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Theorem metider 28690
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 28687 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )
2 xpss 4940 . . . 4  |-  ( X  X.  X )  C_  ( _V  X.  _V )
31, 2syl6ss 3443 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( _V  X.  _V ) )
4 df-rel 4840 . . 3  |-  ( Rel  (~Met `  D )  <->  (~Met `  D )  C_  ( _V  X.  _V ) )
53, 4sylibr 216 . 2  |-  ( D  e.  (PsMet `  X
)  ->  Rel  (~Met `  D ) )
61ssbrd 4443 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( x
(~Met `  D )
y  ->  x ( X  X.  X ) y ) )
76imp 431 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
y )  ->  x
( X  X.  X
) y )
8 brxp 4864 . . . 4  |-  ( x ( X  X.  X
) y  <->  ( x  e.  X  /\  y  e.  X ) )
97, 8sylib 200 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
y )  ->  (
x  e.  X  /\  y  e.  X )
)
10 psmetsym 21319 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  =  ( y D x ) )
11103expb 1208 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x D y )  =  ( y D x ) )
1211eqeq1d 2452 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( (
x D y )  =  0  <->  ( y D x )  =  0 ) )
13 metidv 28688 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  <->  ( x D y )  =  0 ) )
14 metidv 28688 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
y  e.  X  /\  x  e.  X )
)  ->  ( y
(~Met `  D )
x  <->  ( y D x )  =  0 ) )
1514ancom2s 810 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( y
(~Met `  D )
x  <->  ( y D x )  =  0 ) )
1612, 13, 153bitr4d 289 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  <->  y (~Met `  D ) x ) )
1716biimpd 211 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  ->  y (~Met `  D ) x ) )
1817impancom 442 . . 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 459 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  D  e.  (PsMet `  X
) )
21 simprr 765 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
y (~Met `  D
) z )
221ssbrd 4443 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  ( y
(~Met `  D )
z  ->  y ( X  X.  X ) z ) )
2322imp 431 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  y
(~Met `  D )
z )  ->  y
( X  X.  X
) z )
24 brxp 4864 . . . . . . . . 9  |-  ( y ( X  X.  X
) z  <->  ( y  e.  X  /\  z  e.  X ) )
2523, 24sylib 200 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  y
(~Met `  D )
z )  ->  (
y  e.  X  /\  z  e.  X )
)
2621, 25syldan 473 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y  e.  X  /\  z  e.  X
) )
2726simpld 461 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
y  e.  X )
28 simprl 763 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  x (~Met `  D )
y )
2928, 9syldan 473 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x  e.  X  /\  y  e.  X
) )
3029simpld 461 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  x  e.  X )
3126simprd 465 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
z  e.  X )
32 psmettri2 21318 . . . . . 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 1269 . . . . 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 473 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D y )  =  ( y D x ) )
3529, 13syldan 473 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x (~Met `  D ) y  <->  ( x D y )  =  0 ) )
3628, 35mpbid 214 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D y )  =  0 )
3734, 36eqtr3d 2486 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y D x )  =  0 )
38 metidv 28688 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
(~Met `  D )
z  <->  ( y D z )  =  0 ) )
3926, 38syldan 473 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y (~Met `  D ) z  <->  ( y D z )  =  0 ) )
4021, 39mpbid 214 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y D z )  =  0 )
4137, 40oveq12d 6306 . . . . . 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 9684 . . . . . . 7  |-  0  e.  RR*
43 xaddid1 11529 . . . . . . 7  |-  ( 0  e.  RR*  ->  ( 0 +e 0 )  =  0 )
4442, 43ax-mp 5 . . . . . 6  |-  ( 0 +e 0 )  =  0
4541, 44syl6eq 2500 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( ( y D x ) +e
( y D z ) )  =  0 )
4633, 45breqtrd 4426 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  <_  0 )
47 psmetge0 21321 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  z  e.  X )  ->  0  <_  ( x D z ) )
4820, 30, 31, 47syl3anc 1267 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
0  <_  ( x D z ) )
49 psmetcl 21316 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  z  e.  X )  ->  (
x D z )  e.  RR* )
5020, 30, 31, 49syl3anc 1267 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  e.  RR* )
51 xrletri3 11448 . . . . 5  |-  ( ( ( x D z )  e.  RR*  /\  0  e.  RR* )  ->  (
( x D z )  =  0  <->  (
( x D z )  <_  0  /\  0  <_  ( x D z ) ) ) )
5250, 42, 51sylancl 667 . . . 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 932 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  =  0 )
54 metidv 28688 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  z  e.  X )
)  ->  ( x
(~Met `  D )
z  <->  ( x D z )  =  0 ) )
5520, 30, 31, 54syl12anc 1265 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x (~Met `  D ) z  <->  ( x D z )  =  0 ) )
5653, 55mpbird 236 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  x (~Met `  D )
z )
57 psmet0 21317 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x D x )  =  0 )
58 metidv 28688 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  x  e.  X )
)  ->  ( x
(~Met `  D )
x  <->  ( x D x )  =  0 ) )
5958anabsan2 830 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x (~Met `  D
) x  <->  ( x D x )  =  0 ) )
6057, 59mpbird 236 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  x
(~Met `  D )
x )
611ssbrd 4443 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( x
(~Met `  D )
x  ->  x ( X  X.  X ) x ) )
6261imp 431 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
x )  ->  x
( X  X.  X
) x )
63 brxp 4864 . . . . 5  |-  ( x ( X  X.  X
) x  <->  ( x  e.  X  /\  x  e.  X ) )
6462, 63sylib 200 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
x )  ->  (
x  e.  X  /\  x  e.  X )
)
6564simpld 461 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
x )  ->  x  e.  X )
6660, 65impbida 842 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X  <->  x (~Met `  D ) x ) )
675, 19, 56, 66iserd 7386 1  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  Er  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   _Vcvv 3044    C_ wss 3403   class class class wbr 4401    X. cxp 4831   Rel wrel 4838   ` cfv 5581  (class class class)co 6288    Er wer 7357   0cc0 9536   RR*cxr 9671    <_ cle 9673   +ecxad 11404  PsMetcpsmet 18947  ~Metcmetid 28682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-2 10665  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-psmet 18955  df-metid 28684
This theorem is referenced by:  pstmxmet  28693
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