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Theorem metider 26343
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 26340 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )
2 xpss 4967 . . . 4  |-  ( X  X.  X )  C_  ( _V  X.  _V )
31, 2syl6ss 3389 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( _V  X.  _V ) )
4 df-rel 4868 . . 3  |-  ( Rel  (~Met `  D )  <->  (~Met `  D )  C_  ( _V  X.  _V ) )
53, 4sylibr 212 . 2  |-  ( D  e.  (PsMet `  X
)  ->  Rel  (~Met `  D ) )
61ssbrd 4354 . . . . 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 4891 . . . 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 19908 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  y  e.  X )  ->  (
x D y )  =  ( y D x ) )
11103expb 1188 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x D y )  =  ( y D x ) )
1211eqeq1d 2451 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( (
x D y )  =  0  <->  ( y D x )  =  0 ) )
13 metidv 26341 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
(~Met `  D )
y  <->  ( x D y )  =  0 ) )
14 metidv 26341 . . . . . . 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 4354 . . . . . . . . . 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 4891 . . . . . . . . 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 19907 . . . . . 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 1220 . . . . 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 2477 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( y D x )  =  0 )
38 metidv 26341 . . . . . . . . 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 6130 . . . . . 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 9451 . . . . . . 7  |-  0  e.  RR*
43 xaddid1 11230 . . . . . . 7  |-  ( 0  e.  RR*  ->  ( 0 +e 0 )  =  0 )
4442, 43ax-mp 5 . . . . . 6  |-  ( 0 +e 0 )  =  0
4541, 44syl6eq 2491 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( ( y D x ) +e
( y D z ) )  =  0 )
4633, 45breqtrd 4337 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  <_  0 )
47 psmetge0 19910 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  z  e.  X )  ->  0  <_  ( x D z ) )
4820, 30, 31, 47syl3anc 1218 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
0  <_  ( x D z ) )
49 psmetcl 19905 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  z  e.  X )  ->  (
x D z )  e.  RR* )
5020, 30, 31, 49syl3anc 1218 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  e.  RR* )
51 xrletri3 11150 . . . . . 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 913 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  (
x (~Met `  D
) y  /\  y
(~Met `  D )
z ) )  -> 
( x D z )  =  0 )
55 metidv 26341 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  z  e.  X )
)  ->  ( x
(~Met `  D )
z  <->  ( x D z )  =  0 ) )
5620, 30, 31, 55syl12anc 1216 . . 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 19906 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x D x )  =  0 )
59 metidv 26341 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  x  e.  X )
)  ->  ( x
(~Met `  D )
x  <->  ( x D x )  =  0 ) )
6059anabsan2 818 . . . 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 4354 . . . . . 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 4891 . . . . 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 828 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X  <->  x (~Met `  D ) x ) )
685, 19, 57, 67iserd 7148 1  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  Er  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2993    C_ wss 3349   class class class wbr 4313    X. cxp 4859   Rel wrel 4866   ` cfv 5439  (class class class)co 6112    Er wer 7119   0cc0 9303   RR*cxr 9438    <_ cle 9440   +ecxad 11108  PsMetcpsmet 17822  ~Metcmetid 26335
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 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-2 10401  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-psmet 17831  df-metid 26337
This theorem is referenced by:  pstmxmet  26346
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