  RR ) )
2524simp2d 962 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  z  e.  X )
26 simpl 108 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  D  e.  ( *Met `  X ) )
2716adantrr 466 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D y )  e.  RR )
2824simp3d 963 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
y D z )  e.  RR )
29 rexadd 9476 . . . . . 6  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y ) +e ( y D z ) )  =  ( ( x D y )  +  ( y D z ) ) )
30 readdcl 7618 . . . . . 6  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y )  +  ( y D z ) )  e.  RR )
3129, 30eqeltrd 2176 . . . . 5  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y ) +e ( y D z ) )  e.  RR )
3227, 28, 31syl2anc 406 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
( x D y ) +e ( y D z ) )  e.  RR )
3311adantrr 466 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  y  e.  X )
34 xmettri 12300 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  e.  X  /\  z  e.  X  /\  y  e.  X ) )  -> 
( x D z )  <_  ( (
x D y ) +e ( y D z ) ) )
3526, 21, 25, 33, 34syl13anc 1186 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D z )  <_  ( ( x D y ) +e ( y D z ) ) )
36 xmetlecl 12295 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  e.  X  /\  z  e.  X )  /\  (
( ( x D y ) +e
( y D z ) )  e.  RR  /\  ( x D z )  <_  ( (
x D y ) +e ( y D z ) ) ) )  ->  (
x D z )  e.  RR )
3726, 21, 25, 32, 35, 36syl122anc 1193 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D z )  e.  RR )
381xmeterval 12363 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
x  .~  z  <->  ( x  e.  X  /\  z  e.  X  /\  (
x D z )  e.  RR ) ) )
3938adantr 272 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x  .~  z  <->  ( x  e.  X  /\  z  e.  X  /\  (
x D z )  e.  RR ) ) )
4021, 25, 37, 39mpbir3and 1132 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  x  .~  z )
41 xmet0 12291 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x D x )  =  0 )
42 0re 7638 . . . . . . 7  |-  0  e.  RR
4341, 42syl6eqel 2190 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x D x )  e.  RR )
4443ex 114 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  -> 
( x D x )  e.  RR ) )
4544pm4.71rd 389 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  <->  ( (
x D x )  e.  RR  /\  x  e.  X ) ) )
46 df-3an 932 . . . . 5  |-  ( ( x  e.  X  /\  x  e.  X  /\  ( x D x )  e.  RR )  <-> 
( ( x  e.  X  /\  x  e.  X )  /\  (
x D x )  e.  RR ) )
47 anidm 391 . . . . . 6  |-  ( ( x  e.  X  /\  x  e.  X )  <->  x  e.  X )
4847anbi2ci 450 . . . . 5  |-  ( ( ( x  e.  X  /\  x  e.  X
)  /\  ( x D x )  e.  RR )  <->  ( (
x D x )  e.  RR  /\  x  e.  X ) )
4946, 48bitri 183 . . . 4  |-  ( ( x  e.  X  /\  x  e.  X  /\  ( x D x )  e.  RR )  <-> 
( ( x D x )  e.  RR  /\  x  e.  X ) )
5045, 49syl6bbr 197 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  <->  ( x  e.  X  /\  x  e.  X  /\  (
x D x )  e.  RR ) ) )
511xmeterval 12363 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
x  .~  x  <->  ( x  e.  X  /\  x  e.  X  /\  (
x D x )  e.  RR ) ) )
5250, 51bitr4d 190 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  <->  x  .~  x ) )
538, 20, 40, 52iserd 6385 1  |-  ( D  e.  ( *Met `  X )  ->  .~  Er  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448    C_ wss 3021   class class class wbr 3875    X. cxp 4475   `'ccnv 4476   dom cdm 4477   "cima 4480   Rel wrel 4482   ` cfv 5059  (class class class)co 5706    Er wer 6356   RRcr 7499   0cc0 7500    + caddc 7503   RR*cxr 7671    <_ cle 7673   +ecxad 9398   *Metcxmet 11931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-er 6359  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-2 8637  df-xadd 9401  df-xmet 11939
This theorem is referenced by:  blpnfctr  12367  xmetresbl  12368
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