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Theorem xmeter 20008
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 5189 . . . . 5  |-  ( `' D " RR ) 
C_  dom  D
31, 2eqsstri 3386 . . . 4  |-  .~  C_  dom  D
4 xmetf 19904 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
5 fdm 5563 . . . . 5  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
64, 5syl 16 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  dom  D  =  ( X  X.  X ) )
73, 6syl5sseq 3404 . . 3  |-  ( D  e.  ( *Met `  X )  ->  .~  C_  ( X  X.  X ) )
8 relxp 4947 . . 3  |-  Rel  ( X  X.  X )
9 relss 4927 . . 3  |-  (  .~  C_  ( X  X.  X
)  ->  ( Rel  ( X  X.  X
)  ->  Rel  .~  )
)
107, 8, 9mpisyl 18 . 2  |-  ( D  e.  ( *Met `  X )  ->  Rel  .~  )
111xmeterval 20007 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
x  .~  y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x D y )  e.  RR ) ) )
1211biimpa 484 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x  e.  X  /\  y  e.  X  /\  (
x D y )  e.  RR ) )
1312simp2d 1001 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  y  e.  X )
1412simp1d 1000 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  x  e.  X )
15 simpl 457 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  D  e.  ( *Met `  X
) )
16 xmetsym 19922 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x D y )  =  ( y D x ) )
1715, 14, 13, 16syl3anc 1218 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x D y )  =  ( y D x ) )
1812simp3d 1002 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x D y )  e.  RR )
1917, 18eqeltrrd 2518 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( y D x )  e.  RR )
201xmeterval 20007 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
y  .~  x  <->  ( y  e.  X  /\  x  e.  X  /\  (
y D x )  e.  RR ) ) )
2120adantr 465 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( y  .~  x  <->  ( y  e.  X  /\  x  e.  X  /\  ( y D x )  e.  RR ) ) )
2213, 14, 19, 21mpbir3and 1171 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  y  .~  x )
2314adantrr 716 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  x  e.  X )
241xmeterval 20007 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  (
y  .~  z  <->  ( y  e.  X  /\  z  e.  X  /\  (
y D z )  e.  RR ) ) )
2524biimpa 484 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  y  .~  z
)  ->  ( y  e.  X  /\  z  e.  X  /\  (
y D z )  e.  RR ) )
2625adantrl 715 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
y  e.  X  /\  z  e.  X  /\  ( y D z )  e.  RR ) )
2726simp2d 1001 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  z  e.  X )
28 simpl 457 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  D  e.  ( *Met `  X ) )
2918adantrr 716 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D y )  e.  RR )
3026simp3d 1002 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
y D z )  e.  RR )
31 rexadd 11202 . . . . . 6  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y ) +e ( y D z ) )  =  ( ( x D y )  +  ( y D z ) ) )
32 readdcl 9365 . . . . . 6  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y )  +  ( y D z ) )  e.  RR )
3331, 32eqeltrd 2517 . . . . 5  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y ) +e ( y D z ) )  e.  RR )
3429, 30, 33syl2anc 661 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
( x D y ) +e ( y D z ) )  e.  RR )
3513adantrr 716 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  y  e.  X )
36 xmettri 19926 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  e.  X  /\  z  e.  X  /\  y  e.  X ) )  -> 
( x D z )  <_  ( (
x D y ) +e ( y D z ) ) )
3728, 23, 27, 35, 36syl13anc 1220 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D z )  <_  ( ( x D y ) +e ( y D z ) ) )
38 xmetlecl 19921 . . . 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 )
3928, 23, 27, 34, 37, 38syl122anc 1227 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D z )  e.  RR )
401xmeterval 20007 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
x  .~  z  <->  ( x  e.  X  /\  z  e.  X  /\  (
x D z )  e.  RR ) ) )
4140adantr 465 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x  .~  z  <->  ( x  e.  X  /\  z  e.  X  /\  (
x D z )  e.  RR ) ) )
4223, 27, 39, 41mpbir3and 1171 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  x  .~  z )
43 xmet0 19917 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x D x )  =  0 )
44 0re 9386 . . . . . . 7  |-  0  e.  RR
4543, 44syl6eqel 2531 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x D x )  e.  RR )
4645ex 434 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  -> 
( x D x )  e.  RR ) )
4746pm4.71rd 635 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  <->  ( (
x D x )  e.  RR  /\  x  e.  X ) ) )
48 df-3an 967 . . . . 5  |-  ( ( x  e.  X  /\  x  e.  X  /\  ( x D x )  e.  RR )  <-> 
( ( x  e.  X  /\  x  e.  X )  /\  (
x D x )  e.  RR ) )
49 anidm 644 . . . . . 6  |-  ( ( x  e.  X  /\  x  e.  X )  <->  x  e.  X )
5049anbi2ci 696 . . . . 5  |-  ( ( ( x  e.  X  /\  x  e.  X
)  /\  ( x D x )  e.  RR )  <->  ( (
x D x )  e.  RR  /\  x  e.  X ) )
5148, 50bitri 249 . . . 4  |-  ( ( x  e.  X  /\  x  e.  X  /\  ( x D x )  e.  RR )  <-> 
( ( x D x )  e.  RR  /\  x  e.  X ) )
5247, 51syl6bbr 263 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  <->  ( x  e.  X  /\  x  e.  X  /\  (
x D x )  e.  RR ) ) )
531xmeterval 20007 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
x  .~  x  <->  ( x  e.  X  /\  x  e.  X  /\  (
x D x )  e.  RR ) ) )
5452, 53bitr4d 256 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  <->  x  .~  x ) )
5510, 22, 42, 54iserd 7127 1  |-  ( D  e.  ( *Met `  X )  ->  .~  Er  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3328   class class class wbr 4292    X. cxp 4838   `'ccnv 4839   dom cdm 4840   "cima 4843   Rel wrel 4845   -->wf 5414   ` cfv 5418  (class class class)co 6091    Er wer 7098   RRcr 9281   0cc0 9282    + caddc 9285   RR*cxr 9417    <_ cle 9419   +ecxad 11087   *Metcxmt 17801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-2 10380  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-xmet 17810
This theorem is referenced by:  blpnfctr  20011  xmetresbl  20012
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