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Theorem metidv 26341
Description:  A and  B identify by the metric  D if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metidv  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A
(~Met `  D ) B 
<->  ( A D B )  =  0 ) )

Proof of Theorem metidv
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metidval 26339 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  =  { <. a ,  b >.  |  ( ( a  e.  X  /\  b  e.  X
)  /\  ( a D b )  =  0 ) } )
21adantr 465 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  (~Met `  D
)  =  { <. a ,  b >.  |  ( ( a  e.  X  /\  b  e.  X
)  /\  ( a D b )  =  0 ) } )
32breqd 4324 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A
(~Met `  D ) B 
<->  A { <. a ,  b >.  |  ( ( a  e.  X  /\  b  e.  X
)  /\  ( a D b )  =  0 ) } B
) )
4 eleq1 2503 . . . . . . 7  |-  ( a  =  A  ->  (
a  e.  X  <->  A  e.  X ) )
5 eleq1 2503 . . . . . . 7  |-  ( b  =  B  ->  (
b  e.  X  <->  B  e.  X ) )
64, 5bi2anan9 868 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( a  e.  X  /\  b  e.  X )  <->  ( A  e.  X  /\  B  e.  X ) ) )
7 oveq12 6121 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a D b )  =  ( A D B ) )
87eqeq1d 2451 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( a D b )  =  0  <-> 
( A D B )  =  0 ) )
96, 8anbi12d 710 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( ( a  e.  X  /\  b  e.  X )  /\  (
a D b )  =  0 )  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( A D B )  =  0 ) ) )
10 eqid 2443 . . . . 5  |-  { <. a ,  b >.  |  ( ( a  e.  X  /\  b  e.  X
)  /\  ( a D b )  =  0 ) }  =  { <. a ,  b
>.  |  ( (
a  e.  X  /\  b  e.  X )  /\  ( a D b )  =  0 ) }
119, 10brabga 4624 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A { <. a ,  b >.  |  ( ( a  e.  X  /\  b  e.  X
)  /\  ( a D b )  =  0 ) } B  <->  ( ( A  e.  X  /\  B  e.  X
)  /\  ( A D B )  =  0 ) ) )
1211adantl 466 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A { <. a ,  b
>.  |  ( (
a  e.  X  /\  b  e.  X )  /\  ( a D b )  =  0 ) } B  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( A D B )  =  0 ) ) )
133, 12bitrd 253 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A
(~Met `  D ) B 
<->  ( ( A  e.  X  /\  B  e.  X )  /\  ( A D B )  =  0 ) ) )
14 ibar 504 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( A D B )  =  0  <-> 
( ( A  e.  X  /\  B  e.  X )  /\  ( A D B )  =  0 ) ) )
1514adantl 466 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A D B )  =  0  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( A D B )  =  0 ) ) )
1613, 15bitr4d 256 1  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A
(~Met `  D ) B 
<->  ( A D B )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4313   {copab 4370   ` cfv 5439  (class class class)co 6112   0cc0 9303  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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2741  df-rex 2742  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-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-map 7237  df-xr 9443  df-psmet 17831  df-metid 26337
This theorem is referenced by:  metideq  26342  metider  26343  pstmfval  26345  pstmxmet  26346
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