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Theorem xmeter 20804
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 5363 . . . . 5  |-  ( `' D " RR ) 
C_  dom  D
31, 2eqsstri 3539 . . . 4  |-  .~  C_  dom  D
4 xmetf 20700 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
5 fdm 5741 . . . . 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 3557 . . 3  |-  ( D  e.  ( *Met `  X )  ->  .~  C_  ( X  X.  X ) )
8 relxp 5116 . . 3  |-  Rel  ( X  X.  X )
9 relss 5096 . . 3  |-  (  .~  C_  ( X  X.  X
)  ->  ( Rel  ( X  X.  X
)  ->  Rel  .~  )
)
107, 8, 9mpisyl 18 . 2  |-  ( D  e.  ( *Met `  X )  ->  Rel  .~  )
111xmeterval 20803 . . . . 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 1009 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  y  e.  X )
1412simp1d 1008 . . 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 20718 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x D y )  =  ( y D x ) )
1715, 14, 13, 16syl3anc 1228 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x D y )  =  ( y D x ) )
1812simp3d 1010 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( x D y )  e.  RR )
1917, 18eqeltrrd 2556 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  ( y D x )  e.  RR )
201xmeterval 20803 . . . 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 1179 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  x  .~  y
)  ->  y  .~  x )
2314adantrr 716 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  x  e.  X )
241xmeterval 20803 . . . . . 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 1009 . . 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 1010 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
y D z )  e.  RR )
31 rexadd 11443 . . . . . 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 9587 . . . . . 6  |-  ( ( ( x D y )  e.  RR  /\  ( y D z )  e.  RR )  ->  ( ( x D y )  +  ( y D z ) )  e.  RR )
3331, 32eqeltrd 2555 . . . . 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 20722 . . . . 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 1230 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D z )  <_  ( ( x D y ) +e ( y D z ) ) )
38 xmetlecl 20717 . . . 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 1237 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  (
x D z )  e.  RR )
401xmeterval 20803 . . . 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 1179 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  ( x  .~  y  /\  y  .~  z
) )  ->  x  .~  z )
43 xmet0 20713 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x D x )  =  0 )
44 0re 9608 . . . . . . 7  |-  0  e.  RR
4543, 44syl6eqel 2563 . . . . . 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 975 . . . . 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 20803 . . 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 7349 1  |-  ( D  e.  ( *Met `  X )  ->  .~  Er  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3481   class class class wbr 4453    X. cxp 5003   `'ccnv 5004   dom cdm 5005   "cima 5008   Rel wrel 5010   -->wf 5590   ` cfv 5594  (class class class)co 6295    Er wer 7320   RRcr 9503   0cc0 9504    + caddc 9507   RR*cxr 9639    <_ cle 9641   +ecxad 11328   *Metcxmt 18273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-xmet 18282
This theorem is referenced by:  blpnfctr  20807  xmetresbl  20808
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