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Theorem metidv 28106
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 eleq1 2526 . . . . . 6  |-  ( a  =  A  ->  (
a  e.  X  <->  A  e.  X ) )
2 eleq1 2526 . . . . . 6  |-  ( b  =  B  ->  (
b  e.  X  <->  B  e.  X ) )
31, 2bi2anan9 871 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( a  e.  X  /\  b  e.  X )  <->  ( A  e.  X  /\  B  e.  X ) ) )
4 oveq12 6279 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a D b )  =  ( A D B ) )
54eqeq1d 2456 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( a D b )  =  0  <-> 
( A D B )  =  0 ) )
63, 5anbi12d 708 . . . 4  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( ( a  e.  X  /\  b  e.  X )  /\  (
a D b )  =  0 )  <->  ( ( A  e.  X  /\  B  e.  X )  /\  ( A D B )  =  0 ) ) )
7 eqid 2454 . . . 4  |-  { <. a ,  b >.  |  ( ( a  e.  X  /\  b  e.  X
)  /\  ( a D b )  =  0 ) }  =  { <. a ,  b
>.  |  ( (
a  e.  X  /\  b  e.  X )  /\  ( a D b )  =  0 ) }
86, 7brabga 4750 . . 3  |-  ( ( 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 ) ) )
98adantl 464 . 2  |-  ( ( 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 ) ) )
10 metidval 28104 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  =  { <. a ,  b >.  |  ( ( a  e.  X  /\  b  e.  X
)  /\  ( a D b )  =  0 ) } )
1110adantr 463 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  (~Met `  D
)  =  { <. a ,  b >.  |  ( ( a  e.  X  /\  b  e.  X
)  /\  ( a D b )  =  0 ) } )
1211breqd 4450 . 2  |-  ( ( 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
) )
13 ibar 502 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( A D B )  =  0  <-> 
( ( A  e.  X  /\  B  e.  X )  /\  ( A D B )  =  0 ) ) )
1413adantl 464 . 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 ) ) )
159, 12, 143bitr4d 285 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 367    = wceq 1398    e. wcel 1823   class class class wbr 4439   {copab 4496   ` cfv 5570  (class class class)co 6270   0cc0 9481  PsMetcpsmet 18597  ~Metcmetid 28100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-xr 9621  df-psmet 18606  df-metid 28102
This theorem is referenced by:  metideq  28107  metider  28108  pstmfval  28110  pstmxmet  28111
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