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Theorem xmeter 13516
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 4984 . . . . 5  |-  ( `' D " RR ) 
C_  dom  D
31, 2eqsstri 3185 . . . 4  |-  .~  C_  dom  D
4 xmetf 13430 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
53, 4fssdm 5372 . . 3  |-  ( D  e.  ( *Met `  X )  ->  .~  C_  ( X  X.  X ) )
6 relxp 4729 . . 3  |-  Rel  ( X  X.  X )
7 relss 4707 . . 3  |-  (  .~  C_  ( X  X.  X
)  ->  ( Rel  ( X  X.  X
)  ->  Rel  .~  )
)
85, 6, 7mpisyl 1444 . 2  |-  ( D  e.  ( *Met `  X )  ->  Rel  .~  )
91xmeterval 13515 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
x  .~  y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x D y )  e.  RR ) ) )
109biimpa 296 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x  e.  X  /\  y  e.  X  /\  (
x D y )  e.  RR ) )
1110simp2d 1010 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  y  e.  X )
1210simp1d 1009 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  x  e.  X )
13 simpl 109 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  D  e.  ( *Met `  X
) )
14 xmetsym 13448 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x D y )  =  ( y D x ) )
1513, 12, 11, 14syl3anc 1238 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x D y )  =  ( y D x ) )
1610simp3d 1011 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x D y )  e.  RR )
1715, 16eqeltrrd 2253 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( y D x )  e.  RR )
181xmeterval 13515 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
y  .~  x  <->  ( y  e.  X  /\  x  e.  X  /\  (
y D x )  e.  RR ) ) )
1918adantr 276 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( y  .~  x  <->  ( y  e.  X  /\  x  e.  X  /\  ( y D x )  e.  RR ) ) )
2011, 12, 17, 19mpbir3and 1180 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  y  .~  x )
2112adantrr 479 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  x  e.  X )
221xmeterval 13515 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  (
y  .~  z  <->  ( y  e.  X  /\  z  e.  X  /\  (
y D z )  e.  RR ) ) )
2322biimpa 296 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  y  .~  z
)  ->  ( y  e.  X  /\  z  e.  X  /\  (
y D z )  e.  RR ) )
2423adantrl 478 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
y  e.  X  /\  z  e.  X  /\  ( y D z )  e.  RR ) )
2524simp2d 1010 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  z  e.  X )
26 simpl 109 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  D  e.  ( *Met `  X ) )
2716adantrr 479 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D y )  e.  RR )
2824simp3d 1011 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
y D z )  e.  RR )
29 rexadd 9823 . . . . . 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 7912 . . . . . 6  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y )  +  ( y D z ) )  e.  RR )
3129, 30eqeltrd 2252 . . . . 5  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y ) +e ( y D z ) )  e.  RR )
3227, 28, 31syl2anc 411 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
( x D y ) +e ( y D z ) )  e.  RR )
3311adantrr 479 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  y  e.  X )
34 xmettri 13452 . . . . 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 1240 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D z )  <_  ( ( x D y ) +e ( y D z ) ) )
36 xmetlecl 13447 . . . 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 1247 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D z )  e.  RR )
381xmeterval 13515 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
x  .~  z  <->  ( x  e.  X  /\  z  e.  X  /\  (
x D z )  e.  RR ) ) )
3938adantr 276 . . 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 1180 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  x  .~  z )
41 xmet0 13443 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x D x )  =  0 )
42 0re 7932 . . . . . . 7  |-  0  e.  RR
4341, 42eqeltrdi 2266 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x D x )  e.  RR )
4443ex 115 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  -> 
( x D x )  e.  RR ) )
4544pm4.71rd 394 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  <->  ( (
x D x )  e.  RR  /\  x  e.  X ) ) )
46 df-3an 980 . . . . 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 396 . . . . . 6  |-  ( ( x  e.  X  /\  x  e.  X )  <->  x  e.  X )
4847anbi2ci 459 . . . . 5  |-  ( ( ( x  e.  X  /\  x  e.  X
)  /\  ( x D x )  e.  RR )  <->  ( (
x D x )  e.  RR  /\  x  e.  X ) )
4946, 48bitri 184 . . . 4  |-  ( ( x  e.  X  /\  x  e.  X  /\  ( x D x )  e.  RR )  <-> 
( ( x D x )  e.  RR  /\  x  e.  X ) )
5045, 49bitr4di 198 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  <->  ( x  e.  X  /\  x  e.  X  /\  (
x D x )  e.  RR ) ) )
511xmeterval 13515 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
x  .~  x  <->  ( x  e.  X  /\  x  e.  X  /\  (
x D x )  e.  RR ) ) )
5250, 51bitr4d 191 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  <->  x  .~  x ) )
538, 20, 40, 52iserd 6551 1  |-  ( D  e.  ( *Met `  X )  ->  .~  Er  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2146    C_ wss 3127   class class class wbr 3998    X. cxp 4618   `'ccnv 4619   dom cdm 4620   "cima 4623   Rel wrel 4625   ` cfv 5208  (class class class)co 5865    Er wer 6522   RRcr 7785   0cc0 7786    + caddc 7789   RR*cxr 7965    <_ cle 7967   +ecxad 9741   *Metcxmet 13060
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-er 6525  df-map 6640  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-2 8951  df-xadd 9744  df-xmet 13068
This theorem is referenced by:  blpnfctr  13519  xmetresbl  13520
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