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Theorem xmeter 13603
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 4987 . . . . 5  |-  ( `' D " RR ) 
C_  dom  D
31, 2eqsstri 3187 . . . 4  |-  .~  C_  dom  D
4 xmetf 13517 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
53, 4fssdm 5376 . . 3  |-  ( D  e.  ( *Met `  X )  ->  .~  C_  ( X  X.  X ) )
6 relxp 4732 . . 3  |-  Rel  ( X  X.  X )
7 relss 4710 . . 3  |-  (  .~  C_  ( X  X.  X
)  ->  ( Rel  ( X  X.  X
)  ->  Rel  .~  )
)
85, 6, 7mpisyl 1446 . 2  |-  ( D  e.  ( *Met `  X )  ->  Rel  .~  )
91xmeterval 13602 . . . . 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 13535 . . . . 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 2255 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( y D x )  e.  RR )
181xmeterval 13602 . . . 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 13602 . . . . . 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 9839 . . . . . 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 7928 . . . . . 6  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y )  +  ( y D z ) )  e.  RR )
3129, 30eqeltrd 2254 . . . . 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 13539 . . . . 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 13534 . . . 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 13602 . . . 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 13530 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x D x )  =  0 )
42 0re 7948 . . . . . . 7  |-  0  e.  RR
4341, 42eqeltrdi 2268 . . . . . 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 13602 . . 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 6555 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 2148    C_ wss 3129   class class class wbr 4000    X. cxp 4621   `'ccnv 4622   dom cdm 4623   "cima 4626   Rel wrel 4628   ` cfv 5212  (class class class)co 5869    Er wer 6526   RRcr 7801   0cc0 7802    + caddc 7805   RR*cxr 7981    <_ cle 7983   +ecxad 9757   *Metcxmet 13147
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-er 6529  df-map 6644  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-2 8967  df-xadd 9760  df-xmet 13155
This theorem is referenced by:  blpnfctr  13606  xmetresbl  13607
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