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Theorem metider 28577
Description: The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
metider  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  Er  X )

Proof of Theorem metider
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metidss 28574 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )
2 xpss 4952 . . . 4  |-  ( X  X.  X )  C_  ( _V  X.  _V )
31, 2syl6ss 3473 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( _V  X.  _V ) )
4 df-rel 4852 . . 3  |-  ( Rel  (~Met `  D )  <->  (~Met `  D )  C_  ( _V  X.  _V ) )
53, 4sylibr 215 . 2  |-  ( D  e.  (PsMet `  X
)  ->  Rel  (~Met `  D ) )
61ssbrd 4458 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( x
(~Met `  D )
y  ->  x ( X  X.  X ) y ) )
76imp 430 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
y )  ->  x
( X  X.  X
) y )
8 brxp 4876 . . . 4  |-  ( x ( X  X.  X
) y  <->  ( x  e.  X  /\  y  e.  X ) )
97, 8sylib 199 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
y )  ->  (
x  e.  X  /\  y  e.  X )
)
10 psmetsym 21263 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  =  ( y D x ) )
11103expb 1206 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x D y )  =  ( y D x ) )
1211eqeq1d 2422 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( (
x D y )  =  0  <->  ( y D x )  =  0 ) )
13 metidv 28575 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  <->  ( x D y )  =  0 ) )
14 metidv 28575 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
y  e.  X  /\  x  e.  X )
)  ->  ( y
(~Met `  D )
x  <->  ( y D x )  =  0 ) )
1514ancom2s 809 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( y
(~Met `  D )
x  <->  ( y D x )  =  0 ) )
1612, 13, 153bitr4d 288 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  <->  y (~Met `  D ) x ) )
1716biimpd 210 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  ->  y (~Met `  D ) x ) )
1817impancom 441 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
y )  ->  (
( x  e.  X  /\  y  e.  X
)  ->  y (~Met `  D ) x ) )
199, 18mpd 15 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
y )  ->  y
(~Met `  D )
x )
20 simpl 458 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  D  e.  (PsMet `  X
) )
21 simprr 764 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
y (~Met `  D
) z )
221ssbrd 4458 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  ( y
(~Met `  D )
z  ->  y ( X  X.  X ) z ) )
2322imp 430 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  y
(~Met `  D )
z )  ->  y
( X  X.  X
) z )
24 brxp 4876 . . . . . . . . 9  |-  ( y ( X  X.  X
) z  <->  ( y  e.  X  /\  z  e.  X ) )
2523, 24sylib 199 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  y
(~Met `  D )
z )  ->  (
y  e.  X  /\  z  e.  X )
)
2621, 25syldan 472 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y  e.  X  /\  z  e.  X
) )
2726simpld 460 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
y  e.  X )
28 simprl 762 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  x (~Met `  D )
y )
2928, 9syldan 472 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x  e.  X  /\  y  e.  X
) )
3029simpld 460 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  x  e.  X )
3126simprd 464 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
z  e.  X )
32 psmettri2 21262 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
y  e.  X  /\  x  e.  X  /\  z  e.  X )
)  ->  ( x D z )  <_ 
( ( y D x ) +e
( y D z ) ) )
3320, 27, 30, 31, 32syl13anc 1266 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  <_  ( (
y D x ) +e ( y D z ) ) )
3429, 11syldan 472 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D y )  =  ( y D x ) )
3529, 13syldan 472 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x (~Met `  D ) y  <->  ( x D y )  =  0 ) )
3628, 35mpbid 213 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D y )  =  0 )
3734, 36eqtr3d 2463 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y D x )  =  0 )
38 metidv 28575 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
(~Met `  D )
z  <->  ( y D z )  =  0 ) )
3926, 38syldan 472 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y (~Met `  D ) z  <->  ( y D z )  =  0 ) )
4021, 39mpbid 213 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y D z )  =  0 )
4137, 40oveq12d 6314 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( ( y D x ) +e
( y D z ) )  =  ( 0 +e 0 ) )
42 0xr 9676 . . . . . . 7  |-  0  e.  RR*
43 xaddid1 11521 . . . . . . 7  |-  ( 0  e.  RR*  ->  ( 0 +e 0 )  =  0 )
4442, 43ax-mp 5 . . . . . 6  |-  ( 0 +e 0 )  =  0
4541, 44syl6eq 2477 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( ( y D x ) +e
( y D z ) )  =  0 )
4633, 45breqtrd 4441 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  <_  0 )
47 psmetge0 21265 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  z  e.  X )  ->  0  <_  ( x D z ) )
4820, 30, 31, 47syl3anc 1264 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
0  <_  ( x D z ) )
49 psmetcl 21260 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  z  e.  X )  ->  (
x D z )  e.  RR* )
5020, 30, 31, 49syl3anc 1264 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  e.  RR* )
51 xrletri3 11440 . . . . 5  |-  ( ( ( x D z )  e.  RR*  /\  0  e.  RR* )  ->  (
( x D z )  =  0  <->  (
( x D z )  <_  0  /\  0  <_  ( x D z ) ) ) )
5250, 42, 51sylancl 666 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( ( x D z )  =  0  <-> 
( ( x D z )  <_  0  /\  0  <_  ( x D z ) ) ) )
5346, 48, 52mpbir2and 930 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  =  0 )
54 metidv 28575 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  z  e.  X )
)  ->  ( x
(~Met `  D )
z  <->  ( x D z )  =  0 ) )
5520, 30, 31, 54syl12anc 1262 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x (~Met `  D ) z  <->  ( x D z )  =  0 ) )
5653, 55mpbird 235 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  x (~Met `  D )
z )
57 psmet0 21261 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x D x )  =  0 )
58 metidv 28575 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  x  e.  X )
)  ->  ( x
(~Met `  D )
x  <->  ( x D x )  =  0 ) )
5958anabsan2 829 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x (~Met `  D
) x  <->  ( x D x )  =  0 ) )
6057, 59mpbird 235 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  x
(~Met `  D )
x )
611ssbrd 4458 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( x
(~Met `  D )
x  ->  x ( X  X.  X ) x ) )
6261imp 430 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
x )  ->  x
( X  X.  X
) x )
63 brxp 4876 . . . . 5  |-  ( x ( X  X.  X
) x  <->  ( x  e.  X  /\  x  e.  X ) )
6462, 63sylib 199 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
x )  ->  (
x  e.  X  /\  x  e.  X )
)
6564simpld 460 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
x )  ->  x  e.  X )
6660, 65impbida 840 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X  <->  x (~Met `  D ) x ) )
675, 19, 56, 66iserd 7388 1  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  Er  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   _Vcvv 3078    C_ wss 3433   class class class wbr 4417    X. cxp 4843   Rel wrel 4850   ` cfv 5592  (class class class)co 6296    Er wer 7359   0cc0 9528   RR*cxr 9663    <_ cle 9665   +ecxad 11396  PsMetcpsmet 18895  ~Metcmetid 28569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-2 10657  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-psmet 18903  df-metid 28571
This theorem is referenced by:  pstmxmet  28580
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