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Theorem metider 27624
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 27621 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )
2 xpss 5109 . . . 4  |-  ( X  X.  X )  C_  ( _V  X.  _V )
31, 2syl6ss 3516 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( _V  X.  _V ) )
4 df-rel 5006 . . 3  |-  ( Rel  (~Met `  D )  <->  (~Met `  D )  C_  ( _V  X.  _V ) )
53, 4sylibr 212 . 2  |-  ( D  e.  (PsMet `  X
)  ->  Rel  (~Met `  D ) )
61ssbrd 4488 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( x
(~Met `  D )
y  ->  x ( X  X.  X ) y ) )
76imp 429 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
y )  ->  x
( X  X.  X
) y )
8 brxp 5030 . . . 4  |-  ( x ( X  X.  X
) y  <->  ( x  e.  X  /\  y  e.  X ) )
97, 8sylib 196 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
y )  ->  (
x  e.  X  /\  y  e.  X )
)
10 psmetsym 20641 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  =  ( y D x ) )
11103expb 1197 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x D y )  =  ( y D x ) )
1211eqeq1d 2469 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( (
x D y )  =  0  <->  ( y D x )  =  0 ) )
13 metidv 27622 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  <->  ( x D y )  =  0 ) )
14 metidv 27622 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
y  e.  X  /\  x  e.  X )
)  ->  ( y
(~Met `  D )
x  <->  ( y D x )  =  0 ) )
1514ancom2s 800 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( y
(~Met `  D )
x  <->  ( y D x )  =  0 ) )
1612, 13, 153bitr4d 285 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  <->  y (~Met `  D ) x ) )
1716biimpd 207 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  ->  y (~Met `  D ) x ) )
1817impancom 440 . . 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 457 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  D  e.  (PsMet `  X
) )
21 simprr 756 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
y (~Met `  D
) z )
221ssbrd 4488 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  ( y
(~Met `  D )
z  ->  y ( X  X.  X ) z ) )
2322imp 429 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  y
(~Met `  D )
z )  ->  y
( X  X.  X
) z )
24 brxp 5030 . . . . . . . . 9  |-  ( y ( X  X.  X
) z  <->  ( y  e.  X  /\  z  e.  X ) )
2523, 24sylib 196 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  y
(~Met `  D )
z )  ->  (
y  e.  X  /\  z  e.  X )
)
2621, 25syldan 470 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y  e.  X  /\  z  e.  X
) )
2726simpld 459 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
y  e.  X )
28 simprl 755 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  x (~Met `  D )
y )
2928, 9syldan 470 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x  e.  X  /\  y  e.  X
) )
3029simpld 459 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  x  e.  X )
3126simprd 463 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
z  e.  X )
32 psmettri2 20640 . . . . . 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 1230 . . . . 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 470 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D y )  =  ( y D x ) )
3529, 13syldan 470 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x (~Met `  D ) y  <->  ( x D y )  =  0 ) )
3628, 35mpbid 210 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D y )  =  0 )
3734, 36eqtr3d 2510 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y D x )  =  0 )
38 metidv 27622 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  (
y  e.  X  /\  z  e.  X )
)  ->  ( y
(~Met `  D )
z  <->  ( y D z )  =  0 ) )
3926, 38syldan 470 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y (~Met `  D ) z  <->  ( y D z )  =  0 ) )
4021, 39mpbid 210 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y D z )  =  0 )
4137, 40oveq12d 6303 . . . . . 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 9641 . . . . . . 7  |-  0  e.  RR*
43 xaddid1 11439 . . . . . . 7  |-  ( 0  e.  RR*  ->  ( 0 +e 0 )  =  0 )
4442, 43ax-mp 5 . . . . . 6  |-  ( 0 +e 0 )  =  0
4541, 44syl6eq 2524 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( ( y D x ) +e
( y D z ) )  =  0 )
4633, 45breqtrd 4471 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  <_  0 )
47 psmetge0 20643 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  z  e.  X )  ->  0  <_  ( x D z ) )
4820, 30, 31, 47syl3anc 1228 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
0  <_  ( x D z ) )
49 psmetcl 20638 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  z  e.  X )  ->  (
x D z )  e.  RR* )
5020, 30, 31, 49syl3anc 1228 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  e.  RR* )
51 xrletri3 11359 . . . . . 6  |-  ( ( ( x D z )  e.  RR*  /\  0  e.  RR* )  ->  (
( x D z )  =  0  <->  (
( x D z )  <_  0  /\  0  <_  ( x D z ) ) ) )
5242, 51mpan2 671 . . . . 5  |-  ( ( x D z )  e.  RR*  ->  ( ( x D z )  =  0  <->  ( (
x D z )  <_  0  /\  0  <_  ( x D z ) ) ) )
5350, 52syl 16 . . . 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 ) ) ) )
5446, 48, 53mpbir2and 920 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  =  0 )
55 metidv 27622 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  z  e.  X )
)  ->  ( x
(~Met `  D )
z  <->  ( x D z )  =  0 ) )
5620, 30, 31, 55syl12anc 1226 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x (~Met `  D ) z  <->  ( x D z )  =  0 ) )
5754, 56mpbird 232 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  ->  x (~Met `  D )
z )
58 psmet0 20639 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x D x )  =  0 )
59 metidv 27622 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  x  e.  X )
)  ->  ( x
(~Met `  D )
x  <->  ( x D x )  =  0 ) )
6059anabsan2 820 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x (~Met `  D
) x  <->  ( x D x )  =  0 ) )
6158, 60mpbird 232 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  x
(~Met `  D )
x )
621ssbrd 4488 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( x
(~Met `  D )
x  ->  x ( X  X.  X ) x ) )
6362imp 429 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
x )  ->  x
( X  X.  X
) x )
64 brxp 5030 . . . . 5  |-  ( x ( X  X.  X
) x  <->  ( x  e.  X  /\  x  e.  X ) )
6563, 64sylib 196 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
x )  ->  (
x  e.  X  /\  x  e.  X )
)
6665simpld 459 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x
(~Met `  D )
x )  ->  x  e.  X )
6761, 66impbida 830 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X  <->  x (~Met `  D ) x ) )
685, 19, 57, 67iserd 7338 1  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  Er  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   class class class wbr 4447    X. cxp 4997   Rel wrel 5004   ` cfv 5588  (class class class)co 6285    Er wer 7309   0cc0 9493   RR*cxr 9628    <_ cle 9630   +ecxad 11317  PsMetcpsmet 18213  ~Metcmetid 27616
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-2 10595  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-psmet 18222  df-metid 27618
This theorem is referenced by:  pstmxmet  27627
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